ࡱ>  !"drg  !bjbj Pohh=Y:AA8<M$Pytq %(%(%([)FFFwwwwwwwzf}pwYHCFHHwAA%([) yTTTHA%(6[)wTHwTTRr;u[)P,TmJsw y0Py t}Ol}<u}uFhHGJTG<GFFFwwkT@FFFPyHHHH}FFFFFFFFF : Section 4.1: Maximum and Minimum Values SOLs: APC.12: The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, Newton's method, differentials and linear approximations, and optimization problems. Objectives: Students will be able to: Understand extreme values of a function Find critical values of a function Find extreme values of a function Vocabulary: Absolute maximum a value greater than or equal to all other values of a function in the domain Absolute minimum a value less than or equal to all other values of a function in the domain Maximum value the value of the function than is greater than or equal to all other values of the function Minimum value the value of the function than is less than or equal to all other values of the function Extreme values maximum or minimum functional values Local maximum a value greater than or equal to all other functional values in the vicinity of a value of x Local minimum a value less than or equal to all other functional values in the vicinity of a value of x Critical number a value of x such that f(x) = 0 or f(x) does not exist Key Concept: Extreme Value Theorem: If f is continuous on the closed interval [a,b], then f attains an absolute maximum value f(c) and a absolute minimum value f(d) at some numbers c and d in [a,b]. Fermats Theorem: If f has a local maximum or minimum at c, and if f(c) exists, then f(c) = 0. Or rephrased: If f has a local maximum or minimum at c, then c is a critical number of f. [Note: this theorem is not biconditional (its converse is not necessarily true), just because f(c) = 0, doesnt mean that there is a local max or min at c!! Example y = x] Closed Interval Method: To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: Find the values of f at the critical numbers of f in (a,b) (the open interval) Find the values of f at the endpoints of the interval, f(a) and f(b) The largest value from steps 1 and 2 is the absolute maximum value; the smallest of theses values is the absolute minimum value. Let f be defined on an interval I containing c. f(c) is the minimum of f on I if f(c) d" f(x) for all x in I. f(c) is the maximum of f on I if f(c) e" f(x) for all x in I. Together maxes and mins are called extrema. Look at the diagram on page 279. If f is a continuous function on a closed interval [a,b], then f has both a minimum and a maximum on the interval. Note: maximum and minimum values are the  EMBED Equation.3 -values!! Relative extrema (local extrema) occur in an interval where f(c) is an extrema page 280 A critical number is one where the first derivative is zero or undefined page 283 Relative extrema occur only at critical numbers. Look at page 283 for method. Absolute extrema occur at relative extrema or at the endpoints of the interval. Find the relative and absolute extrema for each of the following: 1. f(x) = -2x + 3x 2.  EMBED Equation.3  on [-1,2] 3. f(x) = (1 + x)> 4. f(x) = sin x  cos x on [0,] Homework: pg 285  288: 1, 4, 6, 9, 31, 35, 36, 53, 57, 74 Read: Section 4.2 Section 4.2: The Mean Value Theorem SOLs: APC.10: The student will state (without proof) the Mean Value Theorem for derivatives and apply it both algebraically and graphically. Objectives: Students will be able to: Understand Rolles Theorem Understand Mean Value Theorem Vocabulary: Existence Theorem a theorem that guarantees that there exists a number with a certain property, but it doesnt tell us how to find it. Key Concepts:  EMBED PowerPoint.Show.8  Example: Verify that the mean value theorem (MVT) holds for f(x) = -x + 6x 6 on [1,3]. Find the number that satisfies the MVT on the given interval or state why the theorem does not apply. 1.  EMBED Equation.3  on [0,32] 2. y = ( x 2)- on [2,5] 3.  EMBED Equation.3  on [1,3] 4.  EMBED Equation.3  on [1,9]  EMBED PowerPoint.Show.8  Geometrically this means that at some number c in (a,b), the tangent line to the graph of f(x) must be parallel to the line through (a,f(a)) and (b,f(b)); in this case a horizontal tangent. Example: Determine whether Rolle s Theorem s hypotheses are satisfied &, if so, find a number c for which f (c) = 0. 1. f(x) = x + 9 on [-3,3] 2. f(x) = x - 2x - x + 2 on [-1,2] 3. f(x) = (x - 1) / x on [-1,1] 4. f(x) = sin x on [0,] Homework Problems: pg 295-296: 2, 7, 11, 12, 24 Read: Section 4.3 Section 4.3: How Derivatives Affect the Shape of a Graph SOLs: APC.1: The student will . graph these functions using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing. Objectives: Students will be able to: Understand and use the First Derivative Test to determine mins and maxs Understand and use the Second Derivative Test to determine mins and maxs Vocabulary: Increasing going up or to the right Decreasing going down or to the left Inflection point a point on the curve where the concavity changes Key Concept:  EMBED PowerPoint.Show.8  Using the first derivative to determine increasing and decreasing intervals: a. if f(x) > 0, then f is increasing b. if f(x) < 0, then f is decreasing c. if f(x) = 0, then f is constant Example: Determine analytically where  EMBED Equation.3  is increasing or decreasing. First Derivative Test: Let  EMBED Equation.3  be a critical number (f2 (x) = 0 or undefined) on an open interval containing EMBED Equation.3 . 1. If f2 changes from negative to positive, f(c) is a relative minimum. 2. If f2 changes from positive to negative, f(c) is a relative maximum. Concavity, etc. Let f be a differentiable function on an open interval I. If f2 is increasing, i.e. f2 2 > 0 the graph of f is concave up; if f2 is decreasing, i.e. f2 2 < 0 the graph of f is concave down. An inflection point is a point where concavity changes. If there is an inflection point at  EMBED Equation.3 , then f2 2 (c) = 0 or is undefined. An inflection point does not have to occur when f2 2 (c) = 0 or is undefined. (example - f(x) = x4) look at page 300 for concavity test Example: Determine the concavity and inflection points of  EMBED Equation.3 .  EMBED PowerPoint.Show.8  Second Derivative Test: (another way to determine relative extrema) Let f2 and f2 2 exist at every point in an open interval (a,b) containing c and suppose f (c) = 0. 1. If f2 2 (c) < 0 (function is concave down at x=c), then f(c) is a relative maximum. 2. If f2 2 (c) > 0 (function is concave up at x=c), then f(c) is a relative minimum. Example: Use the second derivative test to identify relative extrema for g(x) = x  sin x for (0,2). Sketch the graph of f(x) over [0,6] satisfying the following conditions: f(0) = f(3) = 3 f(2) = 4 f(4) = 2 f(6) = 0 f(x) > 0 on [0,2) f(x) < 0 on (2,4)(4,5] f(2) = f(4) = 0 f(x) = -1 on (5,6) f(x) < 0 on (0,3) (4,5) f(x) > 0 on (3,4)  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  Homework Problems: pg 304-307: 1, 10, 11, 14, 17, 21, 27, 35, 38, 45, 74 Read: Section 4.4 Section 4.4: Indeterminate Forms and LHospitals Rule SOLs: APC.11: The student will use l'Hopital's rule to find the limit of functions whose limits yield the indeterminate forms: 0/0 and infinity/infinity Objectives: Students will be able to: Use LHospitals Rule to determine limits of an indeterminate form Vocabulary: Indeterminate Form (0/0 or "/") a form that a value cannot be assigned to without more work Key Concept: When taking limits in chapter 2, we ran into the indeterminate forms  EMBED Equation.3  and  EMBED Equation.3 . These forms do not guarantee that a limit exists or what the limit is, if it does exist. Earlier in the year, we used algebraic techniques to rewrite the expressions and take the limit: factor and cancel rationalize the numerator or denominator long division Not all indeterminate forms can be rewritten, especially when they involve an algebraic function and a transcendental function.  EMBED PowerPoint.Show.8  BEWARE: Do NOT apply the quotient rule.  EMBED PowerPoint.Show.8  We also have the indeterminate forms 1", 00 and "0. When you run into one of these forms, you must rewrite the function using natural logs. Examples:  1.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  2.  EMBED Equation.3   EMBED Equation.3    EMBED Equation.3    EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3.   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3    EMBED Equation.3 = EMBED Equation.3    EMBED Equation.3 = EMBED Equation.3   EMBED Equation.3  Homework Problems: pg 313-315: 1, 5, 7, 12, 15, 17, 27, 28, 40, 49, 53, 55 Read: Section 4.5 Section 4.5: Summary of Curve Sketching SOLs: APC.1: The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing. Objectives: Students will be able to: Sketch or graph a given function Vocabulary: Oblique  neither horizontal nor vertical Slant asymptote  a line (y = mx + b) that the curve approaches as x gets very large or very small found if the limit of [f(x)  (mx+b)] = 0 as x approaches " Key Concept: Checklist look at pages 317 318 Domain for which  EMBED Equation.3  values is f(x) defined? x -intercepts where is f(x) = 0? y -intercepts what is f(0)? Symmetry y-axis is f(-x) = f(x)? Origin is f(-x) = -f(x)? Period is there a number p such that f(x + p) = f(x)? Asymptotes Horizontal does  EMBED Equation.3  or  EMBED Equation.3  exist? Vertical for what  EMBED Equation.3  is  EMBED Equation.3 ? for what  EMBED Equation.3  is  EMBED Equation.3 ? Increasing  on what intervals is f (x) e" 0? Decreasing  on what intervals is f (x) d" 0? Critical numbers  where does f (x) = 0 or DNE? Local extrema  what are the local max/min? Use f or f  test. Concavity Up  where is f  (x) > 0? Down  where is f  (x) < 0? Inflection points  where does f change concavity? Sketch the graph of .   