MATH1910Chapter2TestReview

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1 Class: Date: MATH1910Chapter2TestReview Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the slope m of the line tangent to the graph of the function f( x) = 2 7x at the point 1,9. m = 7 m = 2 m = 2 m = 7 m = 9 2. Describe the x-values at which the graph of the function f( x) = x2 x 2 9 given below is differentiabl f( x) is differentiable at x = ±3. f( x) is differentiable everywhere except at x = ±3. f( x) is differentiable everywhere except at x = 0. f( x) is differentiable on the interval 2,2. f( x) is differentiable on the interval 2,. 1

2 3. Find the derivative of the function. f( x) = x 4 f ( x) = 4x 4 f ( x) = 3x 3 f ( x) = 4x f ( x) = 3x f ( x) = 4x 3 4. Find the derivative of the function. f( x) = 1 x 8 f ( x) = 9 x 9 f ( x) = 8 x 7 f ( x) = 8 x 9 f ( x) = 8 x 9 f ( x) = 7 x 9. Use the rules of differentiation to find the derivative of the function y = 7 + 6sinx. y = 7 + 6cos x y = 7 + cos 6x y = 6cos x y = 7cos 6x y = cos 6x 2

3 6. Find the slope of the graph of the function at the given valu f( x) = x when x = 9 3 f ( 9) = 2187 f ( 9) = 729 f ( 9) = 27 f ( 9) = 27 f ( 9) = Find the derivative of the function f( x) = x 9. x 4 f ( x) = x f ( x) = 1 36 x f ( x) = x f ( x) = 1 9 x f ( x) = x 8. Find the derivative of the function f( x) = 2 + 3cos x. 3 x f ( x) = 2 3x 4 3 f ( x) = 2 3x 4 3 f ( x) = 2 3x 4 3 f ( x) = 2 3x 3 4 f ( x) = 2 3x 3 4 3sinx 3sinx + 3sinx 3sinx + 3sinx 3

4 9. Suppose the position function for a free-falling object on a certain planet is given by s( t) = 14t 2 + v 0 t + s 0. A silver coin is dropped from the top of a building that is 1370 feet tall. Find velocity of the coin at impact. Round your answer to the three decimal places ft/sec ft/sec ft/sec ft/sec ft/sec 10. Find the derivative of the algebraic function H( v) = v 3 v H ( s) = 8v 7 + 1v 4 + 9v 2 H s ( ) = 8v 7 + 9v 4 + 1v 2 H ( s) = 8v 7 1v 4 9v 2 H ( s) = 8v 7 + 1v 4 9v 2 H ( s) = 8v 7 + 9v 4 3v Use the Product Rule to differentiate f u f ( u) = 6u f ( u) = 6u f ( u) = 6u f ( u) = 6u f ( u) = 6u u 7 2 u 13 2 u 6 2 u 13 2 u 6 2 u u 6 2 u u 6 2 u ( ) = u u 6. 4

5 12. Use the Product Rule to differentiat f() t = t 3 cos t f( t) = t 3 sint + 3t 4 cos t f ( t) = 3t 2 cos t t 3 sint f ( t) = t 3 sint 3t 4 cos t f ( t) = t 3 sint 3t 4 cos t f ( t) = t 3 sint + 3t 4 sint

6 13. Use the Quotient Rule to differentiate the function f( x) = 8x. x + 3 f ( x) = f ( x) = x f ( x) = f ( x) = x x x x f ( x) = x x x x x

7 14. Use the Quotient Rule to differentiate the function f ( x) = 4+ x. x f ( x) = f ( x) = f ( x) = f x 9 + 8x x 2 x x x 2 x x x 2 ( ) = f ( x) = x x x 2 x x + x 2 x Find the derivative of the function f() t = 1t 3 + 6sec(). t f () t = 4t 2 + 6sec() t tan() t f () t = 3t 2 + 6sec 2 () t f () t = 4t 2 + 6tan() t f () t = 3t 2 + 6sec() t tan() t f () t = 4t 2 6sec() t tan() t 7

