Math 131 (Calculus I) Department of Mathematics North Carolina A&T State University Practice Test 2 (Version A) Name: __________________________________________ Class:___________________ Date:___________________ Score:______________ Group Name:______________ Group Member1:__________________________________________ Group Member2:__________________________________________ Multiple Choice 1. The limit lim h 0 f (4 h) f (4) 9 , means h A. f (4) 9 B. f '(4) 9 C. f (0) 9 D. f '(0) 9 2. Form the figure, we can tell A. f '(2) 1 B. f '(2) 1 C. f '(2) 0 D. f '(2) is undefined 3. Find the second derivative of the function f ( x) cos x , then evaluate f "( ) . A. f "( x) sin x; C. f "( x) sin x; f "( ) 1 f "( ) 0 B. f "( x) cos x; f "( ) 1 D. f "( ) 0 f "( x) cos x; 4. The equation of the tangent line to the graph y 2 x3 x 4 at the point (-1, 3) is A. y 5x 8 B. y 7 x 4 C. y 7 x 4 D. y 5x 8 1 5. If y 2sin x x e2 , then y ' is A. y ' 2cos x 1 B. y ' 2cos x 1 e2 C. y ' 2cos x 1 D. y ' 2cos x e2 6. Find all values of x for which y 2 x3 2 x has horizontal tangents. 3 A. x 0, 1,1 B. x 1,1 C. x 1 D. x 0,1 7. A stone is dropped off a building and its height function s(t ) (in meter) after time t seconds is s(t ) 16t 2 . The velocity of the stone at t 3 sec is A. v(3) 16 m / s B. v(3) 32 m / s C. v(3) 64 m / s D. v(3) 96 m / s 8. If f ( x) x 2e x , then f '( x) is A. f '( x) 2 x e x B. f '( x) 2 xe x C. f '( x) ( x 2 2 x)e x D. f '( x) ( x 2 2 x)e x 9. If f ( x) sin(1 3x) , then f '( x) is A. f '( x) cos(3) B. f '( x) cos(1 3x) C. f '( x) 3cos(1 3x) D. f '( x) 3cos(1 3x) 2 10. If y sin 3x e3 x ln(3x) , then dy is dx A. dy cos 3x e3 x 1 dx 3x B. dy 3cos 3x 3e3 x 1 dx 3x C. dy 3cos 3x 3e3 x 1 dx x D. dy 3cos 3x 3e3 x 1 dx x 11. If g ( x) f (3x) , f (3) 4 and f '(3) 7 , then g '(1) is A. g '(1) 21 B. g '(1) 7 C. g '(1) 12 D. Insufficient information to determine 3 12. If f ( x) g ( x ) , then 2 3 A. f '( x) 3x g '( x ) 3 B. f '( x) 3x g '( x ) C. f '( x) g '(3x 2 ) D. f '( x) 3x 2 g '( x) dy 13. Use implicit derivative to find dx of 2 x 3 y 2 3 A. dy 2 x dx 3 y B. dy 1 dx 3 y C. dy 1 dx 3y D. dy 2 2 dx 3 y 14. If the area of a circle of radius r is A r 2 , then the rate of change of the area A. dA 2 dr dt dt B. dA 2 r 2 dr dt dt C. dA 2 r dr dt dt dA is dt D. dA r 2 dr dt dt 3 Free Response Questions Show all of your work to back up your answer. No Work = No Credit 1. Finding derivative of a function using basic differentiation rules. Find the derivative of the function a. f ( x) 3x3 x22 x b. y x3 (2 x 2 x12 x13 ) c. y 5t 2 sin 5t tan t 2 d. y 3x5 3e x 7ln x e4 x e3 2. Finding derivative of a function using Product/Quotient rules. Find the derivative of the function a. y (2 x3 3) tan x 2x b. f ( x) e sin x 4 c. f ( x) 2x 1 cos x d. y x2 3 ln x 3. One Dimensional Motion. Suppose a stone is thrown vertically upward from the edge of a cliff on Mars (acceleration is 12 ft / s 2 ) with an initial velocity of 60 ft / s from a height of 144 ft above the ground. Newton's laws of motion, the position of the stone (measured as the height above the ground) after t seconds is given by s(t ) 6t 2 60t 144 a. Determine the velocity v(t ) of the stone after t seconds. b. When does the stone reach its highest point? 5 c. What is the height of the stone at the highest point? d. When does the stone strike the ground and with what velocity does the stone strike the ground? e. What is the velocity v and acceleration a at time t 2sec . 