Test # 2 Review. function y sin6x such that dx. per second. Find dy. f(x) 3x 2 6x 8 using the limiting process. dt = 2 centimeters. dt when x 7.
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1 Name: Class: Date: ID: A Test # 2 Review Short Answer 1. Find the slope m of the line tangent to the graph of the function g( x) 9 x 2 at the point 4, 7ˆ. 2. 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 rate.at which the tip of his shadow is moving. 3. Find the derivative of the following function f(x) 3x 2 6x 8 using the limiting process. 4. Find the derivative of the function fx ( ) 7x 3 using the limiting process. 8. A point is moving along the graph of the function y sin6x such that dt = 2 centimeters per second. Find dt when x Find an equation of the tangent line to the graph of the function fx ( ) x 2 5x 2 at the point 5,2ˆ. 6. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. yx ( ) x 4 4x 4 7. All edges of a cube are expanding at a rate of 9 centimeters per second. How fast is the volume changing when each edge is 2 centimeters? 1
2 Name: ID: A 9. A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure). When he is 13 feet from the base of the light, at what rate is the tip of his shadow moving? 14. Assume that x and y are both differentiable functions of t. Find when x =49and dt dt 17 for the equation y x. 15. Use implicit differentiation to find an equation of the tangent line to the ellipse x2 2 y at 1,7 ˆ. 16. Find an equation of the tangent line to the graph of the function given below at the given point. 7x 2 2xy 4y , 2, 1ˆ (The coefficients below are given to two decimal places.) 10. A point is moving along the graph of the function 1 y 9x 2 such that 2 centimeters per 4 dt second. Find when x = 2. dt 11. Assume that x and y are both differentiable functions of t. Find when x =11 and dt dt = 8 for the equation xy Find the points at which the graph of the equation has a vertical or horizontal tangent line. 5x 2 4y 2 10x 24y Find d 2 y in terms of x and y Evaluate for the equation 7xy 21 at the given point 3, 1ˆ. Round your answer to two decimal places. 18. Find by implicit differentiation given that tan 4x yˆ 4x. Use the original equation to simplify your answer. 19. Find by implicit differentiation. sinx 7cos14y Find by implicit differentiation. x 2 5x 9xy y 2 4 x 2 y 2 6 2
3 Name: ID: A 21. Find 2xy 9. by implicit differentiation given that 30. Find the second derivative of the function fx ( ) 3x2 5x 4 x. 22. Find by implicit differentiation. x 4 7x 6xy y Find by implicit differentiation. x 2 y Find the derivative of the function. y cos 2x 4 ˆ Evaluate the derivative of the function at the given point. ft () 7 t 1, 5, 7 ˆ Find the derivative of the function. f( ) 7 5 sin Find the derivative of the algebraic function fx ( ) x 6 ˆ Find an equation of the tangent line to the graph of f at the given point. ft () ( t 5) t 2 ˆ 3,at 2, 3 ˆ 29. Find the derivative of the function. fx ( ) x 8 5 3x 31. Use the Quotient Rule to differentiate the function f() x sinx. x Find the derivative of the function. f() s 9s sins 5coss. 33. Find the derivative of the function Simplify your answer. sinx 2x cosx. 34. Find the derivative of the algebraic function 4 ˆ f() x x3 x Use the quotient rule to differentiate the following 2s function f() s s 5 and evaluate f ( 2) Use the Quotient Rule to differentiate the function f () x 4 x. x Use the Product Rule to differentiate f() s s 5 coss. 38. Find the derivative of the algebraic function Ht () t 6 ˆ 5 t 5 6 ˆ. 39. The volume of a cube with sides of length s is given by V s 3. Find the rate of change of volume with respect to s when s 6 centimeters. 3