8 16. Find the derivative of the function. f() t = 2t 3 sint + t 6 cos t f () t = 6t 6 t 2 sint + 2t t cos t f () t = 6t 2 t 6 sint + 2t t cos t f () t = 6t 2 t 6 sint + 2t t cos t f () t = 2t 3 30t sint + 6t 2 t 6 cos t f () t = 6t 2 t 6 sint 2t t cos t 17. Find an equation of the tangent line to the graph of f at the given point. f( t) = ( t ) t 2 3, at 2, 3 y = 11t + 19 y = t y = 11t + 19 y = 2t 11 y = 11t The length of a rectangle is 4t + 6 and its height is t 4, where t is time in seconds and the dimensions are in inches. Find the rate of change of the area, A, with respect to tim da da da da da = t 4 ( t) square inches/second = t 3 ( t) square inches/second = t 3 ( t) square inches/second = t 4 ( t) square inches/second = t 3 ( 20t + 6) square inches/second 8

9 19. The ordering and transportation cost C for the components used in manufacturing a product is C = x + 1 where C is measured in thousands of dollars and x is the order size in x 2 x + 40,x hundreds. Find the rate of change of C with respect to x for x = 24. Round your answer to two decimal places thousand dollars per hundred 8.04 thousand dollars per hundred 3.28 thousand dollars per hundred 4.92 thousand dollars per hundred 7.96 thousand dollars per hundred 20. When satellites observe Earth, they can scan only part of Earth's surfac Some satellites have sensors that can measure the angle θ shown in the figur Let h represent the satellite's distance from Earth's surface and let r represent Earth's radius. Find the rate at which h is changing with respect to θ when θ = 60 (Assume r = 4460 miles.) Round your answer to the nearest unit mi/radian 10 mi/radian 10 mi/radian 8920 mi/radian 2973 mi/radian 9

10 21. Find the second derivative of the function f( x) = 8x 9. f ( x) = x 9 f ( x) = x 9 13 f ( x) = x 9 f ( x) = x 9 f ( x) = 8x Find the second derivative of the function f( x) = 3x2 + x 4 x f ( s) = 8 x 3 f ( s) = 8 x 3 f ( s) = x + 8 x 3 f ( s) = 4 x 3 f ( s) = 8 x Find the derivative of the algebraic function f( x) = x f ( x) = x x f ( x) = 6x x f ( x) = 30x x f ( x) = 30x 7 x f ( x) = 30x 6 x

11 24. Find the derivative of the function. f( t) = 4 + 7t f ( t) = 280t t 4 f ( t) = 280t t f ( t) = 7t t 7 f ( t) = 280t t f ( t) = 280t t 2. Find the derivative of the function f( t) = ( 1 + 8t) 9 f ( t) = 40 9 ( 1 + 8t) 4 9 f ( t) = 8 9 ( 1 + 8t) 4 9 f ( t) = 1 9 ( 1 + 8t) 4 9 f ( t) = 40 9 ( 1 + 8t) 4 f ( t) = 8( 1 + 8t)

12 26. Find the derivative of the function. f( x) = x 8 3x f ( x) = x7 ( 1x) 2 3x f ( x) = x7 ( 80 3x) 2 3x f ( x) = x7 ( x) 2 3x f ( x) = x7 ( + 3x) 2 3x f ( x) = x7 ( 80 1x) 2 3x 27. Find the derivative of the function. g( x) = x + 6 x g ( x) = g ( x) = g ( x) = g ( x) = g ( x) = x + x x ( 6 + x) 7 + x x x x 2 ( 6 + x) x x x 2 ( 6 + x) x x x 2 ( 6 + x) 7 + x x x 2 ( 6 + x) x

13 28. Find the derivative of the function y = 8cos 4x. y = 4sin 4x y = 8sin 4x y = 32sin 4x y = 32cos 4x y = 32cos 4x 29. Find the derivative of the function. y = cos 2x 4 6 y = 8x 4 cos 2x 4 6 y = 8sin 2x 4 6 y = 8x 3 sin 2x 4 6 y = 8sin 2x 4 6 y = 2sin 2x Find the derivative of the function. f( θ) = 7 sin2 2θ f 28sin2θ cos 2θ ( θ) = f ( θ) = f ( θ) = f ( θ) = 28cos 2θ 28sin2θ cos 2θ 7sin2θ cos 2θ f θ ( ) = 28sin2θ 13