EMBED PowerPoint.Show.8  Sketch a graph of y = f(x) such that Domain: (-",-1) (-1, ") f(2) = 0 f(x) = f(-x) f (x) e" 0 on (1,3] [5,"]   f (x) d" 0 on [3,5] f (3) = 0, local max at x = 3 f (5) = 0, local min at x = 5 f  (x) > 0 on (4,6) f  (4) = 0 f  (6) = 0 f  (x) < 0 on (1,4) (6,")  EMBED PowerPoint.Show.8  Homework Problems: pg 323-324: 3, 13, 18, 37, 58 Read: Section 4.6 Section 4.6: Graphing with Calculus and Calculators SOLs: APC.1: The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions using a graphing calculator. Objectives: Students will be able to: Use the TI-83 or TI-89 to sketch / graph a given function revealing all of it important features Vocabulary: Families of functions functions related to one another by an arbitrary constant Key Concept: Use calculus (previous section material) to complement the calculator Using the calculator by itself can lead to misleading results with an inappropriate viewing window The calculator can help solve the function and its derivatives set equal to zero Find an appropriate viewing window that showcases all of the important features of the graph. You may have to redo this viewing window several times to reveal all the important features Unless otherwise specified, round to the nearest thousandth on paper, keep the full decimal in your calculator for calculations. Families of functions: a collection of functions defined by a formula with one or more arbitrary constants; ex. y = ax + bx + c is a set of parabolas where the relative extrema occurs at  EMBED Equation.3 . Example: Describe the graphs of the family of functions f(x) = x - 3ax Homework Problems: pg 330-331: 1, 4, 12, 15, 16 Read: Section 4.7 Section 4.7: Optimization Problems SOLs: None Objectives: Students will be able to: Use derivatives to solve optimization problems Vocabulary: One-to-one Function never takes on the same value twice (one x for every y) Inverse Function reverse the inputs (domain) and the outputs (range) Cancellation Equations cancels out the function Key Concept:  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  1. A cone with slant height of 6 inches is to be constructed. What is largest possible volume of such a cone? 2. Find the points on the graph of x - 4y = 4 that are closest to the point (5,0). 3. A flyer is to contain 50 square inches of printed matter, with 4-inch margins at the top and bottom and 2-inch margins on each side. What dimensions for the flyer will use the least paper? An open box having a square base is to be constructed from 108 square inches of material. What dimensions will produce a box with maximum volume? Find two positive numbers that minimize the sum of twice the first number plus the second number, if the product of the two numbers is 288. Two posts, one 28 feet tall and the other 12 feet tall, stand 30 feet apart. They are to be held by wires, attached to a single stake, running from ground level to the tops of the poles. Where should the stake be placed to use the least wire? 7. Bill has 4 feet of wire and wants to form a circle and a square from the wire. How much of the wire should he use on the square and how much should be used on the circle to enclose the maximum total area? Homework Problems: pg 336-341: 3, 4, 9, 23, 30, 34, 56 Read: Section 4.8 Chapter 4.8: Applications to Business and Economics SOLs: None Objectives: Students will be able to: Be exposed to use of differentiation in business and economics Vocabulary: Cost function Marginal cost Average cost function Marginal revenue function Marginal profit function Key Concept: Will cover if time permits Homework Problems: pg 346 - 347: 5, 9, 15, 19 Read: Section 4.9 Section 4.9: Newtons Method SOLs: None Objectives: Students will be able to: Use Newtons method to approximate roots of an equation Vocabulary: Iterative process a process that uses the first answer as input to the second equation for the second answer and so on until some limiting parameter is reached ! very useful in spreadsheets and in computer programs Newton s Method  an iterative process that can (but may not) find the roots of an equation Key Concept: Will cover if time permits Homework  Problems: pg 351-353: Read: Section 4.10 Section 4.10: Antiderivatives SOLs: None Objectives: Students will be able to: Understand the concept of an antiderivative Understand the geometry of the antiderivative and that of slope fields Work rectilinear motion problems with antiderivatives Vocabulary: Antiderivative the opposite of the derivative, if f(x) = F(x) then F(x) is the antiderivative of f(x) Integrand what is being taken the integral of [F(x)] Variable of integration what variable we are taking the integral with respect to Constant of integration a constant (derivative of which would be zero) that represents the family of functions that could have the same derivative Key Concept: What is a function that has a derivative of 3x? A function is an antiderivative of f on an open interval I if F(x) = f(x) for all x in I. There is an entire family of antiderivatives for each derivative. The general antiderivative of 3x is x + c. c is called the constant of integration. We use the following symbols:  EMBED Equation.3    Basic Rules: 1. Power Rule  EMBED Equation.3  Ex. Find a)  EMBED Equation.3  b)  EMBED Equation.3  c)  EMBED Equation.3  d)  EMBED Equation.3  e)  EMBED Equation.3  f)  EMBED Equation.3  Other Rules: 2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  Ex. Find g)  EMBED Equation.3  h)  EMBED Equation.3  i)  EMBED Equation.3  j)  EMBED Equation.3  k)  EMBED Equation.3  l)  EMBED Equation.3  Trig Rules: 6.  EMBED Equation.3  7.  EMBED Equation.3  8.  EMBED Equation.3  9.  EMBED Equation.3  10.  EMBED Equation.3  11.  EMBED Equation.3  Ex. Find m)  EMBED Equation.3  n)  EMBED Equation.3  o)  EMBED Equation.3  p)  EMBED Equation.3  q)  EMBED Equation.3  r)  EMBED Equation.3  Initial Conditions: You can solve for a particular  EMBED Equation.3  if you are given certain preliminary information (initial conditions); find  EMBED Equation.3  such that F(1) = 4. Ex. Find the velocity function v(t) and position function s(t) corresponding to the acceleration function a(t) = 4t + 4 given v(0) = 8 and s(0) = 12. Direction (slope) fields:  EMBED PowerPoint.Show.8  Homework Problems: pg 358-360: Day 1: 1-3, 12, 13, 16 Day 2: 25, 26, 53, 61, 74 Read: Review Chapter 4 Chapter 4: Review SOLs: None Objectives: Students will be able to: Know material presented in Chapter 4 Vocabulary: None new Key Concept: Calculus Review: Chapter 4 Non-Calculator For problems 1 and 2, use the function  EMBED Equation.DSMT4 . 1. The function  EMBED Equation.DSMT4  has one relative minimum and two relative maxima one relative minimum and one relative maximum two relative maxima and no relative minimum two relative minima and one relative maximum two relative minima and no relative maximum 2. How many inflection points does  EMBED Equation.DSMT4  have? A. 0 B. 1 C. 2 D. 3 E. 4 3. The maximum and minimum values of the function  EMBED Equation.DSMT4  on the closed interval [0, 4] are respectively A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  4. 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`?`@`S`T`U`V``````````````````aaaFaSaTa[acadahaiara{ssesesesesseh[&0h[&056CJaJh[&0CJaJ!j.hh%1CJEHUaJ#jMA hh%1CJUVaJhY]CJaJ!jA,hh%1CJEHUaJ#j֯MA hh%1CJUVaJjhh%1CJUaJhh%15CJaJh*5CJaJhh%1CJaJh%1CJaJ'rasa{a}a~aaaaaaaaaaaaaaaaaaaab5b:b*CJaJh fh%1CJaJh%1CJaJhh%1CJaJtauavawaxayaza{a|a}a~aaaaaaabMbabbbnbobbb T^`Tgd+Pgd+P$a$gdGH$a$gdrMgdJ$a$gd%1KbLbMbNbVbWbabbbibmbnbob|bbbbbbbbccc8c9cPcQcRcScgchcccccddּͳͨyj^jzhECJEHUj+zA hECJUVaJjxhECJEHUj+zA hECJUVaJjhECJU hECJhEh1%h+PCJaJh+P6CJaJhgCJaJh+PCJ\aJh+P5CJaJh+PCJaJ hg5 h+P5 hJ5hJCJ\aJhssuCJ\aJ$bbbbbbccccUcVccccd;dgdhddddOeee & FgdEgdEgdEgdE ^`gd+Pgd+PdddddeeeeeeTeUelemeneoeteueeeeeeeeeeeeeeeƺuiZj--zA hECJUVaJjhECJEHUj-zA hECJUVaJjhECJEHUj,zA hECJUVaJjhECJEHUj,zA hECJUVaJjhECJEHUj,zA hECJUVaJjhECJUj]}hECJEHUj+zA hECJUVaJ hECJ eeeeeeeeeeeNOfghiz{㷫㜐ufZj.hECJEHUj.zA hECJUVaJjhECJEHUj.zA hECJUVaJj&hECJEHUj.zA hECJUVaJjhECJEHUjh.zA hECJUVaJUjhECJEHUjD-zA hECJUVaJ hECJjhECJUjhECJEHU# EMBED Equation.DSMT4  is the number defined by Rolles Theorem, then for  EMBED Equation.DSMT4  on the interval  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  is A. 1 B. 1 C. 0 D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  5. The acceleration of a particle moving along a straight line is given by  EMBED Equation.DSMT4 , if, when  EMBED Equation.DSMT4 , its velocity,  EMBED Equation.DSMT4 , is 1 and its distance,  EMBED Equation.DSMT4 , is 3, then at any time  EMBED Equation.DSMT4 , the distance  EMBED Equation.DSMT4  is A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  7. Which equation has the slope field shown at the right? A.  EMBED Equation.2  B.  EMBED Equation.2  C.  EMBED Equation.2  D.  EMBED Equation.2  8. Find the limit using L'Hopital's Rule, if applicable.  EMBED Equation.DSMT4  A. 0 B.  