Math 142 Week-in-Review #4 (Sections , 4.1, and 4.2)

Transcription

1 Math 142 WIR, copyright Angie Allen, Fall Math 142 Week-in-Review #4 (Sections , 4.1, and 4.2) Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on sections 4.1 and 4.2). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your suggested and online homework. 1. If j(x) = 2ex 5 8 x 3 + ln 5 x 2 πx 1.83, find d j dx. 2. ( d x 8 + 3x 2 4 ) x 5 dx x x 2 3. Find the equation of the line tangent to the curve y = x2 3x 4e x at x = 0. x 5

2 Math 142 WIR, copyright Angie Allen, Fall Use the limit definition of the derivative to find f (x) if f (x) = 4x x. 5. The position of a particle is given by s = t t 2 +30t, t 0, where t is time in seconds and s is measured in meters. a) Find the velocity after 4 seconds. b) When is the particle at rest?

3 Math 142 WIR, copyright Angie Allen, Fall x 2 + 4x Find lim x 1 (x + 1) 2, if it exists. If it does not exist, use limits to describe the way in which it does not exist. 7. Determine where the function f (x) = 3 x x x 2 is continuous.

4 Math 142 WIR, copyright Angie Allen, Fall The price-demand equation of a store that sells x gourmet blenders per month at a price of p dollars per blender is given by p = 876(0.9985) x. a) Find the average rate of change of price when production increases from 800 blenders to 900 blenders each month, and interpret your answer. b) Find the rate of change of revenue at a production level of 850 blenders, and interpret your answer. 9. Evaluate lim exist. x 5 4 numerically, if it exists. If it does not exist, use limits to describe the way in which it does not (x + 5) 2

5 Math 142 WIR, copyright Angie Allen, Fall A company that makes cameras has a cost function given by C(x), where x is the number of cameras made and C(x) is the total cost of producing x cameras (in dollars). Use the following information to help you answer the questions below. C(21) = $ C (21) = $22.36 C(22) = $ C (22) = $ C(23) = $ C (23) = $21.32 C(24) = $ C (24) = $20.85 a) Find the exact cost of the 23 rd camera. b) Approximate/estimate the cost of the 23 rd camera. c) Find the rate of change of cost when 22 cameras are produced. d) Find the exact cost if 24 cameras are produced. e) Approximate/estimate the cost if 24 cameras are produced. f) Find the marginal cost if 21 cameras are produced, and interpret your answer.

6 Math 142 WIR, copyright Angie Allen, Fall Consider the function g(x) below, and answer the following questions. a) Find lim x 0 g(x), if it exists. b) Find lim g(x), if it exists. x 3 8 g(x) c) Find lim x 2 g(x), if it exists. 8 8 d) Where is g(x) continuous? 8 e) What is the first condition in the definition of continuity to fail at x = 0? f) Find the average rate of change between x = 1 and x = 6. g) Find g ( 5), if it exists. h) Find g ( 2), if it exists. i) For what values of x does g (x) not exist, if any? Explain.

7 Math 142 WIR, copyright Angie Allen, Fall Find the derivative of each of the following functions. Do not simplify your answers. a) f (x) = x ( 2x 2 4x + 7 )( ) 4 x 9 + 2(8x ) 3log 6 x b) f (x) = (x3 7x + π 2 )e x 3 5 x 3 x 4 + 2x For the function f (x) = f (6 + h) f (6) x, find lim, if it exists. h 0 h

8 Math 142 WIR, copyright Angie Allen, Fall If f (x) = (x a)(x b), find lim x a f (x). x a 15. Sketch the derivative of the following function. f(x) x

9 Math 142 WIR, copyright Angie Allen, Fall Consider the function f (x) below and answer the following questions. a) Find lim f (x), if it exists. x 5 x + 5 x 2 x 30 x 4 f (x) = 9 (x+4)/(8 x) 10 3 x 2 17 ( ) 4 < x < 18 e x3 +64 x + 12 log 3 (25 x) x > 18 b) Find lim f (x), if it exists. x 24 c) Find lim f (x), if it exists. x 4 + d) Where is f (x) continuous?

10 Math 142 WIR, copyright Angie Allen, Fall Use the information in the table below regarding the functions f (x) and g(x) to answer the following. a) If m(x) = f (x) g(x), find m (5). x f (x) f (x) g(x) g (x) b) If n(x) = 2 f (x)g(x), find n (0). c) If h(x) = 3x2 g(x) f (x), find h ( 6).

11 Math 142 WIR, copyright Angie Allen, Fall Determine where the function f (x) = 8(x+2)/ x 1 is continuous. log 2 (4x 5) Find lim x 5 x , if it exists. If it does not exist, use limits to describe the way in which it does not exist. x + 5

12 Math 142 WIR, copyright Angie Allen, Fall The following table gives some values of a continuous function, f (x). Use the information to x f (x) a) find lim f (x), if it exists. x 2 b) find the average rate of change on the interval [ 2.1, 2]. c) estimate the instantaneous rate of change at x = 2.

13 Math 142 WIR, copyright Angie Allen, Fall Consider the function g(x) below, and answer the following questions. a) Find lim g(x), if it exists. x 5 b) Find lim g(x), if it exists. x g(x) c) Where is g(x) continuous? 8 8 d) What is the first condition in the definition of continuity to fail at x = 5? 8 e) Find g (0), if it exists. f) Find g (5), if it exists.