14 31. Find the derivative of the function. y = 3 sec2 x y = 6 sec2 x tanx y = 6 sec2 x tan 2 x y = 6 sec x tanx y = 6 sec2 x tanx y = 3 sec2 x tanx 32. Find the derivative of the function. f( t) = sec 2 ( 7πt ) f ( t) = 3π sec 2 ( 7πt ) tan( 7πt ) f ( t) = 70sec 2 ( 7πt ) tan( 7πt ) f ( t) = 70π sec 2 ( 7πt ) tan( 7πt ) f ( t) = 70π sec 2 ( 7πt ) tan( 7πt) f ( t) = 7π sec 2 ( 7πt ) tan( 7πt ) 33. Evaluate the derivative of the function f( t) = 6t + 3 t 1 11 at the point, 8. f ( t) = 7 8 f ( t) = f ( t) = 7 8 f ( t) = f ( t) =

15 34. Evaluate the derivative of the function f( t) = 8t2 + 2 t 1 f ( t) = f ( t) = f ( t) = f ( t) = f ( t) = Find the second derivative of the function. f( x) = 3x f = 63x ( 7 + 3x) x 3 f = 63x 7 + 3x x 3 f = 63x 7 + 3x x 3 f = 63x 7 + 3x x 3 f = 63x 7 + 3x x at the point 4, The displacement from equilibrium of an object in harmonic motion on the end of a spring is y = 1 4 cos 8t 1 sin1t where y is measured in feet and t is the time in seconds. Determine the position 3 of the object when t = π / 12. Round your answer to two decimal places feet 0.21 feet 17.3 feet 0.46 feet 0.11 feet 1

16 37. A buoy oscillates in simple harmonic motion y = Acos ωt as waves move past it. The buoy moves a total of 3. feet (vertically) between its low point and its high point. It returns to its high point every 14 seconds. Write an equation describing the motion of the buoy if it is at its high point at t = 0. y = 3.cos π 7 t y = 1.7cos 7πt y = 1.7cos π 14 t y = 1.7cos π 7 t y = 3.cos 14πt 38. Find by implicit differentiation. x 2 + y 2 = 2 = x y = x y = y x = y x = x y 2 16

17 39. Find by implicit differentiation. x 2 + x + 9xy y 2 = 4 = 2x + 9y 2y 9x = 2x + + 9y 2x 9y = x + + 9y y 9x = 2x + + 9y 2y 9x = 2x + 9y 2y 9x 40. Find by implicit differentiation given that 2xy = 9. = 9y x = 9xy = xy = 9xy = y x 17

18 41. Find by implicit differentiation. x 4 + 7x + 6xy y 7 = 9 = y 4x3 7y 6 6x = 4x y 7y 6 6x = 4x y 6y 6 6x = 4x y 7y 6 6x = 3x y 7y 6 6x 42. Find by implicit differentiation. x 3 + 8x + x 13 y y 6 = 4 = 3x x 12 y 6y x 13 = 3x y y 13x = 3x x 12 y y x 13 = 3x x 13 y 6y x 13 = 3x x 12 y 6y x 13 18

19 43. Find by implicit differentiation. sinx + 7cos 14y = 2 = = = = cos x 98cos 14y cos x 98 sin14y cos x 14 sin14y cos x 98siny = cos x 98 sin14y 44. Evaluate places. 4 4 for the equation x + y = 27 at the given point 1024,1. Round your answer to two decimal = = 4.00 = 0.2 = 0.2 = Find an equation of the tangent line to the graph of the function y 6 2 = ( x ) at the point 8.20,2.00.The coefficients below are given to two decimal places. y = 0.63x y = 3.38x y = 3.38x 2.68 y = 0.63x y = 0.63x

20 46. Use implicit differentiation to find an equation of the tangent line to the ellipse x2 2 + y2 98 y = 11x + 11 y = 2x + 9 y = 8x + 9 y = 9x + 11 y = 4x + 9 = 1 at 1, Find d 2 y d 2 x in terms of x and y given that x2 + 6y 2 = 9. Use the original equation to simplify your answer. y = 1 4y 3 y = 1y 3 y = 4y 3 y = 24y 3 y = 1 24y Find d 2 y in terms of x and y. 2 x 2 + y 2 = 6 d 2 y 2 = 2 x y d 2 y 2 = 6 y 2 d 2 y 2 = d 2 y 2 = x 2 y + y y 2 y x 2 d 2 y 2 = x 2 y + y y 2 20

21 49. Assume that x and y are both differentiable functions of t. Find when x = 49and y = x. = 17 for the equation = = 14 = = = Assume that x and y are both differentiable functions of t. Find xy = 132. when x = 11 and = 8 for the equation = 106 = 88 = 22 3 = 22 3 = A point is moving along the graph of the function y = when x = x such that = 2 centimeters per secon Find = 9 = = = =

