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*Winter Break Packet - Loudoun County Public آ Winter Break Packet Consider this your midterm*

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Winter Break Packet

Consider this your midterm exam review. This packet covers Chapters 1 – 6 and 8. The midterm will also contain material

from Chapter 7 (which we will learn after break). Answers to this packet will be posted, and we will go over questions on

this packet when we return from break. (28 MC, 2 FR)

Will this be collected? _______________________

When? __________________________________

How many points? ________________________

You may NOT use a calculator on this section. Choose the BEST answer for each question.

1) For 0x , the horizontal line 2y is an asymptote for the graph of the function f. Which of the following

statements must be true?

(A) (0) 2f

(B) ( ) 2f x for all 0x

(C) (2)f is undefined

(D) 2

lim ( ) x

f x

(E) lim ( ) 2 x

f x

2) 3 2

3 2

2 3 4 lim

4 3 2 1x

x x x

x x x

(A) 4 (B) 1 (C) 1

4 (D) 0 (E) -1

3) The graph of f’, the derivative of the function f, is shown below. Which of the following is true about f?

(A) f is decreasing for 1 1x

(B) f is increasing for 2 0x

(C) f is increasing for 1 2x

(D) f has a local minimum at 0x

(E) f is not differentiable at 1x and 1x

4) If 3( ) ln 4 xf x x e , then '(0)f is

(A) 2

5 (B)

1

5 (C)

1

4 (D)

2

5 (E) nonexistent

5) The function f has the property that ( )f x , '( )f x , and "( )f x are negative for all real values x. Which of the

following could be the graph of f?

(A) (B) (C)

(D) (E)

6) Using the substitution 2 1u x , 2

0

2 1x dx is equivalent to

(A)

1 2

1 2

1

2 u du

(B) 2

0

1

2 u du

(C)

5

1

1

2 u du

(D) 2

0

u du

(E)

5

1

u du

7) The graph of a function f is shown below. At which value of x is f continuous, but not differentiable?

(A) a (B) b (C) c (D) d (E) e

8) sin 6x x dx

(A) cos 6 sin 6x x x C

(B) 1

cos 6 sin 6 6 36

x x x C

(C) 1

cos 6 sin 6 6 6

x x x C

(D) 1

cos 6 sin 6 6 36

x x x C

(E) 6 cos 6 sin 6x x x C

9) The function f is continuous on the closed interval 2,14 and has values as shown in the table below. Using the

subintervals 2,5 , 5,10 , and 10,14 , what is the approximation of 14

2

( )f x dx found by using a right Riemann

sum?

x 2 5 10 14

( )f x 12 28 34 30

(A) 296 (B) 312 (C) 343 (D) 374 (E) 390

10)

2

2 1

x dx

x x

(A) ln 2 ln 1x x C

(B) ln 2 ln 1 3x x x C

(C) 4ln 2 2ln 1x x C

(D) 4ln 2 2ln 1x x C

(E) 2 2 1

2ln 3 2

x x x C

11) If the line tangent to the graph of the function f at the point 1,7 passes through the point 2, 2 , then

'(1)f is

(A) -5 (B) 1 (C) 3 (D) 7 (E) undefined

12) The derivative of g’ of a function g is continuous and has exactly two zeros. Selected values of g’ are given in the

table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the following

intervals?

x -4 -3 -2 -1 0 1 2 3 4

'( )g x 2 3 0 -3 -2 -1 0 3 2

(A) 2 2x only

(B) 1 1x only

(C) 2x

(D) 2x only

(E) 2x or 2x

13) Let f be the function given below. Which of the following statements are true about f?

2 3 ( )

4 7 3

x if x f x

x if x

I. 3

lim ( ) x

f x

exists

II. f is continuous at 3x

III. f is differentiable at 3x

(A) None

(B) I only

(C) II only

(D) I and II only

(E) I, II, and III

14) What is the slope of the line tangent to the curve 2 23 2 6 2y x xy at the point 3,2 ?

(A) 0 (B) 4

9 (C)

7

4 (D) 4 (E) 13

Free Response. You may NOT use a calculator on this question. Show all of the work which justifies your solutions. Partial credit may be given on this section. Circle your final answer to each section. Water is pumped into an underground tank at a constant rate of 8 gallons per minute (NO, this is NOT a “work”

problem). Water leaks out of the tank at the rate of 1t gallons per minute, for 1200 t minutes. At

time 0t , the tank contains 30 gallons of water. (a) How many gallons of water leak out of the tank from time 0t to 3t minutes? (b) How many gallons of water are in the tank at time 3t minutes? (c) Write an expression for )(tA , the total number of gallons of water in the tank at time t.

(d) At what time t, for 1200 t , is the amount of water in the tank a maximum? Justify your answer using theorems or “tests.”

You MAY use a calculator on this section of the exam. Choose the BEST answer for each question and write it on the

answer sheet provided.

1) The regions A, B, and C in the figure below are bounded by the graph of the function f and the x-axis. If the area

of each region is 2, what is the value of 5

5

( ) 1f x dx

?

(A) -2 (B) -1 (C) 4 (D) 12 (E) 16

2) The graph of f’, the derivative of the function f, is shown below. If (0) 0f , which of the following must be

true?

I. (0) (1)f f

II. (2) (1)f f

III. (1) (3)f f

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

3) The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the

area of the circle at the instant when the circumference of the circle is 20 meters?

(A) 20.04 / secm

(B) 20.4 / secm

(C) 24 / secm

(D) 220 / secm

(E) 2100 / secm

4) The rate of change of the altitude of a hot-air balloon is given by 3 2( ) 4 6r t t t for 0 8t . Which of the

following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

(A) 3.514

1.572

( )r t dt

(B) 8

0

( )r t dt

(C) 2.667

0

( )r t dt

(D) 3.514

1.572

'( )r t dt

(E) 2.667

0

'( )r t dt

5) The velocity, in ft/sec, of a particle moving along the x-axis is given by the function ( ) t tv t e te . What is the

average velocity of the particle from time 0t to time 3t ?

(A) 20.086 ft/sec

(B) 26.447 ft/sec

(C) 32.809 ft/sec

(D) 40.671 ft/sec

(E) 79.342 ft/sec

6) Let f be a differentiable function with (2) 3f and '(2) 5f , and let g be the function defined by

( ) ( )g x xf x . Which of the following is an equation of the line tangent to the graph of g at the point where

2x ?

(A) 3y x

(B) 3 5 2y x

(C) 6 5 2y x

(D) 6 7 2y x

(E) 6 10 2y x

7) Let g be the function given by 2 0

( ) sin

x

g x t dt for 1 3x . On which of the following intervals is g

decreasing?

(A) 1 0x

(B) 0 1.772x

(C) 1.253 2.171x

(D) 1.772 2.507x

(E) 2.802 3x

8) The function f has first derivative given by 3

'( ) 1

x f x

x x

. What is the x-coordinate of the inflection point of

the graph of f?

(A) 1.008 (B) 0.473 (C) 0 (D) -0.278 (E) The graph of f has no inflection point

9) If a trapezoidal sum overapproxima