6 4. Finding derivative of a function using General Power Rule and Chain Rule. a. f ( x) (3x 2 e2 x 1)4 b. y tan(2 x 2 3e x ) c. y 1 3x 2 x 2 d. e. g ( x) ln( 2xx11 ) f ( x) ln( x2e3 x ) f. y cos(3x) ln(2 x 1) 7 5. Calculating the derivatives at a given point using a table. Suppose f and g are differentiable functions with values given below x f ( x) f '( x) g ( x) g '( x) 0 3 3 3 -3 -3 4 -3 2 5 a. If h( x) g ( x) , find h '(0) f ( x) b. If k ( x) f ( f ( x)), k '(0) c. If m( x) g ( f ( x)) , find m '(3) d. n( x) f (3sin x) , find n '( ) . 2 e. If p( x) tan( f ( x) g ( x)) , find p '(3) f. q( x) 3e g( x ) find q '(0) . 8 6. Implicit differentiation. Find the derivative by implicit differentiation. a. x3 2 xy y3 0 b. 2 x2 x2 y cos y 1 9 7. Finding the equation of a Tangent line using implicit differentiation. Consider the implicit function x 2 xy y 2 4 and find the following. a. Show that the point (2, 2) lies on the graph of the above function. dy b. Find the formula for dx as a function both x and y using implicit differentiation. c. Find an equation of the tangent line to the graph at the point (2, 2). 10 8. Related Rates. When a spherical balloon is inflated with gas, its volume increases at the rate of 10 ft 3 / min . Using this information find the following. (Volume of a sphere is V 4 r 3 ). 3 a. Find a formula for the rate of change of volume dv dr in terms of rate of change of radius . dt dt b. How fast the radius of the balloon increasing at the instant the radius is 1 ft . 11 c. How fast the radius of the balloon increasing at the instant the radius is 2 ft . 9. Related Rates. The formula for the volume of a cone is V 1 r 2 h . Find the rate of 3 change of the volume if dr 2 in / min and h 3r when (a) r 3 in dt and (b) r 12 in . 12 Answer Key 1. Finding derivatives of a function using basic differentiation rules. 4 1 3 x 2 x a. f '( x) 9 x 2 c. dy 10t 5cos 5t sec2 t dx dy 10 x 4 1 dx b. d. dy 7 15 x 4 3e x 4e4 x dx x 2. Finding derivative of a function using Product/Quotient rules. a. dy (2 x3 3)sec2 x 6 x 2 tan x dx c. f '( x) 2cos x (2 x 1)sin x cos2 x b. f '( x) (cos x 2sin x)e2 x d. dy 2 x 2 ln x x 2 3 dx x(ln x) 2 3. One dimensional motion a. Velocity v(t ) s '(t ) 12t 60 b. Stone teaches its highest point when time t 5sec . c. Stone's height at the highest point is 294 ft above the ground d. Stone strike the ground when time t 12 sec. e. v(2) 12(2) 60 36 ft / sec Velocity of the stone at impact is 84 ft / sec . a(2) 12 ft / sec 2 4. Finding derivative of a function using General Power Rule and Chain Rule. a. f '( x) 8(3x e2 x )(3x2 e2 x 1)3 b. dy (4 x 3e x )sec2 (2 x 2 3e x ) dx c. dy 3 4x dx 2 1 3x 2 x 2 d. f '( x) e. g '( x) f. dy 2cos(3x) 3ln(2 x 1) sin(3x) dx 2x 1 2 1 2x 1 x 1 2 3 x 13 5. Calculating the derivatives at a given point using a table. a. h '(0) 2 b. k '(0) 9 c. m '(3) 15 d. n '( ) 0 2 e. p '(3) 2 f. q '(0) 6e4 6. Implicit differentiation. a. dy 2 y 3x 2 dx 2 x 3 y 2 b. dy 2 x( y 2) dx x 2 sin y 7. Finding the equation of a Tangent line using implicit differentiation. a. b. dy y 2x dx x 2 y c. The equation of the tangent line at the point (2, 2) is y x 4 . 8. Related Rates. a. dV dr 4 r 2 dt dt b. The radius of the balloon increases at the rate of c. The radius of the balloon increases at the rate of 9. 5 ft / min when the radius is 1 ft 2 5 ft / min when the radius is 2 ft 8 Related Rates. a. b. dV dr 3 r 2 dt dt dV 54 in3 / min dt dV 864 in3 / min dt 14