ࡱ> oj{ bjbj 7 xx)qmTT####?#Ё2C"n<<<<___$u_]N___<<!+~+~+~_l<<+~_+~+~\+~<K=k#caD +~ۀ0Ё+~Nl N+~N+~_"6_+~N_b____uh___Ё____N_________T ]:  AP Calculus Assignments: Derivative Techniques DayTopicAssignment1Basic Differentiation FormulasHW Derivative Techniques - 1 2Product and quotient rulesHW Derivative Techniques - 23Derivatives of exponential and log functionsHW Derivative Techniques - 3 4Derivatives of Trig FunctionsHW Derivative Techniques - 45Rates of ChangeHW Derivative Techniques - 5 6Practice **QUIZ**HW Derivative Techniques - 67The Chain RuleHW Derivative Techniques - 7 8The Chain RuleHW Derivative Techniques - 89Implicit Differentiation HW Derivative Techniques - 9 10More practice **QUIZ**HW Derivative Techniques - 1011Derivatives of inverse functionsHW Derivative Techniques - 1112Still more practice **QUIZ**HW Derivative Techniques - 1213DifferentiabilityHW Derivative Techniques - 1314ReviewHW Derivative Techniques - Rev15**TEST** AP Calculus HW: Derivative Techniques - 1 Differentiate the following: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8. Find the equation of the line tangent to  EMBED Equation.DSMT4  at the point where x = 2. 9. Find the equation of the line tangent to  EMBED Equation.DSMT4  at the point where x = 1. 10. A bug travels along the x-axis. Its position as a function of time is given by  EMBED Equation.DSMT4  (x in meters, t in seconds; its a very fast bug) for t ( 0. a. Find the bugs velocity as a function of time. b. Find the bugs acceleration as a function of time. c. Find the bugs acceleration when its velocity is 0. 11. Solve  EMBED Equation.DSMT4  for q'(x) and prettify it (no complex fractions). (Note: this is an algebra problem; not a calculus problem. Do it anyway. And mind your gs and qs.) 12. Kenny had to take the derivative of  EMBED Equation.DSMT4 . He wrote  EMBED Equation.DSMT4 . What happened? AP Calculus HW: Derivative Techniques - 2 Differentiate using the product and quotient rules. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Differentiate using the most appropriate method. In #6 ( 8, a and b are constants. 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. Find the equation of the tangent line to the graph of  EMBED Equation.DSMT4  at the point where x = 2. 10. Suppose  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Evaluate a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  (This assignment continued on next page.) 11. If  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , evaluate  a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  12. The graphs of f and g are shown at right. Use them to evaluate  a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  13. If h is a differentiable function, find an expression for each of the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  14. a. Use nDeriv on your calculator to graph  EMBED Equation.DSMT4  and its derivative. (Enter Y = nDeriv(e^(X),X,X)) What is the derivative of  EMBED Equation.DSMT4 ? If the graph doesnt do it for you, check the tables. b. Use nDeriv to make a table of values for the derivative of  EMBED Equation.DSMT4  for x = 1, 2, 3, 4, 5, 6, . . . From the table, guess the derivative of  EMBED Equation.DSMT4 . 15. Use your calculator to estimate the value of  EMBED Equation.DSMT4 . 16. Kenny had to take the derivative of  EMBED Equation.DSMT4 . He wrote  EMBED Equation.DSMT4 . What happened? AP Calculus HW: Derivative Techniques - 3 Differentiate the following: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14. If  EMBED Equation.DSMT4  where g(0) = 5 and g'(0) = 2, find f '(0). (This assignment continued on next page.) 15. Review the graphs of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  and then use nDeriv on your calculator to graph the derivatives of those functions. From the graphs, determine the derivatives of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . 16. A problem on a long-ago homework assignment said: Estimate the value of the following limits. Then put the answers in your brain. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  Retrieve those answers from your brain. If you cant find them, redo the problem. 