ࡱ> QS POx[ 4bjbj .ΐΐ,T8]xY y!y!y!y!###\\\\\\\$^Ra\#####\y!y!g\2&2&2&#y!y!\2&#\2&2&V+\y!z$X4\]08]Xpa&.ah+\a+\x##2&#####\\2&###8]####a######### : MAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative Remember: Equation of the tangent line through the point  EMBED Equation.3  having slope m is  EMBED Equation.3 . Find the derivative  EMBED Equation.DSMT4   EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  4. Find the equation of the tangent line to  EMBED Equation.DSMT4  at the point x = 0. 5. Find the equation of the tangent line to  EMBED Equation.DSMT4  at the point (1, 1). Section 3.8 Derivative of Logarithmic and Exponential Functions Remember:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Note:  EMBED Equation.DSMT4 is a constant Practice problems 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.3  Section 3.9 Derivative of Inverse Trigonometric Functions Remember 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  Practice problems:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 , simplify the answer. Given  EMBED Equation.DSMT4 , find  EMBED Equation.DSMT4  The function EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 has an inverse function g(x). Explain why. And find  EMBED Equation.DSMT4  The function  EMBED Equation.DSMT4  has an inverse function g(x). Explain why. Find EMBED Equation.DSMT4 . Find equation of the tangent line to y = g(x) at (7, 1). Find  EMBED Equation.3 , the derivative of the inverse function of g if  EMBED Equation.3 . Section 3.10 Related Rates Practice problems: A puddle is evaporating in such a way that its diameter is decreasing at rate of 0.1 cm/min. At what rate is the area of the puddle decreasing when the diameter is 8 cm. 2. The volume of a cube is increasing at a rate of 10 cm3/min. How fast is the surface area increasing when the length of the edge is 30 cm. 3. Two cars start moving from the same point. One travels South at 60mi/h and the other travels West at 25mi/h. At what rate is the distance between the two cars increasing two hours later? 4. A car is traveling at 50 mph due south at a point 1/2 mile north of an intersection. A police car is traveling at 40 mph due west at a point 1/4 mile east of the same intersection. At that instant, the radar in the police car measurers the rate at which the distance between the cars is changing. What does the radar gun register? 5. A 10-foot ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at the rate of 2 ft/s, how fast is the bottom of the ladder sliding away from the wall when the top of the ladder is 8 ft off the ground. 6. An oil tanker has an accident and oil pours at the rate of 150 gallons per minute. Suppose that the oil spreads onto the water in a circle at a thickness of  EMBED Equation.DSMT4 . Given that  EMBED Equation.DSMT4 gallons. Determine the rate at which the radius of the spill is increasing when the radius reaches 500 ft. Solution: The shape is like a cylinder. We consider the volume equation  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  Also given that  EMBED Equation.DSMT4  Thus  EMBED Equation.DSMT4  7. Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? Solution: We are given  EMBED Equation.3 , where  EMBED Equation.3 . Then we get  EMBED Equation.3 . 8. A particle moves along the curve EMBED Equation.3 . As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instance? Solution: Given  EMBED Equation.3 ,  EMBED Equation.3  Find  EMBED Equation.3  . We write  EMBED Equation.3 . By differentiation we get EMBED Equation.3 . Using the given values we get  EMBED Equation.3 . 4.1-4.4 Application/Optimization Find the open interval where is the function  EMBED Equation.3  concave up, concave down? Find the local extrema of the function  EMBED Equation.3  Find all critical numbers of  EMBED Equation.DSMT4  and also maximum and minimum values if any Graph the function  EMBED Equation.DSMT4 showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing. Find the open interval where the function  EMBED Equation.DSMT4  is increasing or decreasing. Find also local maximum and local minimum if any. A farmer has 2500 ft of fencing and wants to fence of a rectangular field that borders a straight river. He needs not to fence along the river. What are the dimensions of the field that has the largest area? Find the radius of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches. An 8 feet tall fence runs parallel to the side of a house 3 feet away. What is the length of the shortest ladder that clears the fence and reaches the house? Assume that the vertical wall of the house and the horizontal ground have infinite extent. Answer: Length approximately 15 ft. The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue? A right circular cylinder is inscribed in a sphere of radius r. To find the largest possible surface area of such a cylinder. A 8000 cubic foot tank with a square base and an open top is to be constructed of a sheet of steel of a given thickness. Find the length of a side of the square base of the tank with minimum surface area. Find the x-coordinate(s) for the points on the graph of  EMBED Equation.DSMT4 that are closest to the point (0, 5). Find the maximum area of the circle inscribed in an equilateral triangle whose sides have length 2 meters. Round your answer correct to two decimal places if needed. Section 4.5 Linear Approximation We have the linear approximation  EMBED Equation.DSMT4  Practice problems: Find the linear approximation of  EMBED Equation.DSMT4  at  EMBED Equation.DSMT4  and thus approximate the values of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Evaluate  EMBED Equation.DSMT4  using Linear Approximation Prove that EMBED Equation.3  for x close to 0, and illustrate this approximation by drawing the graphs of  EMBED Equation.3  and  EMBED Equation.3  on the same screen. We consider  EMBED Equation.3  Now using linear approximation formula we find  EMBED Equation.3  Section 4.6 Mean Value Theorem First of all we discuss the special case of Mean Value Theorem. The Rolles Theorem: Suppose that f is continuous on the interval  EMBED Equation.DSMT4 , differentiable on  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 then there is a number  EMBED Equation.DSMT4 such that  EMBED Equation.DSMT4  The Mean Value Theorem: Suppose that f is continuous on the interval  EMBED Equation.DSMT4 , differentiable on  EMBED Equation.DSMT4 then there is a number  EMBED Equation.DSMT4 such that  EMBED Equation.DSMT4  Constant Theorem: Suppose that  EMBED Equation.DSMT4 for all x on an open interval I, then f must be a constant, that is  EMBED Equation.DSMT4 , a constant Corollary: Suppose that  EMBED Equation.DSMT4 for all x in some open interval I, then for some constant c,  EMBED Equation.DSMT4  Determine whether Rolles Theorem applies to the function  EMBED Equation.3 on [0, 10]. If so, find the points that are guaranteed to exist by Rolles Thereom. Determine whether Rolles Theorem applies to the function  EMBED Equation.3 on [0, 10]. If so, find the points that are guaranteed to exist by Rolles Thereom. Determine whether Mean Value Theorem applies to the function  EMBED Equation.3 on [0, 9]. Section 4.7 LHopitals Rule We have noticed the following indeterminate forms  EMBED Equation.DSMT4 . The LHopital rule is applicable for  EMBED Equation.DSMT4  if this limit has the form  EMBED Equation.DSMT4 . The LHopital rule is as follows  EMBED Equation.DSMT4 . One may apply the rule repeatedly as long as the indeterminate form appears. Examples with the above indeterminate forms:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , does not exist (DNE)  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , DNE as the result is not a finite number  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  DNE Section 4.8 Integration 1. Find f(x) if  QUOTE   2. Find f(x) if  QUOTE   3. Find g(x) when  EMBED Equation.DSMT4  4. Find f(x), given  EMBED Equation.DSMT4  Find the following indefinite integral      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  Answer: Section 3.7: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  Section 3.8: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. 2 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  Section 3.9: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. 1/3 9.  EMBED Equation.DSMT4  10. 1/76 Section 3.10: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. 65 mph Section 4.1-4.4: 1. Up on  EMBED Equation.DSMT4  2. Max ( 0.8, 1.03) 3. Max (-2, 30.24); min(2, -30.24) 4. 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