Find the volume of the solid generated by revolving the shaded region about the given axis. Use the disc/washer method 1) About the x-axis

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1 Final eam practice for Math 6 Disclaimer: The actual eam is different Find the volume of the solid generated b revolving the shaded region about the given ais. Use the disc/washer method ) About the -ais ) = - 5 Find the volume of the solid generated b revolving the region about the given line. ) The region in the second quadrant bounded above b the curve = -, below b the -ais, and on the right b the -ais, about the line = ) Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves and lines about the -ais. ) = 6, = 6 ) Find the area enclosed b the given curves. ) =, = ) Solve the initial value problem. 5) d d = (7 - )-, (0) = 5) Find. 6) = 5 sin( + 8) 6) Use implicit differentiation to find d/d. 7) + + = 8 7) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 8) 5 - π cos = 6π, slope at (, π) 8) Find the derivative at each critical point and determine the local etreme values. 9) = 5-9)

2 Find the value or values of c that satisf the equation f(b) - f(a) = f'(c) b - a in the conclusion of the Mean Value Theorem for the given function and interval. 0) f() = + 75, [7, 5]. 0) Using the derivative of f() given below, determine the intervals on which f() is increasing or decreasing. ) f () = ( - )(5 - ) ) Using the derivative of f() given below, determine the critical points of f(). ) f () = ( - )( + ) ) Answer the question. ) f() = - +, 5, -5, , 6 5 d - < 0 0 < < = < < < < 5 ) (, -5) -6 t Does lim (-)+ f() eist? Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () ) 6 6 ) -5 g(), = Use implicit differentiation to find d/d. 5) - = 5) 6) + = 6) Use implicit differentiation to find d/d and d/d. 7) - + = 5 7)

3 At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 8) 55 =, tangent at (, ) 8) Solve the problem. 9) A compan knows that the unit cost C and the unit revenue R from the production and R sale of units are related b C = + 0. Find the rate of change of unit revenue 8,000 when the unit cost is changing b $9/unit and the unit revenue is $000. 9) Solve the problem. Round our answer, if appropriate. 0) A man 6 ft tall walks at a rate of ft/sec awa from a lamppost that is 8 ft high. At what rate is the length of his shadow changing when he is 75 ft awa from the lamppost? (Do not round our answer) 0) Find the linearization L() of f() at = a. ) f() = + +, a = ) ) f() = 9 + 5, a = 0 ) You want a linearization that will replace the function over an interval that includes the point 0. To make our subsequent work as simple as possible, ou want to center the linearization not at 0 but at a nearb integer = a at which the function and its derivative are eas to evaluate. What linearization do ou use? ) f() = +, 0 =. ) Use the linear approimation ( + )k + k, as specified. ) Find an approimation for the function f() = for values of near zero. ) - Find d. 5) = 7 + 5) Use implicit differentiation to find d/d. 6) + - = + 6)

4 Determine from the graph whether the function has an absolute etreme values on the interval [a, b]. 7) 7) A) No absolute etrema. B) Absolute minimum onl. C) Absolute minimum and absolute maimum. D) Absolute maimum onl. Find the absolute etreme values of each function on the interval. 8) = 7-8 on [-, 5] 8) 9) F() = -, 0.5 9) Find the derivative at each critical point and determine the local etreme values. 0) = /( - 9); 0 0) ) = -, < 0 + -, 0 ) Find the value or values of c that satisf the equation f(b) - f(a) = f'(c) b - a in the conclusion of the Mean Value Theorem for the given function and interval. ) f() = + +, [-, ]. ) Find all possible functions with the given derivative. ) = + )

5 Sketch the graph and show all local etrema and inflection points. ) = + ) Find an antiderivative of the given function. 5) 8/ 5) Sketch the graph and show all local etrema and inflection points. 6) = ) Find an antiderivative of the given function. 7) ) Find the most general antiderivative. 8) d 8) 0 Solve the initial value problem. 9) d d = + 5, () = 5 9) 5

6 Use a finite approimation to estimate the area under the graph of the given function on the stated interval as instructed. 0) f() = between = 0 and = using the "midpoint rule" with two rectangles of equal width. 0) Solve the initial value problem. ) dr dt = 0t + sec t, r(-π) = ) Graph the integrand and use areas to evaluate the integral. ) 6 - d ) - Evaluate the integral. ) 0 d ) 0 Find the average value of the function over the given interval. ) f() = + on [-7, 7] ) Find the derivative. 5) d sin t dt 6 - u du 5) 0 6) = 9t cos (t8) dt 6) 0 Solve the initial value problem. 7) d = csc, () = -9 7) d 8) d d = 8(6-7)-, (0) = 8) Use the substitution formula to evaluate the integral. 0 r dr 9) 9) r 50) - d 50) 6