22 2. The radius, r, of a circle is decreasing at a rate of centimeters per minut Find the rate of change of area, A, when the radius is 6. da da da da da = 360π sq cm/min = 360π sq cm/min = 60π sq cm/min = 60π sq cm/min = 30π sq cm/min 3. The radius r of a sphere is increasing at a rate of 6 inches per minut Find the rate of change of the volume when r = 11 inches. dv dv dv dv dv = 3630π in 3 / min = 142π in 3 / min = 2904π in 3 / min 1 = 142π in3 / min 1 = 2904π in3 / min 4. A spherical balloon is inflated with gas at the rate of 300 cubic centimeters per minut How fast is the radius of the balloon increasing at the instant the radius is 70 centimeters? dr = 3 98π cm/min dr = 1 98π cm/min dr 3 = 196π cm/min dr = 98π cm/min dr = 196π cm/min 22

23 . A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. If water is flowing into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep. 9 40π ft/min 9 100π ft/min 81 20π ft/min 81 0π ft/min π ft/min 6. A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per secon How fast is the top of the ladder moving down the wall when its base is 13 feet from the wall? Round your answer to two decimal places..00 ft/sec.33 ft/sec 1.71 ft/sec.33 ft/sec 6.00 ft/sec 23

24 7. A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per secon Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 19 feet from the wall. Round your answer to three decimal places rad/sec rad/sec rad/sec rad/sec rad/sec 24

25 8. A man 6 feet tall walks at a rate of 2 ft per second away from a light that is 16 ft above the ground (see figure). When he is 8 ft from the base of the light, find the ratat which the tip of his shadow is moving. 8 ft per minute 4 ft per minute 64 ft per minute 32 ft per minute 16 ft per minute 9. An airplane is flying in still air with an airspeed of 2 miles per hour. If it is climbing at an angle of 21, find the rate at which it is gaining altitud Round your answer to four decimal places mi/hr mi/hr mi/hr mi/hr mi/hr 2

26 MATH1910Chapter2TestReview Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 DIF: Easy REF: Section 2.1 OBJ: Calculate the slope of a line tangent to the graph of a function at a specified point 2. ANS: B PTS: 1 DIF: Medium REF: Section 2.1 OBJ: Identify the x-value (or values) at which a function is differential 3. ANS: E PTS: 1 DIF: Easy REF: Section 2.2 OBJ: Differentiate a function using basic differentiation rules 4. ANS: D PTS: 1 DIF: Medium REF: Section 2.2 OBJ: Differentiate a function using basic differentiation rules. ANS: C PTS: 1 DIF: Easy REF: Section 2.2 OBJ: Differentiate trigonometric functions 6. ANS: E PTS: 1 DIF: Easy REF: Section 2.2 OBJ: Calculate the slope of the graph of a function at a given point 7. ANS: A PTS: 1 DIF: Medium REF: Section 2.2 OBJ: Differentiate a function using basic differentiation rules 8. ANS: B PTS: 1 DIF: Medium REF: Section 2.2 OBJ: Differentiate trigonometric functions 9. ANS: D PTS: 1 DIF: Medium REF: Section 2.2 OBJ: Calculate the velocity for an object falling according to a given position function MSC: Application 10. ANS: D PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Differentiate a function using the product rule 11. ANS: D PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Differentiate a function using the product rule 12. ANS: D PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Differentiate a function using the product rule 13. ANS: A PTS: 1 DIF: Difficult REF: Section 2.3 OBJ: Differentiate a function using the quotient rule 14. ANS: B PTS: 1 DIF: Difficult REF: Section 2.3 OBJ: Differentiate a function using the quotient rule 1. ANS: A PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Differentiate a function using the product rule 16. ANS: B PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Differentiate a function using the product rule 17. ANS: C PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Write an equation of a line tangent to the graph of a function at a specified point 18. ANS: C PTS: 1 DIF: Difficult REF: Section 2.3 OBJ: Interpret a derivative as a rate of change MSC: Application 1

