Math 2413 t2rsu14. Name: 06/06/ Find the derivative of the following function using the limiting process.
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1 Name: 06/06/014 Math 413 trsu14 1. Find the derivative of the following function using the limiting process. f( x) = 4x + 5x. Find the derivative of the following function using the limiting process. f( x) = x 3 1
2 3. The graph of the function f is given below. Select the graph of f.
3 a. d. b. e. c. 3
4 4. Find the derivative of the function. f( x) = x 4 5. Find the derivative of the function f( x) = 7x 3 + 4x Find the derivative of the function f( x) = 4x 4cos( x). 7. Find the slope of the graph of the function at the given value. f( x) = x + 6 when x= 5 x 8. Find the derivative of the function f( x) = x 5 9. x 4 9. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y( x) = x 3 + 1x Suppose the position function for a free-falling object on a certain planet is given by s t ( ) = 1t + v 0 t+s 0. A silver coin is dropped from the top of a building that is 137 feet tall. Find the instantaneous velocity of the coin when t=4. 4
5 11. 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. 1. Find the derivative of the algebraic function H( v) = v 5 ˆ 3 v ˆ. 13. Use the Product Rule to differentiate f( s) = s 5 cos s. 14. Use the Quotient Rule to differentiate the function f( x) = 8x. x Use the Quotient Rule to differentiate the function f( x) = sinx. x Find the derivative of the function. f( s) = 9s sins+ 5cos s. 17. Find the second derivative of the function f( x) = 8x
6 18. Find the second derivative of the function f( x) = 3x + 5x 4 x. 19. Find the derivative of the algebraic function f( x) = x 6 ˆ Find the derivative of the function. f( x) = x 7 ( 5+8x) 3 1. Find the derivative of the function y= 8cos 4x.. Find the derivative of the function. y= cos x 4 ˆ 6 3. Find an equation to the tangent line for the graph of f at the given point. f( x) = 5x 5 ˆ + 5, 1,100 ˆ 4. Find the second derivative of the function f( x) = sin5x 6. 6
7 5. Find dy by implicit differentiation. dx x + y = 5 6. Find dy by implicit differentiation. dx x + 5x+ 9xy y = 4 7. Evaluate dy dx places. for the equation 7xy= 1 at the given point 3, 1ˆ. Round your answer to two decimal 8. Use implicit differentiation to find an equation of the tangent line to the ellipse x + y 98 = 1 at 1,7ˆ. 9. Find d y in terms of x and y given that 3 7xy= 3x 7y. dx 30. Assume that x and y are both differentiable functions of t. Find dy dx when x = 49and dt dt y= x. = 17 for the equation 7
8 31. A point is moving along the graph of the function y= sin6x such that dx dt Find dy dt when x= π 7. = centimeters per second. 3. A spherical balloon is inflated with gas at the rate of 300 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 70 centimeters? 33. A conical tank (with vertex down) is 1 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. 34. A ladder 0 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 feet per second. 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. 8
9 35. 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? 9
10 ID: A Math 413 trsu14 Answer Section 1. ANS: 8x+ 5. ANS: f ( x) = ( x 3) 3. ANS: A 4. ANS: f ( x) = 4x 3 5. ANS: f ( x) = 1x + 8x 6. ANS: f ( x) = 8x+ 4sin( x) 7. ANS: f ( 5) = ANS: f ( x) = x 5 9. ANS: x= 0 and x= ANS: 96 ft/sec 11. ANS: 108 cm 1. ANS: H ( s) = 8v v 4 9v 13. ANS: f s 14. ANS: ( ) = s 5 sins+ 5s 4 cos s f ( x) = x 5 x ˆ ˆ 1
11 ID: A 15. ANS: f ( x) = 16. ANS: f s 17. ANS: 3+x cos x x sinx ˆ x + 3 ( ) = 9s cos s+4sins 13 f ( x) = x ANS: f ( s) = 8 x ANS: f ( x) = 30x 5 x 6 ˆ ANS: f ( x) = x 6 ( 5+8x) ( 35+80x) 1. ANS: y = 3sin4x. ANS: y = 8x 3 sin x 4 ˆ 6 3. ANS: y= 500x ANS: f ( x) = 150x 4 cos 5x 6 900x 10 sin5x 6 5. ANS: dy dx = x y 6. ANS: dy x+ 5+9y = dx y 9x 7. ANS: dy dx = ANS: y= x ANS: d y dx = 0 ˆ
12 ID: A 30. ANS: dy dt = ANS: dy dt = 1cos 6π 7 3. ANS: dr 3 = dt 196π cm/min 33. ANS: 81 50π ft/min 34. ANS: 1.71 ft/sec 35. ANS: 50 3 ft/sec 3
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