7 Find the area of the shaded region. 5) f() = (, 8) 0 5 (0, 0) (-, -) -5-0 g() = 6 5) 5) 5) - = - - = - 5) = sec 5) = cos π π 7

8 Find the volume of the solid generated b revolving the shaded region about the given ais. 5) About the -ais 5) = ) About the -ais 55) = sec π π 56) About the -ais 56) 6 5 = 5 6 Find the volume of the solid generated b revolving the region about the given line. Use the disc/washer method 57) The region in the second quadrant bounded above b the curve = 6 -, below b the -ais, and on the right b the -ais, about the line = 57) Find the volume of the solid generated b revolving the region about the given line. 58) The region in the first quadrant bounded above b the line =, below b -ais, and on the right b the line =, about the line = - 58) 8

9 59) The region in the first quadrant bounded above b the line =, below b the curve =, and on the left b the -ais, about the line = - 59) 60) The region in the second quadrant bounded above b the curve = 9 -, below b the -ais, and on the right b the -ais, about the line = 60) Find the volume of the solid generated b revolving the shaded region about the given ais. 6) About the -ais 6) = sin π Find the volume of the solid generated b revolving the region about the -ais. 6) The region enclosed b = /, = 0, = 8 6) Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves and lines about the -ais. 6) = 5, = - 5, = 6) 6) =, = 0, =, = 6) Use the shell method to find the volume of the solid generated b revolving the shaded region about the indicated line. 65) About the line = 7 65) 6 5 =

10 Find the limit, if it eists. 66) lim h 0 h+ + 66) Find the limit. 67) lim 0 tan 5 67) 68) lim 0 sin 8 68) Use the shell method to find the volume of the solid generated b revolving the shaded region about the indicated line. 69) About the line = 69) = 6 (solid) = 6 (dashed) 5 6 Provide an appropriate response. 70) It can be shown that the inequalit - cos 70) holds for all values of 0. Find lim cos 0 if it eists. Find numbers a and b, or k, so that f is continuous at ever point. 7) -5, < - f() = a + b, - -, > - 7) Find the derivative of the function. 7) = ) Find the derivative. 7) = sec 7) 0

11 Solve the problem. 7) Find the tangent to = cos at = π. 7) 75) Does the graph of the function = + sin have an horizontal tangents in the interval 0 π? If so, where? 75) 76) Find all points on the curve = sin, 0 π, where the tangent line is parallel to the line =. 76)

12 Answer Ke Testname: 6FINALP ) 56 5 π ) 56 π ) 9 5 π ) 6 5) = - (7 - ) ) - 80 sin( + 8) 7) ) -π 9) Critical Pt. derivative Etremum Value 0 min 0 =0 undefined min 0 = 5 = 0 0 local ma ) 5 7 ) Decreasing on (, 5); increasing on (-, ) (5, ) ) -, ) Yes ) 8 5) 6) - - ( + ) 7) d d = - + ; d d = - ( + ) 8) = - + 9) $07.00/unit 0) ft/sec ) L() = 0-5 ) L() = 5 ) ) f() + + 5) 7 + d

13 Answer Ke Testname: 6FINALP 6) ( - ) + - ( - ) 7) A 8) Maimum = (0, 7); minimum = (5, -9) 9) Maimum =, - ; minimum =, - 0) Critical Pt. derivative Etremum Value = 0 Undefined local ma 0 =.5 0 minimum ) Critical Pt. derivative Etremum Value = 0 undefined local min = 0 local ma 9 ) ) + + C ) Local minimum: (-,-) Local maimum: (,) Inflection point: (0,0), (-, - ),(, ) ) /

14 Answer Ke Testname: 6FINALP 6) Local ma: -,-, min: -,-5 Inflection point: -, ) / 8) C 9) = ).5 ) r = 5t + tan t + - 5π ) 8π ) 0 ) cos t 5) 6 - sin t 6) 9 cos ( ) 7) = csc t dt - 9 8) = - (6-7) ) ) 5 5) 97 5) 7 5 5) - 5) π 55) 9π

15 Answer Ke Testname: 6FINALP 56) 8 5 π 57) 60 π 58) 0 7 π 59) 6 5 π 60) 5 π 6) 8π - 6π 6) 96 5 π 6) 5 5 π 6) 56 π 65) 75 π 66) / 67) 5 68) 8 69) 0 7 π 70) 0 7) a = 6, b = 7) = (7 - ) 7) = sec tan 7) = - + π 75) No 76) π,, 5π,- 5