17. Kenny did problem #16 with his calculator in degree mode. What happened? AP Calculus HW: Derivative Techniques - 4 1. Memorize the derivative formulas for the six basic trig functions. Differentiate: 2. y = sinx + cosx 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8. Find the equation of the tangent line to the graph of  EMBED Equation.DSMT4  at the point where x = (/4. 9. Find the equation of the tangent line to the graph of  EMBED Equation.DSMT4  at the point where x = (. 10. If  EMBED Equation.DSMT4 , find expressions for r'(() and r"((). 11. Evaluate  EMBED Equation.DSMT4 . (Try to do this without actually taking 99 derivatives!) 12. a. Draw the graphs of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (g is called the signum function, sgn(x)). b. What is the relationship between the functions f and g above? 13. Without looking at your notes, write down the derivative formulas for the six basic trig functions. If you cant do this, go back to problem #1. 14. Kenny thought he could survive AP Calc without knowing his derivative formulas cold. What happened? AP Calculus HW: Derivative Techniques - 5 1. A bug moves along the x-axis with position  EMBED Equation.DSMT4  for t ( 0 (x in meters, t in seconds.) a. Find the bugs velocity at time t. b. When is the bug at rest? c. When is the bug moving forward? d. Find the bugs acceleration as a function of time. 2. a. Find the rate of change of the circumference of a circle with respect to its radius. b. Find the rate of change of the area of a circle with respect to its radius. 3. In a right triangle, one acute angle measures q and the leg adjacent to that angle measures a. a. Express the length of the opposite leg in terms of a and q. b. Find the rate of change of the opposite leg with respect to a (assuming q remains constant) and evaluate for ( = 0.5 and a = 100 cm. What are the units? c. Find the rate of change of the opposite leg with respect to q (assuming a remains constant) and evaluate for q = 0.5 and a = 100 cm. What are the units? 4. A tank holds 4800 gallons of water that drains through a hole in the bottom in 40 minutes. Torricellis Law says the volume of water in the tank after t minutes is given by  EMBED Equation.DSMT4 . a. Find the rate at which water is draining from the tank after 10 minutes. b. What does the sign of your answer in part a indicate? c. When is the water flowing out of the tank the fastest? Slowest? 5. a. Find the instantaneous rate of change of the volume of a cube as a function of its edge length x, in cm. b. Find the average rate of change of the volume of a cube on the interval 2 cm ( x ( 11 cm. c. For what value of x does the instantaneous rate of change of the volume equal the average rate of change on the interval [2, 11]? 6. Boyles Law says that for an ideal gas at a constant temperature, PV = k (pressure in pascals times volume in liters equals a constant). a. For what value of pressure will the instantaneous rate of change of volume with respect to pressure be the same as the average rate of change of volume on the interval 2 Pa ( P ( 8 Pa? b. A sample of gas at low pressure is steadily compressed (pressure is increased) at a constant temperature for a period of several minutes. Is the volume changing more rapidly at the beginning or the end of the period? Justify your answer. 7. Newtons Law of Gravitation says that two masses M and m attract each other with a force of  EMBED Equation.DSMT4  where G is a constant. Find  EMBED Equation.DSMT4 . Why is it negative? 8. Review the two limit definitions of the derivative f '(a). 9. Kenny was given the function y = xlnx for x ( [1, e] and asked to find the rate of change of y with respect to x. He did  EMBED Equation.DSMT4 . What happened? AP Calculus HW: Derivative Techniques - 6 1. The functions f and g are differentiable for 1 ( x ( 4. The values of the functions and their derivatives at selected points are given in the table below.  a. Evaluate  EMBED Equation.3 . b. Evaluate  EMBED Equation.DSMT4 . c. Evaluate  EMBED Equation.DSMT4 . d. Write an equation for the line tangent to the graph of g at x = 1. e. Estimate g"(2.) 