EMBED Equation.DSMT4  C. 1 D. Limit does not exist Calculator 9. If  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 , then a value for  EMBED Equation.DSMT4  prescribed by the Mean Value Theorem is A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  10. If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  then the curve has no horizontal tangents the curve is concave up for all x the curve is concave down for all x the curve has no inflection point none of these  11. The graph at the right shows the velocity of an object moving along a straight line during the time interval  EMBED Equation.DSMT4 . For what  EMBED Equation.DSMT4  does this object attain its maximum velocity? A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Free Response (1996 BC 4) This problem deals with functions defined by  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is a positive constant and  EMBED Equation.DSMT4 . Sketch the graphs of the two functions  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  on the axes provided.  Find the x-coordinates of all points,  EMBED Equation.DSMT4 , where the line  EMBED Equation.DSMT4  is tangent to the graph of  EMBED Equation.DSMT4 . Are the points of tangency described in part (b) relative maximum points of  EMBED Equation.DSMT4 ? Why? For all values of  EMBED Equation.DSMT4 , show that all inflection points of the graph of  EMBED Equation.DSMT4  lie on the line  EMBED Equation.DSMT4 . Answers: 1. D 2. C 3. A 4. A 5. C 6. E 7. A 8. A 9. B 10. D 11. B (1996 BC 4) a. b. Homework Problems: pg 361-364: Read: Study for Chapter 4 Test After Chapter 4 Test: Homework Problems: None Read: Section 5.1     Chapter 4: Applications of Differentiation This is  EMBED Equation.3 - use log technique  EMBED Equation.3  -  EMBED Equation.3  Now we have  EMBED Equation.3  - Apply Lhopitals We must undo taking the log by taking the antilog This is  EMBED Equation.3 - use log technique  EMBED Equation.3  -  EMBED Equation.3  Now we have  EMBED Equation.3  - Apply Lhopitals This is still  EMBED Equation.3  We must undo taking the log by taking the antilog This is  EMBED Equation.3 - use log technique  EMBED Equation.3  -  EMBED Equation.3  Now we have  EMBED Equation.3  - Apply Lhopitals This is still  EMBED Equation.3  We must undo taking the log by taking the antilog  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Antiderivative Constant of Integration Integral Symbol Integrand Variable of Integration e_ !A$%`;<GH & FgdEgdEgdE   ]^uvwxƺuiZj{A hECJUVaJjhECJEHUj|{A hECJUVaJjhECJEHUjn{A hECJUVaJjahECJEHUjY{A hECJUVaJjhhECJEHUj.zA hECJUVaJjhECJUjhECJEHUj.zA hECJUVaJ hECJ ./01@AXYZ[de|}~㹭㞒wh\j hECJEHUj{A hECJUVaJj*hECJEHUj{A hECJUVaJj_hECJEHUj{A hECJUVaJjhECJEHUj{A hECJUVaJjhECJEHUj{A hECJUVaJ hECJjhECJUjhECJEHU#%&=>?@FG^_`aghŹԪԏthYMjhECJEHUj{A hECJUVaJjhECJEHUj{A hECJUVaJj;hECJEHUj{A hECJUVaJjhECJEHUj'{A hECJUVaJjhECJEHUjr&{A hECJUVaJ hECJjhECJUjݹhECJEHUj{A hECJUVaJ#*+>?@AHI\Ź爵{j`{{jhEEHU j: hEUVmHnHujhEUh`b^hECJhEjhEUmHnHujuhECJEHUj"{A hECJUVaJjOhECJEHUj"{A hECJUVaJjhECJEHUj"{A hECJUVaJ hECJjhECJU \]^_dexyz{<GOPgȾ٭ٝxi]ٝjhECJEHUj'{A hECJUVaJj hECJEHUj&{A hECJUVaJjhECJU hECJjhEEHU j: hEUVmHnHujhEEHU j: hEUVmHnHuhEjhEUjhEEHU j: hEUVmHnHu!ghijop !&'>?ŹԪԏthYj.zA hECJUVaJj:hECJEHUj.zA hECJUVaJjshECJEHUj-zA hECJUVaJjhECJEHUj-zA hECJUVaJjxhECJEHUj-zA hECJUVaJ hECJjhECJUjhECJEHUjz-zA hECJUVaJ?@AFG^_`afg~gh㹭㞒wgXj0zA hECJUVaJjhECJUmHnHujhECJEHUj+.zA hECJUVaJjhECJEHUj.zA hECJUVaJjhECJEHUj-zA hECJUVaJjhECJEHUj.zA hECJUVaJ hECJjhECJUjhECJEHU 7YgiL+./01234Kgd1^gdE & FgdEgdEgdE & FgdEQRijklrs㹭㞒wh\jt hECJEHUj0zA hECJUVaJjhECJEHUj0zA hECJUVaJj=hECJEHUj0zA hECJUVaJjhECJEHUj0zA hECJUVaJj5hECJEHUj'0zA hECJUVaJ hECJjhECJUjhECJEHU#<=TUVW^_vwxyԦԋpdUjpT{A hECJUVaJjhECJEHUjeT{A hECJUVaJjhECJEHUj.T{A hECJUVaJjthECJEHUjT{A hECJUVaJjhECJEHUjT{A hECJUVaJhE hECJjhECJUjhECJEHUj0zA hECJUVaJ!+-/4Z[rstu()@ABC]^u㥙~ocjN&hECJEHUjU{A hECJUVaJj~#hECJEHUjT{A hECJUVaJj hECJEHUjmU{A hECJUVaJjhECJEHUj.T{A hECJUVaJhEjhECJUmHnHu hECJjhECJUjWhECJEHU!uvwxcd~ŹԪԚ~skbYbkQkQhzgCJaJh+P6CJaJh+P5CJaJh+PCJaJh1%h+PCJaJjhECJUmHnHu hE>* h1h1hEj-hECJEHUjU{A hECJUVaJj@+hECJEHUjU{A hECJUVaJ hECJjhECJUj(hECJEHUjU{A hECJUVaJc~ABCEFHIgd+PgdEgdE.35=>BCDFGIJLMOz{|}´㍅}t}iZjh)hhh 6@CJUaJh)hhh 6@CJaJh 6@5CJaJh 6@hn|jhn|Uh1%h+PCJ\aJhzgCJ\aJh+P6CJaJh+P5CJaJh1%h+P5CJ\aJ h+P5\ hzg5\h1%h+P5\hgCJ\aJh+PCJ\aJhv)h+P5CJaJh+PCJaJ#IKLNO{|}QRMNgdbC|$a$QRZ[nygV!jg6h)hhh 6@CJEHUaJ#joB h)hhh 6@CJUVaJ!j#4h)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJ!j$2h)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJh 6@jh)hhh 6@CJUaJ!j+0h)hhh 6@CJEHUaJ#j;B h)hhh 6@CJUVaJh)hhh 6@CJaJ nopq¾¬‰x¾fU¾!j>h)hhh 6@CJEHUaJ#joB h)hhh 6@CJUVaJ!jL<h)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJ!jM:h)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJh 6@h)hhh 6@CJaJjh)hhh 6@CJUaJ!jb8h)hhh 6@CJEHUaJ#j3B h)hhh 6@CJUVaJMNVWjklm¾¾¬¾‰xfU¾!jFh)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJ!jDh)hhh 6@CJEHUaJ#jB h)hhh 6@CJUVaJ!jBh)hhh 6@CJEHUaJ#jJB h)hhh 6@CJUVaJh 6@h)hhh 6@CJaJjh)hhh 6@CJUaJ!j@h)hhh 6@CJEHUaJ#jHB h)hhh 6@CJUVaJ  I J b c d | } ~                gdygdbC|      I J K ^ _ ` a d e x y z {    ¾¬¾¾wh[j#Oh$ h 6@EHUjJ h 6@CJUVaJjLhKh 6@EHUjH h 6@CJUVaJjh 6@U!jJh)hhh 6@CJEHUaJ#jHB h)hhh 6@CJUVaJh 6@h)hhh 6@CJaJjh)hhh 6@CJUaJ!jHh)hhh 6@CJEHUaJ#joB h)hhh 6@CJUVaJ    !!!h1%h+PCJ\aJh+h 6@jh 6@UjQhJ;h 6@EHUjyJ h 6@CJUVaJ          !!!!gd+P$a$gd%1gd%1 ,1h/ =!8"8#$% n''5$"9PNG  IHDR{_gAMA|QIDATH 0 `C* @t5Inb=z8C}JdՈ, 5Ϡ;e$oyxDĄwK~dyhk%+.v+mq;pٗ8BTM",LZnL.Dnuw-ݗ^i] M龐21.m龴;r aBm?>?>?OɃg?3y?????????>?>?>??>?>?<?s'<9~O?>s'??????>??>?>?>?>>O?>O?????>?>?>?>?>? 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H!6 z!X4 ! ! ! ! ! z!= !6 X4 6 !X4 ! ! ! ! ! != 6 _ X4 Z W W W Z _ 6 X4 =  6 X4      6  X4 ! % % % !  = 6 vX4 q m m m q v6 X4      = 36 X4      6 3X4 8 ; ; ; 7 3= 6 X4      6 X4      = J6 X4      6 JX4 N R R R N J= 6 X4      6 X4      = `6 .X4 * & & & * .6 `X4 e i i i e `= 6 X4      6 X4      = w6 EX4 @ = = = @ E6 wX4 |    { w= 6 X4      6 X4      = 6 \X4 W S S S W \6 X4      = 6 X4      6 X4  ! ! !  = 6 rX4 n j j j n r6 X4      = 06 X4      6 0X4 4 8 8 8 4 0= 6 X4      6 X4      = F6 X4      6 FX4 K O O O K F= 6 X4      6 X4      = ]6 3X4 / + + + / 36 ]X4 b e e e b ]=<?M % % $$AA( " FEMF+@ FGDICF(GDIC::EFGDICRp@Times New Roman0 b 0  LTiK0W0  Lz08x` 0 L!0دm50౥(L`Pdv%    TdE@n\KKKAJAEXLTLet   % ( Rp@Times New RomanW0 L|Q|Hm| c0 L!0z08m50 (L`PdvH!\P|p|m|Z|m|dv%    TTk@{]KKKAJAoXLPf % ( Rp@Times New Roman L|Q|Hm| dm50 (L`z08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    T}@V\KKKAJA}XLxbe a function that is        % ( Rp@Times New Roman0  0  L,TiK0W0 Lz08` -0 L!0دm50< (LP dv%    T8^5zKKKAJAv'La) continuous on the closed interval [          % ( Rp@Times New RomanW0 L|Q|Hm| 0 L!0z08m50 (LP dvH!\P|p|m|Z|m|dv%    T`6^VzKKKAJA6vLTa,b  % ( Rp@Times New Roman L|Q|Hm| m50 (Lz08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TTW^^zKKKAJAWvLP] % ( Rp@Times New Roman0 u 0  L.TiK0W0  Lz08` /0 L!0دm50< (LP dv%    TD|EKKKAJA)Lb) differentiable on the open interval (          % ( Rp@Times New RomanW0 L|Q|Hm| v0 L!0z08m50 (LP dvH!\P|p|m|Z|m|dv%    T`F|eKKKAJAFLTa,b  % ( Rp@Times New Roman L|Q|Hm| wm50 (Lz08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TTf|mKKKAJAfLP) % ( Rp@Times New Roman0  40  L+TiK0W0 4 Lz08x ,0 L!0دm50 (Lx dv%    TE;KKKAJAEL|then there is a number       % ( Rp@Times New RomanW0 4L|Q|Hm| 0 L!0z08m50 (Lx dvH!\P|p|m|Z|m|dv%    TT<FKKKAJA<LPc  % ( Rp@Times New Roman L|Q|Hm| m50 (Lz08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TdMpKKKAJAMLTin ( % ( Rp@Times New Roman L|Q|Hm| H! \P|p|m|z08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    T`qKKKAJAqLTa,b  % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TKKKAJA Ld) such that    % ( Rp@Times New Roman0  L0  LTiK0W0 L Lz08 0 L!0دm50P (L dv%    T`KKKAJALTf(b % ( Rp@Times New RomanW0 LL|Q|Hm| 0 L!0z08m50 (L dvH!\P|p|m|Z|m|dv%    TXKKKAJALP)  % ( Rp@Times New Roman L|Q|Hm| m50 (Lz08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TTKKKAJALP  % ( Rp@Times New Roman L|Q|Hm| H! \P|p|m|z084dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    T`KKKAJALTf(a   % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TTKKKAJALP) % ( Rp@Times New Roman0 <  >0#0- @ ſ0-z08xT0I0i0W- LdvdvW@Times New Roman dv%    TdEgKKKAJAELTf (c  % ( Rp@Times New Roman @ @| Q|Hm| I0i0W- Lz08W @Times New Ro dvH!PD|p|m|Z|m|dv%    TdhKKKAJAhLT) =  % ( Rp@Times New Romani0W-@| Q|Hm| W @Timez08dvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    TKKKAJALl---------------     % ( Rp@Times New Roman @| Q|Hm| H! PD|p|m|z08dvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    T.+KKKAJA.L(instantaneous rate of change,         % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z084dvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    TT,:KKKAJA,LPm % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z08dvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    Tx;cKKKAJA;L\tangent % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z08LdvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    Ti=KKKAJAiL= average rate of change,        % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z08تdvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    TT>LKKKAJA>LPm % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z08ddvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    TpMoKKKAJAMLXsecant % ( Rp@Times New Roman P@| Q|Hm| H! PD|p|m|z08 dvH! PD|p|m|Z|m|dvH!PD|p|m|Z|m|dv%    TTpuKKKAJApLP) % ( Rp@Times New Roman0  0  LTiK0W0  Lz08x` 0 L!0دm50౥ (L`P dv%    TXKKKAJALPb  % ( Rp@Times New RomanW0 L|Q|Hm| 0 L!0z08m50 (L`P dvH!\P|p|m|Z|m|dv%    TTKKKAJALP  % ( Rp@Times New Roman L|Q|Hm| m50 (L`z08dvH! \P|p|m|Z|m|dvH!\P|p|m|Z|m|dv%    TXKKKAJALPa  % ( Rp@Times New Roman0   0  T LBTiK0W0  T Lz08x C0  L!0 د m50 ( L0Q dv%    T<E#?KKKAJAE;(Lor equivalently:      % ( Rp@Times New RomanW0  L|Q|Hm| 0  L!0z08m50 ( L0Q  dv H!