4 Name: ID: A 40. A ball is thrown straight down from the top of a 300-ft building with an initial velocity of 12 ft per second. The position function is st () 16t 2 v 0 t s 0. What is the velocity of the ball after 4 seconds? 50. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. yx ( ) x 3 12x Suppose the position function for a free-falling object on a certain planet is given by st () 13t 3 v 0 t s 0. A silver coin is dropped from the top of a building that is 1370 feet tall. Determine the velocity function for the coin. 42. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. yx ( ) 9 x Suppose the position function for a free-falling object on a certain planet is given by st () 12t 2 v 0 t s 0. A silver coin is dropped from the top of a building that is 1372 feet tall. Find the instantaneous velocity of the coin when t Find the slope of the graph of the function fx ( ) x 2x 3 ˆ 7 at x Find the derivative of the function fx ( ) 4x 2 4cos( x). 46. Find an equation of the line that is tangent to the graph of the function f(x) 8x 2 and parallel to the line 16x y Find the derivative of the function fx ( ) 7x 3 4x Find the slope of the graph of the function fx ( ) ( 4x 7) 2 at x Find the derivative of the function fx ( ) 2 3cosx. 3 x 4
5 Test # 2 Review Answer Section SHORT ANSWER 1. ANS: m 8 PTS: 1 DIF: Medium REF: OBJ: Calculate the slope of a line tangent to the graph of a function at a specified point NOT: Section ANS: 16 ft per minute 5 PTS: 1 DIF: Difficult REF: OBJ: Solve a related rate problem involving a man walking away from a light source MSC: Application NOT: Section ANS: f ( x) 6x 6 PTS: 1 DIF: Easy REF: OBJ: Calculate the derivative of a function by the limit process NOT: Section ANS: f ( x) 7 2 7x 3 PTS: 1 DIF: Medium REF: OBJ: Calculate the derivative of a function by the limit process NOT: Section ANS: y 5x 23 PTS: 1 DIF: Medium REF: a OBJ: Write an equation of a line tangent to the graph of a function at a specified point NOT: Section ANS: x 1 PTS: 1 DIF: Difficult REF: OBJ: Calculate the values for which the slope of a function is zero NOT: Section 2.2 1
6 7. ANS: 108 cm 3 /sec PTS: 1 DIF: Easy REF: OBJ: Solve a related rate problem involving the volume of a cube and the length of a side MSC: Application NOT: Section ANS: dt 12cos 6 7 PTS: 1 DIF: Easy REF: 2.6.8a OBJ: Solve a related rate problem involving a point moving along a curve NOT: Section ANS: 50 3 ft/sec PTS: 1 DIF: Difficult REF: a OBJ: Solve a related rate problem involving a man walking away from a light source MSC: Application NOT: Section ANS: dt PTS: 1 DIF: Easy REF: 2.6.6a OBJ: Solve a related rate problem involving a point moving along a curve NOT: Section ANS: dt 22 3 PTS: 1 DIF: Easy REF: 2.6.3b OBJ: Calculate the value of an implicit derivative from given information NOT: Section ANS: There is a horizontal tangent at x 1 and a vertical tangent at y 3. PTS: 1 DIF: Easy REF: OBJ: Identify the points where an implicit function has horizontal and vertical tangent lines 2
7 13. ANS: x 2 d 2 y 2 y y y 2 PTS: 1 DIF: Easy REF: OBJ: Calculate the second derivative implicitly 14. ANS: dt PTS: 1 DIF: Easy REF: 2.6.1a OBJ: Calculate the value of an implicit derivative from given information NOT: Section ANS: y 2x 9 PTS: 1 DIF: Easy REF: a OBJ: Write an equation of a line tangent to the graph of an ellipse at a specified point. 16. ANS: y 2.50x 6.00 PTS: 1 DIF: Medium REF: OBJ: Write an equation of a line tangent to the graph of an implicit function at a specified point. 17. ANS: 0.33 PTS: 1 DIF: Easy REF: OBJ: Evaluate the derivative of an implicit function at a given point 18. ANS: 4x2 x 2 1 PTS: 1 DIF: Medium REF: a OBJ: Differentiate an equation using implicit differentiation 3