2. The graphs of f and g are shown at right. Use them to evaluate a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  3. Differentiate the following: a. y = 3sin x b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f. y = ex g. y = 3x h. y = sec( i. y = e3 j.  EMBED Equation.3  k.  EMBED Equation.DSMT4  l.  EMBED Equation.3  4. The mass m in grams of a spherical snowball is a function of its radius r in centimeters: m = f(r). a. Using correct units, tell what f(4) = 268 means. b. Using correct units, tell what f '(4) = 201 means. Give an alternate notation for f ' in this context. c. Using the values from (a) and (b) above, estimate the mass of the snowball when (1) r = 4.1 cm. (2) r = 3.8 cm. d. What is the appropriate calculus notation for the rate of change of the mass of the snowball with respect to time t ? 5. Let f be a differentiable function of with f(5) = 2 and f '(5) = 3. Find the equation of the line tangent to the graph of g(x) = x2f(x) at x = 5. 6. Let f be a differentiable function of with f(9) = 4 and f '(9) = 2. Find the equation of the line tangent to the graph of  EMBED Equation.3  at x = 9. 7. a. Sketch a graph of a continuous function f having f ' > 0 for all x. b. Does the function you sketched in part (a) have an inverse function? HDYK? c. Hypothesize a general rule about the derivative of a continuous function f and the invertibilty of f. d. Use calculus to prove that the function f(x) = x3 + 4x 5 has an inverse function. e. Use calculus to prove that g(x) = x3 4x 5 is not invertible (unless we restrict the domain). 8. Kenny had to take the derivative of  EMBED Equation.DSMT4 . He wrote  EMBED Equation.DSMT4 . What happened? AP Calculus HW: Derivative Techniques - 7 Differentiate. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13. y = 4cos3(x) 14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17. If  EMBED Equation.DSMT4 , f(2) = 4, f '(2) = 1.5, f '(5) = 6, g(2) = 5 and g'(2) = 0.5, evaluate F '(2). 18. The functions f and g and are shown in the graph at right. Use the graph to evaluate the following:  a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  AP Calculus HW: Derivative Techniques - 8 Differentiate: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  In #13 16, f is a differentiable function. 13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4   17. Selected values of a differentiable function f are shown in the table at right. Use them to evaluate a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  AP Calculus HW: Derivative Techniques - 9 Use implicit differentiation to find  EMBED Equation.DSMT4  for the following: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. Chris and Jess had to find  EMBED Equation.DSMT4  for the function  EMBED Equation.DSMT4 . a. Jess applied implicit differentiation to the problem in its original form. What did she get? b. Chris used exponentiation base e to eliminate the natural log, getting  EMBED Equation.DSMT4 , and then differentiated implicitly. What was his answer? b. Jess and Chris then spent 15 minutes arguing about who was right. Who was? Justify your answer. 4. Find the slope of the graph of  EMBED Equation.DSMT4  at the point where y = 0. 5. Find the slope of the graph of  EMBED Equation.DSMT4  at the point  EMBED Equation.DSMT4 . 6. Find an equation of the tangent line to the graph of  EMBED Equation.DSMT4  at the point in the fourth quadrant where x = 2. 7. The graph of  EMBED Equation.DSMT4  is an ellipse. Find the coordinates of the points where the ellipse has horizontal tangent lines. AP Calculus HW: Derivative Techniques - 10 Yes, its a lot of problems. If you are getting good at this, it shouldnt take so long. If you are not yet good at this, you need the practice. Start differentiating. Differentiate the following. 1.  EMBED Equation.DSMT4  2. (x) = aebx 3. y = 2tan(4x 1) 4.  EMBED Equation.DSMT4  5. y = e2 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. (r) = 2r(r + h) 11. (z) = z2e 3z 12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17. Use the table of values for f, f ', g and g' to evaluate h'(2) for a. h(x) = x2 4g(x) b. h(x) = f(x)g(x) c.  