\P|p|m|Z|m|dv%    T`#?KKKAJA;LTf(b % ( Rp@Times New Roman  L|Q|Hm| m50 ( L0Q z08dvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    TX#?KKKAJA;LP)  % ( Rp@Times New Roman L|Q|Hm| H! \P|p|m|z08dvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    TT#?KKKAJA;LP  % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z084dvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    T`#?KKKAJA;LTf(a  % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08dvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    Td#?KKKAJA;LT) =  % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08LdvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    Tx#E?KKKAJA;L\f (c)(b   % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08ddvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    TTK#X?KKKAJAL;LP  % ( Rp@Times New Roman \L|Q|Hm| H! \P|p|m|z08 dvH! \P|p|m|Z|m|dv H!\P|p|m|Z|m|dv%    TX_#r?KKKAJA_;LPa)  % ( F4(EMF+*@$??!b Ld)??" FEMF+@ Current UserGSummaryInformation(@PowerPoint Document(,DocumentSummaryInformation8leeOh+'0 `h  PowerPoint PresentationoweoweChris Headleese1riMicrosoft PowerPointon@x@@0hDw՜.+,0 ( `/ 0DTimes New Roman0B 0DArialNew Roman0B 0"@ .  @n?" dd@  @@`` 0@1@ 0AA@3 ʚ;ʚ;g4dddd B 0ppp@ <4dddd8pC 0tQE 80___PPT10  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> *"(    6pE  P E  T Click to edit Master title style! !  0XPE   E  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0vE  `` E  Z*   0E  `  E  \*   0$E  `  E  \* H  0޽h ? ̙33 Default Design 91 +?(    <{E N1  JMean Value Theorem (MVT) _  0| ;C 'For a differentiable function f(x), the slope of the secant line through (a, f(a)) and (b, f(b)) equals the slope of the tangent line at some point c between a and b. In other words, the average rate of change of f(x) over [a,b] equals the instantaneous rate of change at some point c in (a,b).@(       17t, x>RB  s *D/ RB  s *D? '| ? RB  s *D ! ? RB  s *D  ?   <t  q _ P(c,f(c)) 0    <p 6 |.  =a 0   <P  Q  _ A(a,f(a)) 0    < F A =b 0 RB  s *D/ RB  s *D? %? f2  0"` | f2  0"` w# RB   s *D  9 RB ! s *D  ?  " < P:!  NP100   # <  d  NP200   $ < 9 4  =a 0  & < /*  =y 0  ) <8 F 2 =b 0  * BCYDEFԔ]EqY@  S" b f2   0"`  f2   0"` / RB + s *D)  RB , s *D1F+ f2 - 0"` . <    _ B(b,f(b)) 0   / BCDE(FԔ oN@k&iv @     RB 0 s *D  RB 1 s *D1  RB 2 s *D1/  RB 4 s *D 9  5 <p * y  Nc200   6 <8 3 uO  Nc100  RB 7 s *D <  8 <  l  =x 0  9 < /6 =y 0  : <i  x  =x 0  ; <Xn B 4 Mc00  RB < s *D6 , ? 0r ST+K Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) then there is a number c in (a,b) such that f(b)  f(a) f (c) = --------------- (instantaneous rate of change, mtangent = average rate of change, msecant) b  a or equivalently: f(b)  f(a) = f (c)(b  a)  FW   [E,&8B@xXH  0޽h ? ̙3380___PPT10.r0 e,   QOn-screen Shown-s,: Times New RomanArialDefault DesignSlide 1  Fonts UsedDesign Template Slide Titles FMicrosoft Equation 3.0 DS Eq_1119783994OF Ǿ,T Ǿ,TOle CompObjfObjInfo!uation Equation.39q0mIyI y=x 25 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native "L_1119784103@F Ǿ,T Ǿ,TOle $CompObj %fObjInfo!'Equation Native (H_1119784134$F Ǿ,T Ǿ,TOle *,ItI y=x+1x FMicrosoft Equation 3.0 DS Equation Equation.39qpII y=x 12 +2x"2() 1CompObj#%+fObjInfo&-Equation Native ._1254475654|+dO) Ǿ,T ,T3 dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_bB Chris HeadleeChris HeadleeOh+'0 `Ole 1EPRINT(*DCompObj2{ObjInfo)-4      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~VWel1'% EMFD? \KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$=='% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDICJFGDICF(GDICUFGDICF(GDICsu3FGDICRp@Times New Roman0?\00p8o 0> 0z08x8tL8 0L0$دm50 (L8< dv%    Tx~+KKKAJA~%L\Rolle s    % ( Rp@Times New Roman L|dQ|Hm| ee8tL8 0L0z08m50 (L8< dv)!\P|p|m|Z|m|dv%    Txh+KKKAJA%L\Theorem. % ( F(GDIC::FGDICRp@Times New Roman0A  0  LTiK0W0  Lz08x` 0 L!0دm50౥(L`Pdv%    TdE@n\KKKAJAEXLTLet   % ( Rp@Times New RomanW0 L|dQ|Hm|A ee0 L!0z08m50 (L`Pdv)!\P|p|m|Z|m|dv%    TTk@{]KKKAJAoXLPf % ( Rp@Times New Roman L|dQ|Hm|A eem50 (L`z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T}@V\KKKAJA}XLxbe a function that is        % ( Rp@Times New Roman0  0  L,TiK0W0 Lz08` -0 L!0دm50< (LP dv%    T8^5zKKKAJAv'La) continuous on the closed interval [          % ( Rp@Times New RomanW0 L|dQ|Hm| ee0 L!0z08m50 (LP dv)!\P|p|m|Z|m|dv%    T`6^VzKKKAJA6vLTa,b  % ( Rp@Times New Roman L|dQ|Hm| eem50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TTW^^zKKKAJAWvLP] % ( Rp@Times New Roman0A  0  L.TiK0W0  Lz08` /0 L!0دm50< (LP dv%    TD|EKKKAJA)Lb) differentiable on the open interval (          % ( Rp@Times New RomanW0 L|dQ|Hm|A ee0 L!0z08m50 (LP dv)!\P|p|m|Z|m|dv%    T`F|eKKKAJAFLTa,b  % ( Rp@Times New Roman L|dQ|Hm|A eem50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TTf|mKKKAJAfLP) % ( Rp@Times New Roman0  0  LTiK0W0 Lz08 0 L!0دm50 (Lx dv%    TdKKKAJALTc)  % ( Rp@Times New RomanW0 L|dQ|Hm| ee0 L!0z08m50 (Lx dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(a   % ( Rp@Times New Roman L|dQ|Hm| eem50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TdKKKAJALT) =  % ( Rp@Times New Roman L|dQ|Hm| ee)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T` &KKKAJA LTf(b % ( Rp@Times New Roman \L|dQ|Hm| ee)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TT'.KKKAJA'LP) % ( Rp@Times New Roman0A  L0  L5TiK0W0 L Lz08x 60 L!0دm50P (L dv%    TE;KKKAJAEL|then there is a number       % ( Rp@Times New RomanW0 LL|dQ|Hm|A ee0 L!0z08m50 (L dv)!\P|p|m|Z|m|dv%    TT<FKKKAJA<LPc  % ( Rp@Times New Roman L|dQ|Hm|A eem50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TdMpKKKAJAMLTin ( % ( Rp@Times New Roman L|dQ|Hm|A ee)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`qKKKAJAqLTa,b  % ( Rp@Times New Roman \L|dQ|Hm|A ee)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TKKKAJA Ld) such that     % ( Rp@Times New Roman \L|dQ|Hm|A ee)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    Td,KKKAJALTf (c   % ( Rp@Times New Roman \L|dQ|Hm|A ee)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    Tl-[KKKAJA-LX) = 0  % ( F(GDICF(GDIC,zF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, 9NB䠼CBwhC@( !b $$==_888% '%  ;C6EX4EJPW[[6YX4YTNHCC="Y$P=<? *{ % % $$AA( " FEMF+@ FGDICF(GDICZgF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, 8? CkC C@!b $$==_888% %  ;6eX4kpppke6X4   =UY$U+=<?[e % % $$AA( " FEMF+@ FGDICF(GDIC@BaFEMF+*@$??@\P^UU?AA@@?@@?@@@, 9B CU5B2C@!b $$==_888% %  ;6X4 6X4 =E6EX4A= ==AE6EX4JN NNJE=6X4 6X4    =6KX4GC CCGK6X4 = 6 X4 6X4    =6X4 6X4 =6X4   6X4 =6X4 6X4 =6X4 6X4 =F6X4 6FX4KO OOKF=6X4 6X4 =M6MX4HD DDHL6LX4QU UUQM= 6X4 6 X4  =<?> Cb % % $$AA( " FEMF+@ FGDICF(GDIC aFEMF+*@$??@\P^UU?AA@@?@@?@@@,  IC C9HCC@!b $$==_888% %  ; 6 X4      6 X4      = E6 EX4 A = = = A E6 EX4 J N N N J E= 6 X4      6 X4      = 6 KX4 G C C C G K6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = F6 X4      6 FX4 K O O O K F=~ 6~ X4~      6 X4    ~ ~ =~ M6~ MX4~ H D D D H L6 LX4 Q U U U~ Q~ M=~ 6~ X4~      6 X4    ~ ~ =<? b % % $$AA( " FEMF+@ FGDICF(GDIC3^TFGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT>dHyKKKAJA>vLPa  % ( F(GDIC`FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTf{KKKAJAxLPb  % ( F(GDIC0FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT%KKKAJALPy  % ( F(GDICX|FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT_tKKKAJAqLPx  % ( F(GDICrX|FGDICRp@"Arial0A 1< >0#0- E 00-z08xT0I0i0W- E Ldvdv 80(8\t(0dv%    TT~_tKKKAJA~qLPc  % ( F(GDICaFEMF+*@$??@L@^UU?A@@@@ , 9CrCrlCV=C@ !b $$==_888% %  ;6X4"'**6*X4*&!=6PX4LH#H(H,L,P6+X4+'#=6X4 %)--6,X4,($=6gX4b!_&_+_.c.g6.X4.*%!=$6X4#',0/6/$X4/)+-&-"-)$= 6 ~X4 y$u)u-u1y1~60X40,(#  =!;6! X4!%*.22 62;X41@.C)C%C!@!;="6#X4#'+03363X43/*&""=#R6$ X4$(,155 64RX44V0Z,Z'Z#V#R=%6%X4%).26665X452-(%%=&h6'6X4'2*./.4.727767iX47m3q.q*q&m&h=<?b % % $$AA( " FEMF+@ FGDICF(GDIC? FEMF+*@$??@ 4(UU@A33@ , 䘃BCJCC@  !b $$==_888C33% % W$>  % % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@  ICUCuGCUCpFCuCpFC'CpFCوCuGCiC ICiCzJCiCUKCوCUKC'CUKCuCzJCUC ICUC@ !b $$==%  ;UP r _ _ _ r & & &     <>  % $$AAF`TEMF+@ @4^UU?A@  $$==_888% % UP  r _ _ _ r & & &     % % $$AA( " FEMF+@ FGDICF(GDIC= DFEMF+*@$??@ BCA}BC9NyB7C9NyBC9NyBCA}BV=CBV=CwBV=C丆BC丆BC丆B7CwBCBC@!b $$==%  ;UP     . ..<>= C % $$AAF`TEMF+@@4^UU?A@$$==_888% % UP< E     . ..% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@ C`ClC`CrLCACrLC9CrLC1TClCrCCrC4CrCVUC1TCVUC9CVUCAC4C`CC`C@!b $$==%  ;UP &&<NNN<&<>  % $$AAF`TEMF+@@4^UU?A@$$==_888% % UP~  &&<NNN<&% % $$AA( " FEMF+@ FGDICFGDICF(GDICF(GDIC'zF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, CC/CJhC@!b $$==_888% %  ;6X4  6X4=Y$}-=<?%{ % % $$AA( " FEMF+@ FGDICF(GDICZfF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, |C C:C C@!b $$==_888% %  ;6X46X4=Y$`(=<?[d % % $$AA( " FEMF+@ FGDICF(GDIC;C=dFEMF+*@$??@\P^UU?AA@@?@@?@@@, (CC9C9nC@!b $$==_888% %  ;;6X46:X462226;=6X46X4=565X49===9464X40,,,05=w6X46wX4rooosw=<?9A>a % % $$AA( " FEMF+@ FGDICF(GDIC?aFEMF+*@$??@\P^UU?AA@@?@@?@@@, rC C9CC@!b $$==_888% %  ;/6.X4.2}6};}??6?X4?<72//=.C6.CX4.>1:6:::>>>B6>BX4>G;K6K2K.G.C=-6-X4-15:>>6>X4>;61--=-6,HX4,D0@5@9@=D=H6>X4>:51--=,6,X4,/49==6=X4= 950, ,=<?>b % % $$AA( " FEMF+@ FGDICF(GDIC+^LFGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT6d@yKKKAJA6vLPa  % ( F(GDIC`FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTf{KKKAJAxLPb  % ( F(GDIC +FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTKKKAJALPy  % ( F(GDICX|FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT_tKKKAJAqLPx  % ( F(GDIClX|FGDICRp@"Arial0 B< >0#0- E 00-z08xT0I0i0W- E Ldvdv 80(8\t(0dv%    TTx_tKKKAJAxqLPc  % ( F(GDIC|aFEMF+*@$??@L@^UU?A@@@@, C CCSC@!b $$==_888% %  ;6X46X4=6SX4NKKKOS6X4=6X46X4=6jX4eaabej6X4='6X46'X4,0//,'=6X4|xxx|6X4=>6 X4 6>X4CFFFB>=6X46X4=T6"X4#6UX4Y]]]YT=6X46X4=k69X45111596kX4pttspk=<?zb % % $$AA( " FEMF+@ FGDICF(GDIC: DFEMF+*@$??