27 19. ANS: D PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Interpret a derivative as a rate of change MSC: Application 20. ANS: A PTS: 1 DIF: Difficult REF: Section 2.3 OBJ: Create a function in application and interpret its derivative as a rate of change MSC: Application 21. ANS: C PTS: 1 DIF: Difficult REF: Section 2.3 OBJ: Calculate the second derivative of a function 22. ANS: A PTS: 1 DIF: Medium REF: Section 2.3 OBJ: Calculate the second derivative of a function NOT: Section ANS: C PTS: 1 DIF: Difficult REF: Section 2.4 OBJ: Differentiate a function using the chain rule 24. ANS: B PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a function using the chain rule 2. ANS: A PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a function using the chain rule 26. ANS: E PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a function using the chain rule and product rule 27. ANS: E PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a function using the chain rule and quotient rule 28. ANS: C PTS: 1 DIF: Easy REF: Section 2.4 OBJ: Differentiate a function using the chain rule 29. ANS: C PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a trigonometric function using the chain rule 30. ANS: C PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a trigonometric function using the chain rule 31. ANS: D PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a trigonometric function using the chain rule 32. ANS: C PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Differentiate a trigonometric function using the chain rule 33. ANS: E PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Evaluate the derivative of a function at a point 34. ANS: E PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Evaluate the derivative of a function at a point 3. ANS: B PTS: 1 DIF: Difficult REF: Section 2.4 OBJ: Calculate the second derivative of a function using the chain rule 36. ANS: E PTS: 1 DIF: Easy REF: Section 2.4 OBJ: Evaluate functions in applications MSC: Application 37. ANS: D PTS: 1 DIF: Medium REF: Section 2.4 OBJ: Write an equation describing simple harmonic motion MSC: Application 2

28 38. ANS: B PTS: 1 DIF: Easy REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 39. ANS: D PTS: 1 DIF: Easy REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 40. ANS: E PTS: 1 DIF: Easy REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 41. ANS: D PTS: 1 DIF: Medium REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 42. ANS: A PTS: 1 DIF: Medium REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 43. ANS: B PTS: 1 DIF: Medium REF: Section 2. OBJ: Differentiate an equation using implicit differentiation 44. ANS: C PTS: 1 DIF: Easy REF: Section 2. OBJ: Evaluate the derivative of an implicit function at a given point 4. ANS: A PTS: 1 DIF: Medium REF: Section 2. OBJ: Write an equation of a line tangent to the graph of an implicit function at a specified point 46. ANS: B PTS: 1 DIF: Easy REF: Section 2. OBJ: Write an equation of a line tangent to the graph of an ellipse at a specified point. 47. ANS: A PTS: 1 DIF: Medium REF: Section 2. OBJ: Calculate the second derivative implicitly 48. ANS: E PTS: 1 DIF: Easy REF: Section 2. OBJ: Calculate the second derivative implicitly 49. ANS: A PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Calculate the value of an implicit derivative from given information 0. ANS: D PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Calculate the value of an implicit derivative from given information 1. ANS: E PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Solve a related rate problem involving a point moving along a curve 2. ANS: C PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Solve a related rate problem involving the area of a circle and its radius MSC: Application 3. ANS: C PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Solve a related rate problem involving the volume of a sphere and its MSC: Application 4. ANS: C PTS: 1 DIF: Easy REF: Section 2.6 OBJ: Solve a related rate problem involving the volume of a sphere and its radius MSC: Application. ANS: D PTS: 1 DIF: Difficult REF: Section 2.6 OBJ: Solve a related rate problem involving a cone MSC: Application 6. ANS: C PTS: 1 DIF: Medium REF: Section 2.6 OBJ: Solve a related rate problem involving a moving ladder MSC: Application 3

29 7. ANS: B PTS: 1 DIF: Medium REF: Section 2.6 OBJ: Solve a related rate problem involving a moving ladder and its internal angle MSC: Application 8. ANS: E PTS: 1 DIF: Difficult REF: Section 2.6 OBJ: Solve a related rate problem involving a man walking away from a light source MSC: Application 9. ANS: E PTS: 1 DIF: Medium REF: Section 2.6 OBJ: Solve a related rate problem involving the altitude of an airplane MSC: Application 4

m2413c2 the limiting process. 4. Use the alternative form of the derivative to find the derivative of the function at. a. b. c. d. e.

m2413c2 the limiting process. 4. Use the alternative form of the derivative to find the derivative of the function at. a. b. c. d. e. m2413c2 Multiple Choice Identify the choice that best completes the statement or answers the question 1 Find the derivative of the following function using the limiting process 2 Find the derivative of