EMBED Equation.DSMT4  d. h(x) = xeg(x) e. h(x) = f(g(x)) f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  Find y' for each of the following: 18.  EMBED Equation.DSMT4  19.  EMBED Equation.DSMT4  AP Calculus HW: Derivative Techniques - 11 1. If f is a one-to-one function, f(3) = 5 and f '(3) = 2, which of the following is true? (A) (f 1)'(3) =  EMBED Equation.3  (B) (f 1)'(5) =  EMBED Equation.3  (C) (f 1)'(5) = 2 (D)(f 1)'(2) =  EMBED Equation.3  (E) (f 1(5))' =  EMBED Equation.3  2. If f(x) = x3 + x 7, find (f 1)'(3). 3. If  EMBED Equation.3 , x > 0, and g(x) = f 1(x), find g'(15). 4. If x = y3 +2y + 4, find  EMBED Equation.DSMT4 . (This assignment continued on next page.)  5. The function f is one-to-one and differentiable. Selected values of f are shown in the table. Estimate the value of  EMBED Equation.3 .  6. The function f is graphed at right. The graph consists of two line segments. Find  EMBED Equation.3  without graphing f 1. 7. Use the method we used in class to find the derivative of sin1x to find derivatives for a. tan1x (memorize the answer) and b. ln x (I know you already know the answer; practice the technique.) AP Calculus HW: Derivative Techniques - 12 Differentiate the following: 1. y = 3sin (4x) 2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. f(x) = tan (2x) 7.  EMBED Equation.DSMT4  8. y = tan1 x 9. f(x) = x2ln x 10. f(x) = ex cos (2x) 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13. f(x) = 6x2e0.5x 14.  EMBED Equation.DSMT4  15. y = sin1 x 16. f(x) = ln(x2 3x + 4) 17. Use the graph at right to evaluate the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  Find y' for each of the following: 18.  EMBED Equation.DSMT4  19.  EMBED Equation.DSMT4  AP Calculus HW: Derivative Techniques - 13 1. Determine if  EMBED Equation.DSMT4 is differentiable at x = 1. 2. Find the values of a and b so that  EMBED Equation.DSMT4  is differentiable at x = 1. 3. Let  EMBED Equation.DSMT4 . Find '(0) and "(0). 4. Differentiate the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  5. Find the equation of the tangent line to  EMBED Equation.DSMT4  at the point (2, (4) AP Calculus Review: Derivative Techniques 1. Differentiate the following. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  2. Find the slope of the graph of  EMBED Equation.DSMT4  at the point (8, 64). 3. Find an equation of the tangent line to  EMBED Equation.DSMT4  at the point where x = 4. 4. If (1) = 3 and '(1) =  EQ \f(1,2) , find an equation of the tangent line to y = (x) where x = 1. (This assignment continued on next page.) x(2(101234f(x)462(1(4(20f '(x)1(1(2(322.53g(x)7530(1(2(4g '(x)(1(1.5(2(0.5(1.5(1(2 5. Suppose f and g are differentiable functions. Selected values of f and g and their first derivatives f ' and g' are shown in the table. Evaluate the following: a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  6. If  EMBED Equation.DSMT4 , evaluate  EMBED Equation.DSMT4 . 7. Find the equation of the tangent line to the graph of  EMBED Equation.DSMT4  at the point where y = e2. 8. The itsy bitsy spider is climbing up the water spout. Her position in centimeters as a function of time in seconds is given by y(t) = 150tan1(t), t ( 0. a. Find Itsys velocity as a function of time. b. Find Itsys acceleration as a function of time. c. Is Itsys speed increasing or decreasing? Justify your answer. 9. Find the equation of the tangent line to the graph of x3 3xy + y2 +19 = 0 at the point in the second quadrant where x = 4. 10. If f(x) = 2x5 + x3 + 1, find (f 1)'(2). 11. Let  EMBED Equation.DSMT4 . For what values of a and b will  EMBED Equation.DSMT4  be differentiable at x = 3? 12. A the volume of a right circular cone with a slant height of l, in meters, and a base angle of ( is given by  EMBED Equation.DSMT4 . a. Find the rate of change of the volume with respect to the slant height if the base angle is held constant. b. Find the rate of change of the volume with respect to the base angle if the slant height is held constant. c. What are the units of the answer in part b above? 13. Find the points where the graph of  EMBED Equation.DSMT4  has horizontal or vertical tangents. Answers to selected problems: HW 1 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. 