@4(UU@A33@XLC٠CrtC㘜CWCCljCCrCCC"CUCC@!b $$==_888C33% % U88 Egc0]) % % $$AA( " FEMF+@ FGDICF(GDIC=DFEMF+*@$??@ qCCV=C % $$AAF`TEMF+@@4^UU?A@$$==_888% % UP<E 7     *7*N*aaaN7% % $$AA( " FEMF+@ FGDICF(GDIC9>@EFEMF+*@$??@  4C`CC`CCACC9CC1TCCrC4CrCCrC9vC1TC9vC9C9vCACC`C4C`C@ !b $$==%  ;UP  #555# <>9>?D % $$AAF`TEMF+@!@4^UU?A@ !$$==_888% % UP7<@E  #555# % % $$AA( " FEMF+@ FGDICF(GDICz FEMF+*@$??@" 9C9vCC9vChCzChCrChCjjCCC9CCTnCCCjjCCrCCzCTnC9vC9C9vC@"!b $$==%  ;UP <>z  % $$AAF`TEMF+@#@4^UU?A@"#$$==_888% % UPx  % % $$AA( " FEMF+@ FGDICF(GDIC7 FEMF+*@$??@$@4^UU?A@%, )C`CC`C@%$!b $$==_888% % W$6 ~% % $$AA( " FEMF+@ FGDICFGDICF(GDICF(GDIC&zF EMF+*@$??@&^DZ?Ad8^@ A?( UU@UUUUUU@', GD䠼CGDwhC@'&!b $$==_888% %  ;161X411111161X4111111=1Y$12=<?${ % % $$AA( " FEMF+@ FGDICF(GDICZgF EMF+*@$??@(^DZ?Ad8^@ A?( UU@UUUUUU@), >D CrdyD C@)(!b $$==_888% %  ;/6>X4 >>>> >>6/X4//////==Y$R>=+=<?[e % % $$AA( " FEMF+@ FGDICF(GDIC:<aFEMF+*@$??@*\P^UU?AA@@?@@?@@@+, rND CVND2C@+*!b $$==_888% %  ;363X433333363X4333333=3F63FX43A3=3=3=3@3E63EX43I3M3N3N3J3F=363X433333363X433 3 3 33=363KX43G3C3C3C3G3K63X4333333=3 63 X433333363X43 33333 =363X433333363X4333333=363X43 3333 363X4333333=363X433333363X4333333=363X433333363X4333333=3F63X433333363FX43K3O3O3O3K3F=363X433333363X4333333=3M63MX43H3D3D3D3G3L63LX43Q3U3U3U3R3M=3 63X433333363 X4333333 =<?8 =b % % $$AA( " FEMF+@ FGDICF(GDIC aFEMF+*@$??@,\P^UU?AA@@?@@?@@@-, UpD C9pDC@-,!b $$==_888% %  ; <6 <X4 <$<)<-<1<1<61<X41<-<)<$< < <= <F6 <FX4 0z08x8tL8 0L0$دm50((L(dv%    TT8dByKKKAJA8vLPa  % ( F(GDIC`FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTf{KKKAJAxLPb  % ( F(GDIC *FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTKKKAJALPy  % ( F(GDICX|FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT_tKKKAJAqLPx  % ( F(GDIClZ}FGDICRp@"Arial0A j< >0#0- E 00-z08xT0I0i0W- E Ldvdv 80(8\t(0dv%    TTw`uKKKAJAwrLPc  % ( F(GDIC{I}aFEMF+*@$??@.L@^UU?A@@@@/, ^DrC^DrC@/.!b $$==_888% %  ;767X477777767X4777777=767PX47L7H7H7H7L7P67X4777777=767X477777767X4777777=<?yK~b % % $$AA( " FEMF+@ FGDICF(GDIC8 KFEMF+*@$??@04(UU@A33@1XLND䀇CbSDCYDC^DC=dD9&CHjDCU]pDC@10!b $$==_888C33% % U87 K34;6o7v9}:<% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@2 VpDUCHepDUCpDuCpD'CpDوCHepDiCVpDiCcqDiChqDوChqD'ChqDuCcqDUCVpDUC@2!b $$==%  ;UP )<<;;;<&)<&@<&R<R<R<@<)<<>  % $$AAF`TEMF+@3@4^UU?A@23$$==_888% % UP  )<<;;;<&)<&@<&R<R<R<@<)<% % $$AA( " FEMF+@ FGDICF(GDIC7 >FEMF+*@$??@4 NDC[NDCND7CNDCNDC[NDV=CNDV=C ODV=CUODCUODCUOD7C ODCNDC@4!b $$==%  ;UP 33}3}3}3 3 3 3 3 3333<>7 = % $$AAF`TEMF+@5@4^UU?A@45$$==_888% % UP6 ? 33}3}3}3 3 3 3 3 3333% % $$AA( " FEMF+@ FGDICF(GDICxDKFEMF+*@$??@6 ^D9C^D9C8^Ds+C8^DC8^DC^D0C^D0C;_D0C9_DC9_DC9_Ds+C;_D9C^D9C@6!b $$==%  ;UP 7K7K7^7u777777u7^7K7K<>xD~J % $$AAF`TEMF+@7@4^UU?A@67$$==_888% % UPvCL 7K7K7^7u777777u7^7K7K% % $$AA( " FEMF+@ FGDICF(GDICOJLFEMF+*@$??@8@4^UU?A@9, rSDGClDGC@98!b $$==_888% % W$MHL44;% % $$AA( " FEMF+@ FGDICFGDICF(GDIC-FFGDICRp@Times New Roman0.\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TX "!>KKKAJA :LPA. % ( F(GDIC FFGDICRp@Times New Roman0.\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TX">KKKAJA:LPB. % ( F(GDIC!FFGDICRp@Times New Roman0.\00p8o 0> 0z08x8tL8 0L0$دm50( (L( dv%    TX">KKKAJA:LPC. % ( F(GDICFFGDICRp@Times New Roman0.\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TX" >KKKAJA:LPD. % ( F(GDIC FGDICRp@Times New Roman0A  0  L(TiK0W0  Lz08x` )0 L!0دm50౥ (L`P dv%    T+KKKAJA+ L`Case 1:    % ( Rp@Times New RomanW0 L|(Q|Hm|A 000 L!0z08m50 (L`P dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(x  % ( Rp@Times New Roman L|(Q|Hm|A 00m50 (L`z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TKKKAJAL) = k (constant) [picture A]       % ( Rp@Times New Roman0A  L0 0 L9TiK0W0 L0 Lz08x :0 L!0دm50P (L dv%    T+KKKAJA+ L`Case 2:    % ( Rp@Times New RomanW0 LL|(Q|Hm|A 000 L!0z08m50 (L dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(x  % ( Rp@Times New Roman L|(Q|Hm|A 00m50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TdKKKAJALT) >  % ( Rp@Times New Roman L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(a  % ( Rp@Times New Roman \L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TKKKAJALp) for some x in (     % ( Rp@Times New Roman \L|(Q|Hm|A 00)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTa,b  % ( Rp@Times New Roman \L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TbKKKAJALp) [picture B or C]    % ( Rp@Times New Roman0  0  LSTiK0W0  Lz08xt T0 L!0دm50< (Ld dv%    ToKKKAJAo5LExtreme value theorem guarantees a max value somewher                 % ( Rp@Times New RomanW0 L|(Q|Hm| 000 L!0z08m50 (Ld dv)!\P|p|m|Z|m|dv%    TpKKKAJALXe in [   % ( Rp@Times New Roman L|(Q|Hm| 00m50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`1KKKAJALTa,b  % ( Rp@Times New Roman L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T2KKKAJA2 L`]. Since   % ( Rp@Times New Roman0.Z\00p8o [0> 0z08Z8tZL8 0LZ0$د`0(Ldv%    T`oKKKAJAoLTf(a  % ( Rp@Times New Roman [L|(Q|Hm|A 008tZL8 0LZ0z08`0 (Ldv)!\P|p|m|Z|m|dv%    TdKKKAJALT) =  % ( Rp@Times New RomanZL8L|(Q|Hm|A 00`0 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(b  % ( Rp@Times New Roman L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TKKKAJALx), then at some c in (       % ( Rp@Times New Roman \L|(Q|Hm|A 00)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTa,b  % ( Rp@Times New Roman \L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T\KKKAJA-L) this max must occur. Fermat s Theorem says           % ( Rp@Times New Roman0  0  LTiK0W0  Lz08x` 0 L!0دm50౥ (L`P dv%    Tlo!=KKKAJAo9LXthat   % ( Rp@Times New RomanW0 L|(Q|Hm| 000 L!0z08m50 (L`P dv)!\P|p|m|Z|m|dv%    Td!=KKKAJA9LTf (c   % ( Rp@Times New Roman L|(Q|Hm| 00m50 (L`z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    Tl!=KKKAJA9LX) =0.  % ( Rp@Times New Roman0  L0 0 L9TiK0W0 L0 Lz08x :0 L!0دm50P (L dv%    T+OkKKKAJA+g L`Case 3:    % ( Rp@Times New RomanW0 LL|(Q|Hm| 000 L!0z08m50 (L dv)!\P|p|m|Z|m|dv%    T`OkKKKAJAgLTf(x  % ( Rp@Times New Roman L|(Q|Hm| 00m50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TdOkKKKAJAgLT) <  % ( Rp@Times New Roman L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`OkKKKAJAgLTf(a  % ( Rp@Times New Roman \L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TOkKKKAJAgLp) for some x in (     % ( Rp@Times New Roman \L|(Q|Hm| 00)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`OkKKKAJAgLTa,b  % ( Rp@Times New Roman \L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TOckKKKAJAgLp) [picture C or D]    % ( Rp@Times New Roman0A  0  LSTiK0W0  Lz08xt T0 L!0دm50< (Ld dv%    TomKKKAJAo5LExtreme value theorem guarantees a min value somewher                % ( Rp@Times New RomanW0 L|(Q|Hm|A 000 L!0z08m50 (Ld dv)!\P|p|m|Z|m|dv%    Tpm KKKAJALXe in [  % ( Rp@Times New Roman L|(Q|Hm|A 00m50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T` m-KKKAJA LTa,b  % ( Rp@Times New Roman L|(Q|Hm|A 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T.mKKKAJA. L`]. Since   % ( Rp@Times New Roman0.Z\00p8o [0> 0z08Z8tZL8 0LZ0$دm50P(Ldv%    T`oKKKAJAoLTf(a  % ( Rp@Times New Roman [L|(Q|Hm| 008tZL8 0LZ0z08m50 (Ldv)!\P|p|m|Z|m|dv%    TdKKKAJALT) =  % ( Rp@Times New RomanZL8L|(Q|Hm| 00m50 (Lz08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTf(b  % ( Rp@Times New Roman L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TKKKAJALx), then at some c in (       % ( Rp@Times New Roman \L|(Q|Hm| 00)! \P|p|m|z084dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T`KKKAJALTa,b  % ( Rp@Times New Roman \L|(Q|Hm| 00)! \P|p|m|z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    T\KKKAJA-L) this min must occur. Fermat s Theorem says          % ( Rp@Times New Roman0A  0  LTiK0W0  Lz08x` 0 L!0دm50౥ (L`P dv%    TloKKKAJAoLXthat   % ( Rp@Times New RomanW0 L|(Q|Hm|A 000 L!0z08m50 (L`P dv)!\P|p|m|Z|m|dv%    TdKKKAJALTf (c   % ( Rp@Times New Roman L|(Q|Hm|A 00m50 (L`z08dv)! \P|p|m|Z|m|dv)!\P|p|m|Z|m|dv%    TlKKKAJALX) =0.  % ( F(GDICF(GDIC".zF EMF+*@$??@:^DZ?Ad8^@ A?( UU@UUUUUU@;,  D䠼C8 DJhC@;:!b $$==_888% %  ;i"6l"X4l"q"w"}"""6"X4"z"t"n"i"i"=I"Y$w"}"=<?#,{ % % $$AA( " FEMF+@ FGDICF(GDICZfF EMF+*@$??@<^DZ?Ad8^@ A?( UU@UUUUUU@=, U D C;D C@=<!b $$==_888% %  ;; 6.X4......6; X45 0 0 0 5 ; =|.Y$.|.(=<?[d % % $$AA( " FEMF+@ FGDICF(GDICC*EaFEMF+*@$??@>\P^UU?AA@@?@@?@@@?, D C9DzC@?>!b $$==_888% %  ;+$6,$X4,$0$}4$}9$}=$=$6<$X4<$8$3$/$+$+$=,$B6,$BX4,$>0$:5$:9$:=$>=$B6=$BX4=$G9$K5$K0$K,$G,$B=-$6-$X4-$0$5$:$=$=$6=$X4=$:$5$0$-$-$=-$6.$HX4.$D2$@6$@;$@?$D?$H6>$X4>$:$5$1$-$-$=.$6.$X4.$2$7$;$?$?$6?$X4?$ ;$7$2$.$ .$=/$6/$X4/$3$7$<$@$@$6@$X4@$<$7$3$/$/$=/$60$ X40$4$8$=$A$A$ 6@$X4@$<$8$3$/$/$=0$60$X40$4$9$=$A$A$6A$X4A$=$9$4$0$0$=<?A*Fb % % $$AA( " FEMF+@ FGDICF(GDIC*aFEMF+*@$??@\P^UU?AA@@?@@?@@@, 2D C2DzC@!b $$==_888% %  ;,6,X4,,},},},,6,X4,,,,,,=,C6,CX4,>,:,:,:,=,B6,BX4,G,K,K,K,H,C=,6,X4,,,,,,6,X4,,,,,,=,6,HX4,D,@,@,@,D,H6,X4,,,,,,=,6,X4,,,,,,6,X4, ,,,, ,=,6,X4,,,,,,6,X4,,,,,,=,6, X4,,,,,, 6,X4,,,,,,=,6,X4,,,,,,6,X4,,,,,,=<?*b % % $$AA( " FEMF+@ FGDICF(GDIC5^VFGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT@dJyKKKAJA@vLPa  % ( F(GDIC`FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TTf{KKKAJAxLPb  % ( F(GDIC3FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT'KKKAJALPy  % ( F(GDICX|FGDICRp@"Arial0#\00p8o 0> 0z08x8tL8 0L0$دm50((L(dv%    TT_tKKKAJAqLPx  % ( F(GDICSX{|FGDICRp@"Arial0A < >0#0-  00-z08xT0I0i0W-  Ldvdv `0`\t(0dv%    TT^_htKKKAJA^qLPc  % ( Rp@"Arial 0@|( Q|Hm| 00I0i0W-  Lz08 0 `\t(0dv`)!PD|p|m|Z|m|dv%    TTijoyKKKAJAiwLP1 % ( F(GDICbd`FEMF+*@$??