8 19. ANS: cosx 98sin 14y PTS: 1 DIF: Medium REF: OBJ: Differentiate an equation using implicit differentiation 20. ANS: 2x 5 9y 2y 9x PTS: 1 DIF: Easy REF: OBJ: Differentiate an equation using implicit differentiation 21. ANS: y x PTS: 1 DIF: Easy REF: OBJ: Differentiate an equation using implicit differentiation 22. ANS: 4x3 7 6y 7y 6 6x PTS: 1 DIF: Medium REF: OBJ: Differentiate an equation using implicit differentiation 23. ANS: x y PTS: 1 DIF: Easy REF: OBJ: Differentiate an equation using implicit differentiation 24. ANS: y 8x 3 sin 2x 4 ˆ 6 PTS: 1 DIF: Medium REF: OBJ: Differentiate a trigonometric function using the chain rule NOT: Section 2.4 4
9 25. ANS: f () PTS: 1 DIF: Medium REF: OBJ: Evaluate the derivative of a function at a point NOT: Section ANS: f 28sin2 cos2 ( ) 5 PTS: 1 DIF: Medium REF: OBJ: Differentiate a trigonometric function using the chain rule NOT: Section ANS: f ( x) 30x 5 x 6 ˆ 4 4 PTS: 1 DIF: Difficult REF: OBJ: Differentiate a function using the chain rule NOT: Section ANS: y 11t 19 PTS: 1 DIF: Medium REF: a OBJ: Write an equation of a line tangent to the graph of a function at a specified point 29. ANS: f ( x) x7 ( 80 51x) 2 5 3x PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using the chain rule and product rule NOT: Section ANS: f () s 8 x 3 PTS: 1 DIF: Medium REF: OBJ: Calculate the second derivative of a function 5
10 31. ANS: f () x 3 x 2 cosx 2x sinx ˆ x 2 3 ˆ 2 PTS: 1 DIF: Difficult REF: OBJ: Differentiate a function using the quotient rule 32. ANS: f () s 9s coss 4sins PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using the product rule 33. ANS: 1 2x cosx 2sinx ( 2x cosx) 2 PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using the product rule 34. ANS: f () x 84 36x 3x2 ( x 6) 2 PTS: 1 DIF: Difficult REF: OBJ: Differentiate a function using the quotient rule 35. ANS: f ( 2) PTS: 1 DIF: Difficult REF: OBJ: Differentiate a function using the quotient rule and evaluate the derivative 6
11 36. ANS: f () x 9 8x x 2 x 2 9 ˆ 2 ˆ PTS: 1 DIF: Difficult REF: OBJ: Differentiate a function using the quotient rule 37. ANS: f () s s 5 sins 5s 4 coss PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using the product rule 38. ANS: H () s 11t 10 36t 5 25t 4 PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using the product rule 39. ANS: 108 cm 2 PTS: 1 DIF: Medium REF: OBJ: Interpret a derivative as a rate of change MSC: Application NOT: Section ANS: The velocity after 4 seconds is 140 ft per second. PTS: 1 DIF: Difficult REF: OBJ: Derive the free-fall position function and evaluate velocity at different points MSC: Application NOT: Section ANS: vt () 39t 2 PTS: 1 DIF: Medium REF: a OBJ: Write the velocity function for a specified position function MSC: Application NOT: Section ANS: The graph has no horizontal tangents. PTS: 1 DIF: Difficult REF: OBJ: Calculate the values for which the slope of a function is zero NOT: Section 2.2 7
12 43. ANS: 96 ft/sec PTS: 1 DIF: Medium REF: c OBJ: Interpret a derivative as a rate of change MSC: Application NOT: Section ANS: f () PTS: 1 DIF: Medium REF: OBJ: Calculate the slope of the graph of a function at a given point NOT: Section ANS: f ( x) 8x 4sin( x) PTS: 1 DIF: Medium REF: OBJ: Differentiate trigonometric functions NOT: Section ANS: 16x y 8 0 PTS: 1 DIF: Medium REF: OBJ: Write an equation of a line tangent to the graph of a function that is parallel to a given line NOT: Section ANS: f ( x) 21x 2 8x PTS: 1 DIF: Medium REF: OBJ: Differentiate a function using basic differentiation rules NOT: Section ANS: f () 4 72 PTS: 1 DIF: Medium REF: OBJ: Calculate the slope of the graph of a function at a given point NOT: Section ANS: f ( x) sinx 3x PTS: 1 DIF: Medium REF: OBJ: Differentiate trigonometric functions NOT: Section ANS: x 0an 8 PTS: 1 DIF: Medium REF: OBJ: Calculate the values for which the slope of a function is zero NOT: Section 2.2 8
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