MTH132 Exam 1 Covers: Page Total. Max

Transcription

1 Name: PID: A Section #: Instructor: Page Total Score Max Instructions 1. You will be given exactly 90 minutes for this exam.. No calculators, phones, or any electronic devices. Put them away out of sight. 3. Nothing on your desk or lap but a pen or pencil and this booklet. 4. No outside scratch paper: use the back of the exam pages if needed. 5. Make sure you try all the problems: do not linger too long on hard ones. 6. Show your reasoning and calculations. Unsupported answers will not receive full credit. 7. Cross out (or erase) any incorrect statements in your answers. 8. Please raise your hand if you have any questions and we will come to you. 9. Do not open this booklet until you are instructed to do so. I have read and understood all of the above instructions: Signature: 1 /10/14

2 1. Evaluate the following its. (a) (8 points) x x x x x x + x x x x + = = 0 0 (b) (8 points) x 4 x 4 x 8x x 6x + 8 x 8x x 6x + 8 = x 4 x(x 4) (x 4)(x ) = x 4 x (x ) = 8 = 4 4 (c) (8 points) t 5 t 5 t 5 t 5 t 5 t 5 = (t 5)( t + 5) t 5 ( t 5)( t + 5) = t 5 (t 5)( t + 5) t 5 = t 5 ( t + 5) = = /10/14

3 sin(y + 4y) (d) (8 points) y 0 y sin(y + 4y) sin(y + 4y) = y 0 y y 0 y + 4y y + 4y y y + 4y y + 4 = 1 = = y 0 y y 0 ( π(x ) ) (e) (8 points) sin x 1 (x + 1)3 + 1 Because sine is a continuous function, ( π(x sin ) x 1 (x + 1)3 + 1 ) ( = sin x 1 ( π(x ) (x + 1)3 + 1 )) = sin π 3 3 =. (8 points) Use interval notation to indicate where the function f(x) = 8 x + 1 x + 3 is continuous. (, ) (, 8] 3 /10/14

4 3. (1 points) Find all vertical and horizontal asymptotes for the function f(x) = x + 4x 1 x. x VA: 0 = x x = (x )(x + 1). x = and x = 1 x + 4x 1 + (4/x) ((1/x ) HA: x x = x x 1 (1/x) (/x =.y = ) H.A.: x = and x = 1 V.A.: y = 4. (1 points) Use the formal it definition of the derivative to find the derivative of the function f(x) = 3 x f f(x) + h) f(x) 3 (x+h) (x) = h 0 = 3 x = h h 0 h ((3 x) ( 3 (x + h) ) h 0 (3 x) ( 3 (x + h) ) h = h 0 h h(3 x) ( 3 (x + h) ) = (3 x) d dx ( 3 x ) = (3 x) 4 /10/14

5 5. Find the derivative of each of the following functions. You need not simplify your answer. (a) (8 points) f(x) = x 8 4x + 3 f (x) = (4x + 3)x (x 8)4 (4x + 3) f (x) = (4x + 3)x (x 8)4 (4x + 3) (b) (8 points) g(x) = 3 x cos x g (x) = 3 x( sin x) + (cos x) 1 3 x 3 g (x) = 3 x(sin x) + (cos x) 1 3 x 3 (c) (8 points) h(x) = (x 4 + x) tan x h (x) = (x 4 + x) sec x + (tan x)(4x 3 + ) h (x) = (x 4 + x) sec x + (tan x)(4x 3 + ) 5 /10/14

6 (d) (8 points) F (x) = sec x x 3 + x 1/4 F (x) = (x 3 + x 1/4 ) sec x tan x (sec x)( 3x x 3 4 (x 1/4 ) ) F (x) = (x 3 +x 1/4 ) sec x tan x (sec x)( 3x x 3 4 (x 1/4 ) ) 6. Let f(x) = sin x 6x (a) (8 points) Find f (x) f (x) = 1 x cos x (sin x) 6 x f (x) = 1 x cos x (sin x) 6 x (b) (6 points) Find the equation of the tangent line to f(x) at x = π. f (π) = 1 π cos π (sin π) 6 π = π 6π = 1 6π An equation for the line tangent to the graph of y = f(x) at the point where x = π is y f(π) = f (π)(x π); that is, y = 1 6π (x π) y = 1 6π (x π) 6 /10/14

7 7. A ball is thrown upward from the top of a building 19 feet tall with the initial velocity 64 ft/sec. The height of the ball above ground level then satisfies the equation y(t) = 16t + 64t + 19 (a) (6 points) Find the maximum height of the ball. At the maximum height the velocity of the ball is 0. The velocity is y (t) = 3t So = y (t) = 3t Thus t =. So the height of the ball at t = is y() = 16(4) + 64() + 19 = = 56ft. 65 (b) (6 points) Find the velocity when the ball hits the ground. The ball hits the ground when y(t) = 0 so 0 = 16t + 64t + 19 = t + 4t + 1 = (t 6)(t + ). Thus y(6) = 0 The velocity of the ball upon impact is y (6) = 3(6) + 64 = (10 points) A particle moves along the x axis with a velocity of v(t) = t 5 + 6t + 1 for t 0. Show that there is a time when the particle is stationary. The function v is continuous on [0, ), v(0) = 1 and v() = = 7 So by the Intermediate Value Theorem, there is a c (0, ) such that v(c) = 0. Consequently the particle has velocity 0 when t = c and hence is stationary. 7 /10/14

8 Quick Answer Questions: Answer each of the following questions. No work is required. No partial credit available. y f 3 9. (6 points) The graph of the function f is shown in the figure to the right. Complete each of the following assertions using DNE for does not exist. 1 x a b (a) f(x) = x a (b) f(x) = x a + (c) x a f(x) = (d) f(x) = x b One point for each correct answer. (e) f(x) = 1 x b + (f) x b f(x) = DNE 10. (6 points) Let f(x) = 3x + 1 for all real x and let ε > 0. For which of the following choices of δ is f(x) 7 < ε whenever x < δ? Encircle each possible choice. (a) ε 4 (b) ε (c) ε ε+1 Give 6 points if answer (a) is encircled. encircled answers, but with a minimum score of 0. Take off 1 point for any other (d) ε+1 ε (e) 3ε 8 /10/14