@L@^UU?A@@@@, ǥDoCUD|C@!b $$==_888% %  ;&6&X4&&"&'&+&+&6*&X4*&&&"&&&&=&r6&@X4&;&7#&7(&7,&;,&@6+&rX4+&v'&z#&z&z&v&r=&6&X4& &$&)&-&-&6,&X4,&)&$&&&&=&6&VX4&R!&N&&N*&N.&R.&V6-&X4-&*&%& &&&=&6&X4&"&'&+&/&/&6.&X4.&+&&&!&&&=&6&mX4&h#&e(&e,&e0&i0&m6/&X4/&,&'&"&&&= &*6 &X4 &$&)&-&1&1&60&*X40&/-&3(&3#&3 &/ &*=!&6!&X4!&%&{*&{.&{2&2&61&X41&.&)&$&!&!&="&A6"&X4"& &&+&/&3& 3&62&AX42&F/&I*&I%&I"&F"&A=#&6#&X4#&'&,&0&4&4&64&X43&0&+&'&#&#&=$&X6$&&X4$&!(&-&1&5&!5&&65&XX45&\1&`,&`(&`$&\$&X=%&6%&X4%&)&.&2&6&6&66&X46&2&-&)&%&%&=&&n6&&<X4&&8*&4/&43&47&87&<67&nX47&s3&w.&w*&w&&s&&n='&6'&X4'&+&0&4&8&8&68&X48&4&/&+&'&'&=<?`ea % % $$AA( " FEMF+@ FGDICF(GDICAOFEMF+*@$??@4(UU@A33@th DzCODCDUewCmDq{CD CVu!DC9%DC)DhC 0DUC2DrdC@!b $$==_888C33% % UD@P 1$$+v%oT&2'V(f)Hv*,,% % $$AA( " FEMF+@ FGDICF(GDICmHJFEMF+*@$??@@4^UU?A@, xDhC 2DhC@!b $$==_888% % W$kFJ&{,% % $$AA( " FEMF+@ FGDICF(GDIC'.FEMF+*@$??@ r2DU-CdT2DU-C 2DC 2DoC 2D!CdT2DDZCr2DDZC 3DDZCW3D!CW3DoCW3DC 3DU-Cr2DU-C@!b $$==%  ;UP ,~,~{,{,{,,,,,,,,~,~<>'- % $$AAF`TEMF+@ @4^UU?A@ $$==_888% % UP&/ ,~,~{,{,{,,,,,,,,~,~% % $$AA( " FEMF+@ FGDICF(GDIC@'G/FEMF+*@$??@  DU-CDU-CrPD”CrPDzCrPD2CDȖCDȖCJDȖCD2CDzCD”CJDU-CDU-C@ !b $$==%  ;UP 5$~$~ $ $ $$5$K$]$]$]$K$~5$~<>@'F. % $$AAF`TEMF+@ @4^UU?A@  $$==_888% % UP?&H/ 5$~$~ $ $ $$5$K$]$]$]$K$~5$~% % $$AA( " FEMF+@ FGDICF(GDIC`gFEMF+*@$??@  9DzC+vDzC+Du*{C+D|C+D ~C+vDU5C9DU5CF.DU5CxD ~CxD|CxDu*{CF.DzC9DzC@ !b $$==%  ;UP -&&&&&&-&D&W&W&W&D&-&<>`f % $$AAF`TEMF+@ @4^UU?A@  $$==_888% % UP^g -&&&&&&-&D&W&W&W&D&-&% % $$AA( " FEMF+@ FGDICF(GDICBIFEMF+*@$??@ b'DǡC 'DǡC&D2C&DC&DC 'D9&Cb'D9&C'D9&C(DC(DC(D2C'DǡCb'DǡC@!b $$==%  ;UP )-)-)?)U)k)})})})k)U)?)-)-<>BH % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPAJ )-)-)?)U)k)})})})k)U)?)-)-% % $$AA( " FEMF+@ FGDICF(GDICEaFEMF+*@$??@L@^UU?A@@@@, g'D9ޯCg'D¢C@!b $$==_888% %  ;)6)X4))))))6)X4))))))=)6)NX4)I)E)E)E)I)N6)X4))))))=) 6)X4))))))6) X4)))))) =)6)dX4)`)\)\)\)`)d6)X4))))))=<?Db % % $$AA( " FEMF+@ FGDICF(GDICX|FGDICRp@"Arial0 < >0#0- ` 00-z08xT0I0i0W- ` Ldvdv `0`\t(0dv%    TT_tKKKAJAqLPc  % ( Rp@"Arial 0@|( Q|Hm|A 00I0i0W- ` Lz08 0 `\t(0dv`)!PD|p|m|Z|m|dv%    TTjyKKKAJAwLP2 % ( F(GDIC1FEMF+*@$??@@4^UU?A@,  DyC9%DyC@!b $$==_888% % W$0 #f)% % $$AA( " FEMF+@ FGDICFGDICF4(EMF+*@$??!b Ld)??" 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S  0E  ``  @*   0(4E  `   B*   0p^E  `  E  B* H  0޽h ? ̙33 Default Design5 9515 KZ4(    <XQ ^!  jRolle s Theorem    0LRQ ST+ |~Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) c) f(a) = f(b) then there is a number c in (a,b) such that f (c) = 0H tE, @xXF  ;e Y ; efB  6D;}fB  6D MfB  6D w!xfB   6D    Hp^Q "`+ =a 0    HQ "`C =b 0   HЂQ "`_[ =y 0   H܊Q "`e =x 0   HT"`K Mc00  fB  6D-fB  6DԔ{t2  6"`w/t2  6"`g-t2  6"` ) F :  W : fB 6 6D|:}|fB 7 6D fB 8B 6D KfB 9 6D * 3+  : HI"`t =a 0  ; HT"`   =b 0  < H"`Z =y 0  = H8O"`X   =x 0  > H"`9 Mc00  fB ? 6D. D S B C#DEFAԔ#>v  @  c"$` & =t2 A 6"` (: Et2 B 6"`,-It2 C 6"` *`B H 0D T  F /; Z ;/fB  6D;}fB  6D/rfB  6D !fB  6D   HHQ "`P =a 0   HUQ "`h  =b 0    HQ "`[6 =y 0  ! H`Q "` =x 0  " H "`k Mc00  fB # 6Dm F S BCDEFAԔJ@  c"$` ct2 % 6"`/t2 & 6"`-t2 ' 6"`Qo`B J 0DpSp R <|KQ gN 6A.  S <8nQ h|tO 6B.  T <Q g@ @ N 6C.  U <@kQ hO 6D. v V <   dCase 1: f(x) = k (constant) [picture A] Case 2: f(x) > f(a) for some x in (a,b) [picture B or C] Extreme value theorem guarantees a max value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this max must occur. Fermat s Theorem says that f (c) =0. Case 3: f(x) < f(a) for some x in (a,b) [picture C or D] Extreme value theorem guarantees a min value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this min must occur. Fermat s Theorem says that f (c) =0.H3 )  : 'Y>Y> F  : X : fB ( 6Dj :k }fB ) 6D fB * 6D   fB + 6D  , H9"` p  =a 0  - H"` =b 0  . 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Ta % % $$AA( " FEMF+@ FGDICF(GDIC\&yFGDICRp@Arial0\lX0L?l> 0z088t8 00$4`04(L(dv%    TlcrKKKAJApLXx = c % ( F(GDICIYFEMF+*@$??@L@^UU?AA@@@, RDljC9GDɫC@( !b $$==_888% '%  ;46F4X4C4>4:47454846y4X4{444444=!4'63X433333364X4444"4#4#!4'=36{3 X4x3$s3%o3#k3 j3m363X4333333=V3X63X43 3 33336H3OX4J3KO3JS3LW3OX3TV3X=262RX42U2V2T2Q2L2H62X4222222=26I2X4G2B2>2:292<26|2X42|2{2}222=$2"61vX41z1{1y1v1q1m62X422"2&2'2$2"=<?KZ % % $$AA( " FEMF+@ FGDICF(GDIC SFEMF+*@$??@L@^UU?AA@@@, U=DCUy1D?C@( !b $$==_888% '%  ;p/6-/X4*/+///3/8/;/6~/X4//|/ x/"s/!p/=/6.(X4.$.....6/~X4//// //=.6[.X4X.Y.].a.f.i.6.X4......=5.[6-X4-----.6C.QX4E.UD.Z@.]=._7._5.[=-6-eX4-a-\-Y-W-X-[6-X4------=c-.6 -X4--"-&-+-.-6p-%X4s-(r-.n-0k-3e-2c-.=,6,8X4,5,/,-,*,+,/6-X4 - ---,,=,6P,X4M,N,R,V,[,^,6,X4,,,,,,=<?T % % $$AA( " FEMF+@ FGDICF(GDIC50MFGDICRp@Arial0\lX0L?l> 0z088t8 00$4`04(L(dv%    T@7FKKKAJA@DLlf > 0 for x > c % ( F(GDICr$@FGDICRp@Arial0\lX0L?l> 0z088t8 00$4`04(L(dv%    T}*9KKKAJA}7Llf < 0 for x < c % ( F(GDICUFGDICRp@Arial0# \lX0L? l> 0z08 8t 8 0 0$4m504d(L(dv%    THKKKAJA LdRelative Min     % ( FGDICF(GDIC&F(GDICOFGDICRp@Arial0 H@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TTpuKKKAJApLP- % ( F(GDIC QFGDICRp@Arial0 nH@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TT'1KKKAJA'LP0  % ( F(GDIC FGDICRp@Arial0 H@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TTKKKAJALP+  % ( F(GDIC'FGDICRp@Arial0 yHX0 < TiK0W0HXM<z08D0蜥!04m504(Pdv%    T|3xKKKAJA3L\Sign of    % ( Rp@ArialW0HXL|Q|Hm| z0蜥!0z08@m504 (Pdv+!\P|p|m|Z|m|dv%    TdyKKKAJAyLTf (x  % ( Rp@Arial蜥L|Q|Hm| {m504 (z08̐dv+! \P|p|m|Z|m|dv+!\P|p|m|Z|m|dv%    TTKKKAJALP) % ( F(GDICOFGDICRp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    Tl]KKKAJA]LXx > c    % ( F(GDIC QFGDICRp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    TlAKKKAJALXx = c    % ( F(GDIC FGDICRp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    TlKKKAJALXx < c    % ( F(GDIC'FGDICRp@Arial0 H@0 < TiK0W0H@M<z08 0蜥!04m504d(L(dv%    T3KKKAJA3 LdRelative Max     % ( F(GDIC&FEMF+*@$??@<0H@A@, 9!BU=C96CU=C@!b $$==_8882% % W$$_% % $$AA( " FEMF+@ FGDICF(GDIC'FEMF+*@$??@4(DZ?A@, 9!BC96CC@!b $$==_888% % W$&_% % $$AA( " FEMF+@ FGDICF(GDIC&FEMF+*@$??@ <0H@A@!, 9!B9.C96C9.C@! !b $$==_8882% % W$$_% % $$AA( " FEMF+@ FGDICF(GDIC&+FEMF+*@$??@"<0H@A@#, 9!BU=C9!B9.C@#"!b $$==_8882% % W$$,% % $$AA( " FEMF+@ FGDICF(GDICFEMF+*@$??@$4(DZ?A@%, PCCU=CPCC9.C@%$!b $$==_888% % W$- - % % $$AA( " FEMF+@ FGDICF(GDIC  FEMF+*@$??@&4(DZ?A@', 9CU=C9C9.C@'&!b $$==_888% % W$ % % $$AA( " FEMF+@ FGDICF(GDICOQFEMF+*@$??@(4(DZ?A@), 9CU=C9C9.C@)(!b $$==_888% % W$MR% % $$AA( " FEMF+@ FGDICF(GDICFEMF+*@$??@*<0H@A@+, 96CU=C96C9.C@+*!b $$==_8882% % W$__% % $$AA( " FEMF+@ FGDICFGDICF(GDICCF(GDIClFGDICRp@Arial0 H@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TTKKKAJALP+  % ( F(GDIC%nFGDICRp@Arial0 H@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TTCMKKKAJACLP0  % ( F(GDIC'FGDICRp@Arial0 H@0 t<TiK0W0H@Lt<z080蜥!04m504d(L(dv%    TTKKKAJALP- % ( F(GDICDFGDICRp@Arial0 HX0 < TiK0W0HXM<z08D0蜥!04m504(Pdv%    T|OKKKAJAOL\Sign of    % ( Rp@ArialW0HXL|Q|Hm| 0蜥!0z08@m504 (Pdv+!\P|p|m|Z|m|dv%    TdKKKAJALTf (x  % ( Rp@Arial蜥L|Q|Hm| m504 (z08̐dv+! \P|p|m|Z|m|dv+!\P|p|m|Z|m|dv%    TTKKKAJALP) % ( F(GDIClFGDICRp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    TlzKKKAJAzLXx > c    % ( F(GDIC%nFGDICRp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    Tl3]KKKAJA3LXx = c    % ( F(GDIC'FGDICRp@Arial0 $H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    TlKKKAJALXx < c    % ( F(GDICDFGDICRp@Arial0 H@0 < TiK0W0H@M<z08 0蜥!04m504d(L(dv%    TOKKKAJAO LdRelative Min     % ( F(GDICCFEMF+*@$??@,<0H@A@-, 9DiClDiC@-,!b $$==_8882% % W$@G$&;% % $$AA( " FEMF+@ FGDICF(GDICDFEMF+*@$??@.4(DZ?A@/, 9D9ClD9C@/.!b $$==_888% % W$BG$&;% % $$AA( " FEMF+@ FGDICF(GDICCFEMF+*@$??@0<0H@A@1, 9DClDC@10!b $$==_8882% % W$@G$&;% % $$AA( " FEMF+@ FGDICF(GDICCGFEMF+*@$??@2<0H@A@3, 9DiC9DC@32!b $$==_8882% % W$@IG$G$% % $$AA( " FEMF+@ FGDICF(GDICFEMF+*@$??@44(DZ?A@5, 7DiC7DC@54!b $$==_888% % W$--% % $$AA( " FEMF+@ FGDICF(GDIC%'FEMF+*@$??@64(DZ?A@7, IDiCIDC@76!b $$==_888% % W$$(\2\2% % $$AA( " FEMF+@ FGDICF(GDICknFEMF+*@$??@84(DZ?A@9, $[DiC$[DC@98!b $$==_888% % W$jn66% % $$AA( " FEMF+@ FGDICF(GDICFEMF+*@$??@:<0H@A@;, lDiClDC@;:!b $$==_8882% % W$&;&;% % $$AA( " FEMF+@ FGDICFGDICF(GDICJFGDICRp@Arial|/0UWlW\/0Vz08tl0l4V4m504d(t(dv%    33T PmKKKAJA h@LExamine the behavior of the first derivative close to the critic     % ( Rp@ArialWL|Q|Hm| 50l4Vz08@m504 (t(dv+!\P|p|m|Z|m|dv%    33TPmKKKAJAhL|al value. If the sign  % ( Rp@Arial|/0SUlU\/0Tz08l0l4T4m504d(t(dv%    33T nKKKAJA @Lof the first derivative changes before and after the critical va      % ( Rp@ArialUL|Q|Hm| ɐ0l4Tz08@m504 (t(dv+!\P|p|m|Z|m|dv%    33TnKKKAJALxlue, then a relative   % ( Rp@Arial0 CH̐0 =GTiK0W0H̐M=z080H0蜥!04m504 (t< dv%    33Tx hKKKAJA L\extrema   % ( Rp@ArialW0H̐L|Q|Hm| D0蜥!0z08@m504 (t< dv+!\P|p|m|Z|m|dv%    33TpBKKKAJAp?Lhas occurred. Most folks like to use a table like those below.  % ( F4(EMF+*@$??!b Ld)??" FEMF+@ CompObjE{ObjInfo37GPicturesCurrent User69HG%_3 Chris HeadleeChris HeadleeOh+'0 X`p  Slide 1Chris HeadleeoChris Headleeo17iMicrosoft PowerPointP@`@N@ C[6f՜.+,0SummaryInformation(J(PowerPoint Document(8:Y3DocumentSummaryInformation8O_1119769975=Fd,Td,Tw( / 0LDArialttt0B 0@ .  @n?" dd@  @@`` ( E` 0AA @3ʚ;ʚ;g4\d\d B 0pw p@ <4dddd8pC 0tL 80___PPT10   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6  `}  T Click to edit Master title style! !  0kQ   `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0Q  ^ ` Q  X*  0Q  ^  Q  Z*  0Q  ^ ` Q  Z*H  0޽h ? 3380___PPT10.K tv Default Design& %% ?_ %(   E <t 6J  [)First Derivative Test for Critical Points**8 d    jF &N     <d g `B  0D13 9`B   0D1     BCDEFAAQY'O@    n2   0"`  ZB   s *D     <t v W 7x = c ZB  B s *D  ZB  s *D @ 2    <hXt    V f > 0 for x < c    < t     V f < 0 for x > c    <Ct  b  > Relative Max   8  s ) j &N     v s`B  0D13 9`B   0D1     BtC&DEFAA&&C&t@   * t2 !  