6 m/s2 11.  EMBED Equation.DSMT4  HW 2 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10a. 8 b. 3 c. 12/25 11a. 1 b. 9/4 12a. 0 b. 2/3 13a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  HW 3 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14. 3 HW 4 2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  11. cos x 12b. g = f ', or  EMBED Equation.DSMT4  14. Kenny failed the course miserably. But because his school foolishly paid for the AP exam, Kenny took it anyway. He got a 1. Then he died horribly. HW 5 1a. v(t) = 3t2 24t + 36 b. t = 2 and t = 6 c. [0, 2) ( (6, () d. a(t) = 6t 24 2a.  EMBED Equation.DSMT4 = 2( b.  EMBED Equation.DSMT4 = 2(r (Note:  EMBED Equation.DSMT4 = C) 3a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4 cm/cm c.  EMBED Equation.DSMT4  cm/rad 4a. 180 gal/min b. water flowing out of the tank c. at t = 0; at t = 40 5a.  EMBED Equation.DSMT4  cm3/cm b.  EMBED Equation.DSMT4  cm3/cm c. 7 cm 6a. 4 Pa b. At the beginning since as P increases,  EMBED Equation.DSMT4  will decrease. 7.  EMBED Equation.DSMT4 ; as r increases, F decreases. HW 6 1a. 2 b. 4 c. 9/4 d. y 2 = 5(x 5) e. 3/2 2a. 1/6 b. 7/24 3a. 3cos x b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d. 18u4 e. 6s 4 f. ex g. 3xln3 h. sec ( tan ( i. 0 j. 2tcos t t2sin t k.  EMBED Equation.DSMT4  l. 5x4 + 4x x2 4a. When the radius is 4 cm, the mass is 268 g. b. When the radius is 4 cm, the rate of change of the mass with respect to radius is 201 g/cm. c1) 288.1 g 2) 227.8 g 5. y 50 = 95(x 5) 6.  EMBED Equation.DSMT4  7b. Yes; f is one-to-one by the HLT. c. If f ' is always positive or always negative, f will have an inverse function. d. f '(x) = 3x2 + 4 which is positive for all x so f has an inverse. e. g'(x) = 3x2 4 which is neither always positive nor always negative so f is not invertible. 8. Kenny died horribly. Because it was the second time he had made the same really annoying mistake, his irate math teacher had his body blasted into its individual atoms which were in turn reduced by fission to so many hydrogen atoms. The energy thus released was used to disperse those atoms to the far reaches of the universe. Kenny was never heard from again. HW 7 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. y' = tan( 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17. 3 18a. 0 b. 4/3 c. 3/4 d. undefined e. 2 HW 8 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17a. 3 b. 3 HW 9 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. Horizontal tangents at  EMBED Equation.DSMT4  (and vertical tangents at  EMBED Equation.DSMT4  if you want a little more practice) HW 10 1. y' = 3cos 6x 2. '(x) = abebx 3. y' = 8sec2(4x 1) 4. y' = 2x2/3 + 6x3 5. y' = 0 6. y' = 2xsin x2 7. y' = 2cos x sin x 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. '(r) = 4r + 2(h 11. '(z) = 2ze 3z  3z2e 3z 12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17a. 12 b.  4.5 c.  12 d. 5e 3 e.  3 f. 11/32 g. 7/9 h. 16 18.  EMBED Equation.DSMT4  19.  EMBED Equation.DSMT4  HW 11 1. (B) 2. 1/13 3. 4/33 4. 1/5 5. 2/5 6. 2 7a.  EMBED Equation.DSMT4  HW 12 1. y' = 12cos (4x) 2. f '(x) = 6e3x 3.  EMBED Equation.DSMT4  4. y' = 2x3 4x2 + 6x 5 5.  EMBED Equation.DSMT4  6. f '(x) = 2sec2 (2x) 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. f '(x) = x(1 + 2ln x) 10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  17a. 4 b. 9/8 c. 2 d. 16 e. 19/2 f. 3/2 18.  EMBED Equation.DSMT4  19.  EMBED Equation.DSMT4  HW 13 1. No. 2. a = 6, b = 3 3. '(0) = 0, "(0) DNE 4a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  Review 1a. y' =  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  2. 32/3 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5a. 4 b. 3 c. (5 d. (20 e. (147 f. (1/2 6. 80 7.  EMBED Equation.DSMT4  8a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. Speed is decreasing b/c v and t have opposite signs when t > 0. 9. (13/6 10. 1/13 11.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  12a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c. m3/radian 13. Horizontal tangent at (2, (2.054); vertical tangents at (/145:;Eh 1 M a   ! % > ? 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