Preparing for the AP Exam

Preparing for the AP Exam

[Notes/Highlighting]

Multiple Choice Questions

Use scratch paper as necessary to solve each problem. Pick the best answer from the choices provided.

1.Let.Which of the following is true?

Idoes not exist.

IIfis continuous atx= 3.

III The linex= 3 is a vertical asymptote.

(A)I only

(B)II only

(C)III only

(D)I and II only

(E)I and III only

Correct Answer

B

2.is

(A)0

(B)1

(C)−∞

(D)∞

(E)nonexistent because sin(x)oscillates between −1 and 1.

3.is

(A)0

(B)

(C)

(D)

(E)nonexistent

Correct Answer

C

4.

Using the table of values off(x),the average rate of change offon the interval [−2,4] is

(A)

(B)1

(C)

(D)

(E)12

5.Suppose,,andf(x)≤h(x)≤g(x)for allx.Which of the following must be true?

If(2)=5

IIg(x)≤6 for allx≥2

III

(A)I only

(B)II only

(C)III only

(D)I and III only

(E)None of the statements must be true.

Correct Answer

E

6.Iffis continuous on [2,6],withf(2)= 20 andf(6)= 10,then the Intermediate Value Theorem says which of the following is true?

If(x)= 25 does not have a solution on [2,6].

IIf(x)= 17 has a solution on [2,6].

IIIf(x)= 0 has a solution on [2,6].

(A)I only

(B)II only

(C)III only

(D)I and II only

(E)I,II,and III

7.is

(A)0

(B)

(C)

(D)∞

(E)−∞

Correct Answer

E

8.is

(A)−∞

(B)−3

(C)0

(D)3

(E)∞

9.Iffis continuous for allx,the maximum number of horizontal asymptotes that the graph offcan have is

(A)0

(B)1

(C)2

(D)3

(E)There is no maximum number.

Correct Answer

C

10.Iffis continuous for allx,the maximum number of vertical asymptotes that the graph offcan have is

(A)0

(B)1

(C)2

(D)3

(E)There is no maximum number.

11.is

(A)−∞

(B)−4

(C)−2

(D)0

(E)nonexistent

Correct Answer

C

12.If the graph offhas atangent linefor allx,withf(1.99)= 4.988 andf(2.01)= 5.004,then which of the following is the best approximation to the tangent line whenx= 2?

(A)y= 0.8x+ 3.4

(B)y= 0.8x+5

(C)y= 1.6x+ 1.8

(D)y= 1.6x+5

(E)y= 2.0x+1

13.

The graph of a functionfis shown above.Which of the following is true?

I

II

III

(A)I only

(B)II only

(C)III only

(D)I and II only

(E)I,II,and III

Correct Answer

D

14.

The graph offgiven above has a tangent line at everyxin(−2,3)exceptx=

(A)−1

(B)0

(C)1

(D)2

(E)The graph has a tangent line at eachx.

15.If the domain offis [1,∞)withf(1)= 0,and the liney= 3 is a horizontal asymptote for the graph off,which of the following must be true?

I The graph offnever meets the liney= 3.

II

IIIfis an increasing function.

(A)I only

(B)II only

(C)I and II only

(D)I and III only

(E)I,II,and III

Correct Answer

B

16.is

(A)−∞

(B)

(C)0

(D)

(E)∞

17.

Iffis continuous on [1,8] and some values offare given in the table above,then which of the following must be true?

If(x)= −3 has a solution in [1,8].

IIf(x)= 0 has a solution in [1,8].

IIIf(x)= 9 has a solution in [1,8].

(A)II only

(B)III only

(C)I and II only

(D)II and III only

(E)I,II,and III

Correct Answer

D

18.is

(A)−∞

(B)−1

(C)0

(D)1

(E)∞

19.If,thenis

(A)4

(B)2

(C)1

(D)0

(E)nonexistent

Correct Answer

E

20.is

(A)−∞

(B)−3

(C)0

(D)3

(E)∞

Free Response Questions

Show all of your work and clearly label any functions, graphs, tables, or other objects that you use. On the AP, your work will be scored on the correctness and completeness of your methods, as well as your actual answer. You will usually not be given credit for answers that don’t include supporting work.

1.Let.

(a)What is the average rate of change offon the interval?

(b)What is?

(c)Is the linex= 0 a vertical asymptote off?Justify your answer using limits.

(d)Use the Squeeze Theorem to show the liney= 0 is a horizontal asymptote off.

Correct Answer

(a)

(b)

(c)No,,so neither the left-hand limit nor the right hand limit is infinite,which is needed for the graph to have a vertical asymptote.

(d)We know −1 ≤ sinx≤ 1,so ifx> 0,then,and since,the Squeeze Theorem implies.This means the liney= 0 is a horizontal asymptote.

2.Let

(a)Are the linesx= 5 andx= −5 vertical asymptotes off?Justify your answer.

(b)Identify all horizontal asymptotes off.Justify your answer.

(c)Is there a value ofAthat makesfcontinuous atx= 5?

(d)Is there a value ofBthat makesfcontinuous atx= −5?

3.Letf(x)be a function defined for allx,with −5≤f(x)≤10.Also,does not exist butf(0)= 3.

(a)Letg(x)=xf(x).Showgis continuous atx= 0.

(b)Does the graph ofghave a tangent line at(0,0)?Explain.

Correct Answer

(a)Since −5 ≤f(x)≤ 10,ifx> 0 then −5x≤xf(x)≤ 10x.Thus by the Squeeze Theorem.Next,ifx< 0,then −5x≥xf(x)≥ 10x.Applying the Squeeze Theorem again,.Thus.Checking the functional value,we haveg(0)= 0 · 3 = 0.Thus,sogis continuous atx= 0.

(b)No.,which does not exist.

4.Let.

(a)Use the definition to showfis continuous atx= 4.

(b)What is the average rate of change offon the interval [0,0.004]?

(c)Is there an instantaneous rate of change offwhenx= 0?Explain