6"`V s `B "  0Dd dA  #  Btt 6   7x = c `B $ B 0D  `B % 0DZ   &  B<\t V9   V f > 0 for x > c  '  Bct  h  V f < 0 for x < c  (  B`t ^M   > Relative Min  . h w %  ^  #"r %   2  <t ?6 %  M- @` 1  <xGt ?6   M0 @` 0  <t ?e6   M+ @` /  <pLt ?6 e  Sign of f (x) @` .  <Wt ?w % 6  Qx > c @` -  < t ?w 6  Qx = c @` ,  <d?ew 6  Qx < c @` +  <Ti?w e6  V Relative Max   @``B 3  0o ?w % w ZB 4  s *1 ?6 % 6 `B 5  0o ? % `B 6  0o ?w ZB 7  s *1 ?ew e ZB 8  s *1 ?w  ZB 9  s *1 ?w  `B :  0o ?% w % . h  y N  ]  #"_  N%  I  <?8 N  M+ @` J  <,G?$8   M0 @` K  <O?8 $  M- @` L  <? 8   Sign of f (x) @` M  < ?y N8  Qx > c @` N  <?$y 8  Qx = c @` O  <@׿?y $8  Qx < c @` P  <P? y 8  V Relative Min   @``B Q  0o ? y Ny ZB R  s *1 ? 8 N8 `B S  0o ?  N `B T  0o ? y   ZB U  s *1 ?y  ZB V  s *1 ?$y $ ZB W  s *1 ?y  `B X  0o ?Ny N  _ 0/ CExamine the behavior of the first derivative close to the critical value. If the sign of the first derivative changes before and after the critical value, then a relative extrema has occurred. Most folks like to use a table like those below.4AH  0޽h ? 3380___PPT10.K tvr0y e3    =On-screen Show -s3: ArialDefault DesignSlide 1  Fonts UsedDesign Template Slide TitlesOle WCompObj<>XfObjInfo?ZEquation Native [h FMicrosoft Equation 3.0 DS Equation Equation.39qiLlIЙI fx()=x1+x 2 FMicrosoft Equation 3.0 DS Equation Equation.39q_1119770026 JBFd,Td,TOle ]CompObjAC^fObjInfoD`i~ILI c FMicrosoft Equation 3.0 DS Equation Equation.39qiI|I cEquation Native a,_1119770072GFd,Td,TOle bCompObjFHcfObjInfoIeEquation Native f,_1119770474ELFd,Td,TOle gCompObjKMhfObjInfoNjEquation Native k,_1119770654QFd,Td,T FMicrosoft Equation 3.0 DS Equation Equation.39qimIyI c FMicrosoft Equation 3.0 DS Equation Equation.39qOle lCompObjPRmfObjInfoSoEquation Native p|i`mIyI fx()=x 23 1"x()t dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_' Chris HeadleeChris Headlee_1255682889XdO)d,T,TOle rEPRINTUWTCompObjs{l1'% EMFT\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$=='% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDICD3FGDICRp@Arial0?)\lX0L?*l> 0z08)8t)8 0)0$4m504d (L( dv%    TH8+KKKAJA%*LSecond Derivative Test for Critical Points           % ( F(GDICUF(GDICFEMF+*@$??@DZ?Ad8^@ A?( UU@UUUUUU@, 9BCCPCCXaC@( !b $$==_888% '%  [L  " [8 [-   j- [ k j % % $$AA( " FEMF+@ FGDICF(GDICwUFEMF+*@$??@DZ?Ad8^@ A?( UU@UUUUUU@, xCsC[CsC@!b $$==_888% %  [LxT    $ % % $$AA( " FEMF+@ FGDICFGDICF(GDIC BFEMF+*@$??@@4^UU?A@XLJ6CCZDC䐮CjRC9CǁiCCrLC#C96CC䀠C)C@!b $$==_888% % U8C] >  v % % $$AA( " FEMF+@ FGDICF(GDIC&FEMF+*@$??@ kiCChCCfC%CfC[CfC ChCVCkiCVC}jCVCkC CkC[CkC%C}jCCkiCC@!b $$==%  ;UP ygg$g:yLLL:$<>% % $$AAF`TEMF+@@4^UU?A@$$==_888% % UP' ygg$g:yLLL:$% % $$AA( " FEMF+@ FGDICF(GDIC"~FEMF+*@$??@D8~UU?A@ , UUiCr|CUUiCsC@ !b $$==_888% %  ;(6(X4-000-(6(X4#   #(=I6IX4NRRRNI6IX4EAAAEI=k6kX4osssok6kX4fbbbfk=6X46X4=6X46X4=6X46X4=6X46X4=6X46X4     =363X47;;;7363X4.***.3=T6TX4Y]]]YT6TX4PLLLPT=v6vX4z~~~zv6vX4qmmmqv=6X46X4=6X46X4=6X46X4=6X46X4=6X4!%%%!6X4=>6>X4BFFFB>6>X495559>=_6_X4dgggd_6_X4ZWWWZ_=6X46X4|xxx|=6X46X4=6X46X4=6X46X4=6X4  6X4='6'X4,///,'6'X4""'=H6HX4MQQQMH6HX4D@@@DH=j6jX4nrrrnj6jX4eaaaej=6X46X4=6X46X4=6X46X4=6X46X4=6X46X4  =262X46:::6262X4-***-2=S6SX4X\\\XS6SX4OKKKOS=u6uX4y}}}yu6uX4plllpu=6X46X4=6X46X4=6X46X4=6X46X4=6X4 $$$ 6X4==6=X4AEEEA=6=X484448==^6^X4cfffc^6^X4YVVVY^=6X46X4{www{=6X46X4=6X46X4=<?  % % $$AA( " FEMF+@ FGDICF(GDICuFGDICRp@Arial0\lX0L?l> 0z088t8 00$4`04(L(dv%    Tl|KKKAJALXx = c % ( F(GDIC5FGDICRp@Arial0# \lX0L? l> 0z08 8t 8 0 0$4m504d(L(dv%    T+KKKAJA LdRelative Max     % ( F(GDICbF(GDICFEMF+*@$??@ DZ?Ad8^@ A?( UU@UUUUUU@ , U3DC3DXaC@  !b $$==_888% %  [L ,,[-[,,,j,&-k,j % % $$AA( " FEMF+@ FGDICF(GDICwbFEMF+*@$??@ DZ?Ad8^@ A?( UU@UUUUUU@ , )DsCWDsC@  !b $$==_888% %  [Lxa Z*55Z*Z*5555 % % $$AA( " FEMF+@ FGDICFGDICF(GDICLsFEMF+*@$??@@4^UU?A@XLO,DCx2DCǭ8DϸC?DϸCqpEDϸCU LDCURDC@!b $$==_888% % U8Ls +,$./U124% % $$AA( " FEMF+@ FGDICF(GDICnuFEMF+*@$??@ q>DkCv>DkCU]>DCU]>DǭCU]>D_Cv>DCq>DCmW?DC?D_C?DǭC?DCmW?DkCq>DkC@!b $$==%  ;UP /////$/6/6/6/$////<>nt % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPlu /////$/6/6/6/$////% % $$AA( " FEMF+@ FGDICF(GDICs}FEMF+*@$??@D8~UU?A@, >DٹC>DC@!b $$==_888% %  ;/46/4X4/8/</</</8/46/4X4///+/+/+///4=/U6/UX4/Z/]/]/]/Z/U6/UX4/P/M/M/M/P/U=/v6/vX4/{////{/v6/vX4/r/n/n/n/r/v=/6/X4//////6/X4//////=/6/X4//////6/X4//////=<?q~ % % $$AA( " FEMF+@ FGDICF(GDICyFGDICRp@Arial0\lX0L?l> 0z088t8 00$4`04(L(dv%    Tl KKKAJALXx = c % ( F(GDICEFGDICRp@Arial0# \lX0L? 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If the sign of  % ( Rp@Arial|/0QSlS\/0Rz08tl0l4R4m504d(t(dv%    33T cKKKAJA {@Lthe second derivative is negative, then the function is concave   % ( Rp@ArialSL|Q|Hm| 0l4Rz08@m504 (t(dv_!\P|p|m|Z|m|dv%    33TcKKKAJA{Ltdown at that point  % ( Rp@Arial|/0LNlN\/0Mz08tl0l4M4m504d(t(dv%    33T KKKAJA @Land a relative maximum has occurred. If the sign of the second    % ( Rp@ArialNL|Q|Hm| 0l4Mz08@m504 (t(dv_!\P|p|m|Z|m|dv%    33TKKKAJALhderivative is   % ( Rp@Arial|/0QSlS\/0Rz08tl0l4R4m504d(t(dv%    33T KKKAJA @Lpositive, then the function is concave up at that point and a re  % ( Rp@ArialSL|Q|Hm| 0l4Rz08@m504 (t(dv_!\P|p|m|Z|m|dv%    33TKKKAJALtlative minimum has  % ( Rp@Arial0 H@0 < TiK0W0H@ M<z08 0蜥!04m504d(L(dv%    33T {KKKAJA L`occurred.  % ( F(GDICGOFGDICRp@Arial|/0(*l*\/0)z08Ll0l4)4m504g(Lgdv%    THR/KKKAJAR,*LNote: rate of change of first derivative            % ( Rp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    TR2GKKKAJARDLtis always negative.      % ( F(GDICQOFGDICRp@Arial|/0(*l*\/0)z08Ll0l4)4m504g(Lgdv%    TH\/KKKAJA\,*LNote: rate of change of first derivative            % ( Rp@Arial0 H@0 <TiK0W0H@M<z080蜥!04m504d(L(dv%    T\2GKKKAJA\DLtis always positive.      % ( F4(EMF+*@$??!b Ld)??" 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FGDICFGDICF(GDICXKFGDICRp@Times New Roman0?2\Ēl03Ē> 0z08 28t28 020$lm50 (`( dv%    TcCKKKAJAc=3LGraphing Summary using Information from Derivatives            % ( F(GDIC5\FGDICRp@Times New Roman0j L<g>0#0-0g8<gg0-z08 T0I0i0 -0gdvdv @Times New Roman} dv%    T`@b_KKKAJA@|LTf(x  % ( Rp@Times New Romang8<g@|` Q|Hm|} ``I0i0 -0gz08  @Times New Ro} dv !KPD|p|m|Z|m|dv%    T|`bKKKAJA`|L\) = x  % ( Rp@Times New Romani0 -@|` Q|Hm|} ``  @Timez08$dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TTdwKKKAJAtLP4  % ( Rp@Times New Roman @|` Q|Hm|} `` !K PD|p|m|z08<dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TTbKKKAJA|LP  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08Ȏdv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TXbKKKAJA|LP4x % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08Tdv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TTdwKKKAJAtLP3  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TdbBKKKAJA|LT= x % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08ldv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TTCdKwKKKAJACtLP3  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    T`LbiKKKAJAL|LT(x  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08|gdv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TTibwKKKAJAj|LP  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08 gdv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    TXbKKKAJA|LP4)  % ( Rp@Times New Roman0} < g>0#0-0g8< g0-z08 T0I0i0 -0gdvdv @Times New Romanj bdv%    33Td@hKKKAJA@LTf (x  % ( Rp@Times New Romang8< g@|` Q|Hm|j c``I0i0 -0gz08  @Times New Roj bdv !KPD|p|m|Z|m|dv%    33TxiKKKAJAiL\) = 4x  % ( Rp@Times New Romani0 -@|` Q|Hm|j d``  @Timez08$dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    33TXKKKAJALP3  % ( Rp@Times New Roman @|` Q|Hm|j e`` !K PD|p|m|z08dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    33TTKKKAJALP  % ( Rp@Times New Roman P@|` Q|Hm|j f`` !K PD|p|m|z08<dv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%    33T`KKKAJALT12x % ( Rp@Times New Roman P@|` Q|Hm|j g`` !K PD|p|m|z08Ȏdv !K PD|p|m|Z|m|dv !KPD|p|m|Z|m|dv%  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Q|Hm|} `` !K PD|p|m|z08<dv !K PD|p|m|Z|m|dvx !KPD|p|m|Z|m|dv%    T|KKKAJA Lh24x = 12x (x   % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08Ȏdv !K PD|p|m|Z|m|dvx !KPD|p|m|Z|m|dv%    TT|KKKAJA}LP  % ( Rp@Times New Roman P@|` Q|Hm|} `` !K PD|p|m|z08Tdv !K PD|p|m|Z|m|dvx !KPD|p|m|Z|m|dv%    TXKKKAJALP2)  % ( F(GDICF4(EMF+*@$??@&^DZ?A@@@d8^@ A?( UU@UUUUUU33@'<09*DUCviDUCiDDcDD@'&( !b $$==_88833% '33%  ;y* 6* X4* * * * *! *! 6y*! 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Suppose that           % ( Rp@Times New Roman0l \0 TiK0W0ؐ z08lp0o!0tTm50tl(oddv%    T`~KKKAJALTlim % ( Rp@Times New RomanW0L|Q|Hm|l ]0o!0z08m50t (oddv !\P|p|m|Z|m|dv%    T`~KKKAJALTf(x  % ( Rp@Times New RomanoL|Q|Hm|l ^m50t (oz08$dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    Tl~KKKAJALX) = 0 % ( Rp@Times New Roman L|Q|Hm|l _ ! \P|p|m|z08<dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`~KKKAJALTand  % ( Rp@Times New Roman \L|Q|Hm|l ` ! \P|p|m|z08Tdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`~KKKAJALTlim % ( Rp@Times New Roman \L|Q|Hm|l a ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`~KKKAJALTg(x   % ( Rp@Times New Roman \L|Q|Hm|l b ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    Tl~4KKKAJALX) = 0  % ( Rp@Times New Roman0l k0  TiK0W0 z08 0p 0o!0Tm50t\((dv%    33TxKKKAJAL\or that   % ( Rp@Times New Roman0l 0 "TiK0W0 z08p#0o!0tTm50t8 (odv%    T`KKKAJALTlim % ( Rp@Times New RomanW0L|Q|Hm|l 0o!0z08m50t (odv !\P|p|m|Z|m|dv%    T`KKKAJALTf(x  % ( Rp@Times New RomanoL|Q|Hm|l m50t (oz08$dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TdKKKAJALT) =  % ( Rp@Times New Roman L|Q|Hm|l  ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Ȫdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT KKKAJA LP" % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08૥dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`KKKAJALTand  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`KKKAJALTlim % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`KKKAJALTg(x   % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    Td'KKKAJALT) =  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08}dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT(5KKKAJA(LP % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Pdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT<MKKKAJA<LP" % ( Rp@Times New Roman0l 0 @7TiK0W0 @z08p80o!0tTm50t (x dv%    T Q0KKKAJA,#L(We have an indeterminate form for          % ( Rp@Times New RomanW0L|Q|Hm|l 0o!0z08m50t (x dv !\P|p|m|Z|m|dv%    T`Rt0KKKAJAR,LTlim % ( Rp@Times New RomanoL|Q|Hm|l m50t (z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T|{0KKKAJA{,L\f(x)/g(x     % ( Rp@Times New Roman L|Q|Hm|l  ! \P|p|m|z08$dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T|0KKKAJA,L\)) Then  % ( Rp@Times New Roman0l ૥0 &TiK0W0૥ z08p'0o!0Tm50tj ( dv%    T`IeKKKAJAaLTf(x   % ( Rp@Times New RomanW0૥L|Q|Hm|l 0o!0z08$m50t ( dv !\P|p|m|Z|m|dv%    TI<eKKKAJAaLp)  % ( Rp@Times New RomanoL|Q|Hm|l m50t (z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT=IFeKKKAJA=aLPf % ( Rp@Times New Roman L|Q|Hm|l  ! \P|p|m|z08<dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTEIMeKKKAJAEaLP  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Ȫdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TXNIbeKKKAJANaLP(x  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Tdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`cIveKKKAJAcaLT)  % ( Rp@Times New Roman0V ૥0 TiK0W0૥ڐ z08 p 0o!0Tm50tj ( dv%    T`s^zKKKAJAsvLTlim % ( Rp@Times New RomanW0૥L|Q|Hm|V 0o!0z08$m50t ( dv !\P|p|m|Z|m|dv%    T^zKKKAJAv L`---------   % ( Rp@Times New RomanoL|Q|Hm|V m50t (z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TX^zKKKAJAvLP=  % ( Rp@Times New Roman L|Q|Hm|V  ! \P|p|m|z08<dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`^)zKKKAJAvLTlim % ( Rp@Times New Roman \L|Q|Hm|V  ! \P|p|m|z08Tdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T|0^rzKKKAJA0vL\--------    % ( Rp@Times New Roman0l ૥0 #TiK0W0૥ z08p$0o!0Tm50tj ( dv%    T`sKKKAJALTg(x   % ( Rp@Times New RomanW0૥L|Q|Hm|l 0o!0z08$m50t ( dv !\P|p|m|Z|m|dv%    Ts9KKKAJALp)  % ( Rp@Times New RomanoL|Q|Hm|l m50t (z08dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT:sFKKKAJA:LPg  % ( Rp@Times New Roman L|Q|Hm|l  ! \P|p|m|z08<dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTGsNKKKAJAGLP  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Ȫdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TXOscKKKAJAOLP(x  % ( Rp@Times New Roman \L|Q|Hm|l  ! \P|p|m|z08Tdv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTdskKKKAJAdLP) % ( Rp@Times New Roman0l 0 $0TiK0W0 $z08Xp10o!0tTm50t\(Pdv%    T\ KKKAJA -Lif the limit on the right side exists (or is              % ( Rp@Times New RomanW0L|Q|Hm|l 0o!0z08m50t (Pdv !\P|p|m|Z|m|dv%     TTKKKAJALP  TTKKKAJALP"  % ( Rp@Times New RomanoL|Q|Hm|l m50t (z08 dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTKKKAJALP) % ( F(GDICFGDICRp@Times New Roman0#\T0 0> 0z088t8 00$tT`0t(Pdv%    TXKKKAJALPx  % ( Rp@WingdingsL|Q|Hm|V 8t8 00z08`0t (Pdv !\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@Times New Roman8L|Q|Hm|V `0t (z08 dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTKKKAJALPa  % ( F(GDICFGDICRp@Times New Roman0#\T0 0> 0z088t8 00$tT`0t(Pdv%    TXKKKAJALPx  % ( Rp@WingdingsL|Q|Hm|l 8t8 00z08`0t (Pdv !\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@Times New Roman8L|Q|Hm|l `0t (z08 dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTKKKAJALPa  % ( F(GDICloFGDICRp@Times New Roman0#\T0 0> 0z088t8 00$tT`0t(Pdv%    TXxvKKKAJAxLPx  % ( Rp@WingdingsL|Q|Hm|V 8t8 00z08`0t (Pdv !\P|p|m|Z|m|dv%    TT{KKKAJALP % ( Rp@Times New Roman8L|Q|Hm|V `0t (z08 dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTvKKKAJALPa  % ( F(GDICo3FGDICRp@Times New Roman0#\T0 0> 0z088t8 00$tT`0t(Pdv%    TXv KKKAJALPx  % ( Rp@WingdingsL|Q|Hm|l 8t8 00z08`0t (Pdv !\P|p|m|Z|m|dv%    TT {KKKAJA LP % ( Rp@Times New Roman8L|Q|Hm|l `0t (z08 dv ! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTv'KKKAJALPa  % ( F(GDICFGDICRp@Times New Roman0#\T0 0> 0z088t8 00$tT`0t(Pdv%    TXKKKAJALPx  % ( Rp@WingdingsL|Q|Hm|V 8t8 00z08`0t (Pdv !\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@Times New Roman8L|Q|Hm|V `0t (z08 dv ! 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FEMF+@  dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_" Chris HeadleeChris HeadleeOh+'0 X`p  Slide 1Chris HeadleeoObjInfo"PicturesCurrent User!$GSummaryInformation((Chris Headleeo4riMicrosoft PowerPointP@{@X@@vӲ՜.+,0    QOn-screen Show -s+": ArialTimes New RomanDefault DesignSlide 1PowerPoint Document(#%s+"DocumentSummaryInformation8_1119855192(F,T,TOle  ( `/ 0DArialNew ttd(0B 0DTimes New Roman0B 0@ .  @n?" dd@  @@`` x h8h 0AA@3ʚ;ʚ;g4YdYd B 0&ppp@ <4dddd8pC 0t<( 80___PPT10   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6t  `}  T Click to edit Master title style! !  0w  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  00~ ^ `  X*  0 ^   Z*  0 ^ `  Z*H  0޽h ? 3380___PPT10. Default Design:  2g@"(  F % lJ  7  fT G  8# +  ZB 9 s *D G ZB : s *DZ G Z ZB ; s *D- G - ZB < s *DG ZB = s *DG ZB > s *DG ZB ? s *D G ZB @ s *D G ZB A s *DG ZB B s *DiG iZB C s *D<G <ZB D s *D G ZB E s *Dx G x ZB F s *DKG KZB G s *DG ZB H s *D< G < ZB I s *D G  ZB J s *DG ZB K s *DG ZB L s *DG ZB M s *D G `B N 0Do  O <lH@ ;y0 P < ęmJ@ ;x0`B Q 0Do% ##T ` 0 R# + $ TB S c $D0 0TB T c $D TB U c $Dq qTB V c $D TB W c $D TB X c $D TB Y c $D TB Z c $D TB [ c $D TB \ c $D TB ] c $DH HTB ^ c $D TB _ c $D TB ` c $D TB a c $D' 'TB b c $DP PTB c c $D? ?TB d c $D TB e c $Dh hTB f c $D TB g c $D` `H  0޽h ? 3380___PPT10. r0 e!  Fonts UsedDesign Template Slide Titles FMicrosoft Equation 3.0 DS Equation Equation.39q,|I4I x="b2aCompObj')fObjInfo*Equation Native H_1254470254Q/dO),T,TOle EPRINT,.CCompObj{ObjInfo-1 dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_ Chris HeadleeChris HeadleeOh+'0 `h  l1'% EMFC\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$==% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDICJFGDICF(GDICUFGDICF(GDIC^FGDICRp@Times New Roman z08t0$ङ200D$Bङdv%    TX a"KKKAJA LP1. % ( Rp@Times New Roman|-wPU !f`|-w 0w PU !f|6w;w 0w @G-nwNnw 8Dtz08Њ 0hD@Times New Romanm5000t8 x dv%    T<aKKKAJA<@LAssign variables to all given quantities to be determined. When               ! % ( Rp@Times New Roman0 d\0 @TiK0W0d\\z08Њ8A0!0P0m50P (h( dv%    T<KKKAJA<@Lfeasible, make a sketch. 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FEMF+@ PicturesCurrent User03GSummaryInformation(@PowerPoint Document(24% PowerPoint PresentationoweoweChris Headleese1riMicrosoft PowerPointon@Kw@@@b՜.+,0    QOn-screen Shown-s%: ( `/ 0DTimes New Roman0B 0DArialNew Roman0B 0@ .  @n?" dd@  @@`` 0  0AA@3 ʚ;ʚ;g4YdYd B 0ppp@ <4dddd8pC 0tL(80___PPT10  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> *"(    6s P  T Click to edit Master title style! !  0v   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  04} ``  Z*   0 `   \*   0 `   \* H  0޽h ? ̙33 Default Design&  &(    H&PH@___PPT9" DAssign variables to all given quantities to be determined. When feasible, make a sketch. Label the picture with the quantities. Find a primary expression for the quantity to be optimized. Reduce this primary equation to one having a single independent variable  this may involve the use of  secondary equations (restrictions) relating the independent variable of the original equation. Determine the domain for the independent variable, usually an interval. Find the critical points and end points. Use the techniques of calculus to determine the desired maximum or minimum value. Answer the question (including units). 2p 0p @`  B˙#F9j `,Procedures for Solving Optimization Problems- -H  0޽h ? ̙3380___PPT10.yOr0 eDocumentSummaryInformation8_1255867187[9dO),Tb,TOle EPRINT68{Times New RomanArialDefault DesignSlide 1  Fonts UsedDesign Template Slide Titles dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q $_  chrisheadleechrisheadll 1-% EMF{'F, EMF+@XXF\PEMF+"@ @ $@ 0@?!@ @ F(GDIC FXLEMF+*@$?? @$;E E!b $$>>'% % V0 LF]LF]% % $$AA" FEMF+@ FGDICF(GDICSFGDICF(GDICFGDICF(GDIC FGDICRp@"Arial70MAl0ܬ0,  0m0-g0-Sj00eU0ܡ0-SA0eU00eU0DdK0~ 0I-S 0SA0U0%0<h0pArialS, YIdv%    T)r@ \@)`LtOptimization ExampleA3J*.337..J3. % ( F(GDIC lFGDICRp@"Arial0l0 0!m0-g0-Ko00eU0ܡ0-SA0eU00eU0DdK0~ 0I-K 0SA0U0%0<h0pArialK0YIdv%    T a ?@ \@. 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S  0ܒ ^ `  >*  0 ^   @*  0 ^ `  @*H  0޽h ? 3380___PPT10.p6-O Default Designo 0   o(    <N~ HOptimization Example w  0s~  A farmer whose property borders I-81 wants to maximize the size of the pasture for cattle he can create with his limited resources. He can only afford to buy materials for 2400 feet of fencing. How big can he make his pasture? N8     0` `  0 `B  0DjJ` ` `B  0DԔ     <A~   Z Govt Fencing      <~E  TFarmer s Field    <~L@  rA = l " w and l + 2w = 2400 so l = 2400  2w A = (2400  2w) " w = 2400w  2w dA --- = 2400  4w = 0 when 4w = 2400 or w = 600 ft dw A = 600 " (2400  1200) = 720,000 sq ft_ A F) @A@`A ` A`,_<*H  0޽h ? 3380___PPT10. 0.r0{ 0    JOn-screen Show. 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