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Final Exam

V.63.0121.2011Spring: Calculus I

May 12, 2011

PLEASE READ THE FOLLOWING INFORMATION.

• This is a 90-minute exam. Calculators, books, notes, and other aids are not allowed.

• You may use the backs of the pages or the extra pages for scratch work. Do not unstaple

or remove pages as they can be lost in the grading process.

• Please do not put your name on any page besides the first page.

MC Calculus I Final Examination Spring 2011 MC

MC (36 points). This part consists of 18 multiple choice problems. Nothing more than theanswer is required; consequently no partial credit will be awarded.

1. Find the domain and the range of the function f(x) = 1 +√

x.

A© Domain (0,∞), range [0,∞)

B© Domain [0,∞), range (1,∞)

C© Domain [1,∞), range [0,∞)

D© Domain [0,∞), range [1,∞)

E© Domain [1,∞), range (1,∞)

2. The limit limt→0

(

1

t− 1

t2 + t

)

is

A© −1

B© 0

C© ∞

D© −∞

E© 1

3. The limit limx→0

x2 sin

(

1

x

)

is

A© 1

B© −∞

C© ∞

D© 0

E© −1


4. The limit limx→∞

(√

9x2 + x − 3x) is

A© −1

3

B© 1

3

C© ∞

D© −∞

E© 1

6

5. The limit limn→−∞

2 − e1/n

3 + e2/nis

A© 2

3

B© 1

4

C© 0

D© e

E© −e

6. The limit limx→0

x2

ex − x − 1is

A© −1

B© 1

C© −2

D© 2

E© 0


7. Which of the following must be true?

I. If f is differentiable at a, then f is continuous at a.

II. If f is continuous at a, then f is differentiable at a.

III. If f is differentiable at a, then f ′ is differentiable at a.

A© I only

B© II only

C© III only

D© I and II only

E© II and III only

8. If y =1 − ln x

1 + ln x, what is y′?

A© − 2

x(1 + ln x)2

B© − 2 ln x

x(1 + ln x)2

C© 2 ln x

(1 + ln x)2

D© − 2

(1 + ln x)2

E© 2 ln x

x(1 + ln x)2

9. If f(x) = (x2 + 1)2 arctan x, what is f ′(x)?

A© (1 + x2)(1 + 2 arctan x)

B© 1 + 4x arctan x

C© 4x(1 + x2) arctan x

D© (1 + x2)(1 + 4x arctan x)

E© 1 + 2(1 + x2) arctan x


10. If f(x) =

2∫

√x

ln tdt, what is f ′(x)?

A© ln√

x

B© − ln√

x

C© ln 2 − ln√

x

D© ln√

x

2√

x

E© − ln√

x

2√

x

11. Suppose that f(x) is a differentiable function. If f(x)y + f(y)x = 10, what is dy/dx?

A© −f(y) + f ′(x)y

f(x) + f ′(y)x

B© −f(x) + f ′(y)x

f(y) + f ′(x)y

C© −f(x) + f ′(y)x + f ′(y)x

f(x)

D© − f(x)

f(x) + f ′(y)x + f ′(y)x

E© f(y) − f ′(x)y

f(x) + f ′(y)x

12. Let f(x) = ex2

. The linear approximation of f(x) near a = 1 is

A© e(x − 1) + 1

B© 2xex2

(x − 1) + e

C© e(x − 1) + e

D© 2e(x − 1) + e

E© ex2

(x − 1) + e


13. If f(x) =√

x − 2, what is the inverse of f?

A© f−1(x) = x2 + 2

B© f−1(x) = x2 − 2, x ≥ 0

C© f−1(x) = x2 + 2, x ≥ 2

D© f−1(x) = −x2 + 2

E© f−1(x) = x2 + 2, x ≥ 0

14. Let f(x) = cos x, 0 ≤ x ≤ 1. Find the Riemann sum for a regular partition of size 5 takingthe sample points to be the left endpoints.

A© 1

5

(

1 + cos

(

1

5

)

+ cos

(

2

5

)

+ cos

(

3

5

)

+ cos

(

4

5

)

+ cos 1

)

B© 1

5

(

cos

(

1

5

)

+ cos

(

2

5

)

+ cos

(

3

5

)

+ cos

(

4

5

)

+ cos 1

)

C© 1

5

(

cos

(

2

5

)

+ cos

(

3

5

)

+ cos

(

4

5

)

+ cos 1

)

D© 1

5

(

1 + cos

(

1

5

)

+ cos

(

2

5

)

+ cos

(

3

5

)

+ cos

(

4

5

))

E© 1

5

(

cos

(

1

5

)

+ cos

(

2

5

)

+ cos

(

3

5

)

+ cos

(

4

5

))

15. Let f ′′(x) = 12 sin(2x) + 27e3x and f(0) = 0, f(π) = 3e3π. Find the function f(x).

A© −3 sin(2x) + 3e3x +3x

π− 3

B© 3 sin(2x) + 3e3x +3x

π− 3

C© 3 sin(2x) + 3e3x − 3x

π− 3

D© −3 sin(2x) + 3e3x − 3x

π− 3

E© −3 sin(2x) + 3e3x +3x

π+ 3


16. The indefinite integral

∫

1 + x sin x

xdx is

A© ln x + cos x + C

B© ln x − cos x + C

C© ln |x| + cos x + C

D© ln |x| + cos |x| + C

E© ln |x| − cos x + C

17. The indefinite integral

∫

(2x + 1)5x2+x+1dx is

A© 5x2+x+1

ln 5+ C

B© 5x2+x+1 + C

C© 5x2+x+1

ln x+ C

D© 5x2+x+1 ln 5 + C

E© 5x2+x+1 ln x + C

18. The definite integral

π/10∫

0

cos(5x)dx is

A© −1

5

B© 1

5

C© 0

D© −1

E© 1

FR1 Calculus I Final Examination Spring 2011 FR1

FR (34 points). Problems FR1–FR4 are free response questions. Put your answers in the boxes(where provided) and your work/explanations in the space below the problem.

• Read and follow the instructions of every problem.

• Show all of your work for purposes of partial credit. Full credit may not be given for

an answer alone.

• Justify your answers. Full sentences are not necessary, but English words help. Whenin doubt, do as much as you think is necessary to demonstrate that you understand theproblem, keeping in mind that your grader will be necessarily skeptical.

FR1 (5 points). Find the derivative of f(x) =1

1 − xusing the definition of the derivative. No

credit will be given for shortcut methods such as the quotient rule.

f ′(x) =


FR2 (9 points). Let

f(x) =x2 − 1

|x − 1|(a) Find lim

x→1+f(x) and lim

x→1−f(x).

limx→1+

f(x) =

limx→1−

f(x) =

(b) Does limx→1

f(x) exist? Explain why.

(c) Sketch the graph of f(x).


FR3 (12 points). Let f(x) =x2

(x − 1)2. One can see that f ′(x) = − 2x

(x − 1)3, f ′′(x) =

2(2x + 1)

(x − 1)4.

(a) Find the vertical and horizontal asymptote(s).

Vertical asymptote(s) =

Horizontal asymptote(s) =

(b) Find the interval(s) of increase or decrease.

Interval(s) of increase =

Interval(s) of decrease =


(c) Find the interval(s) of concavity and the inflection point(s).

Interval(s) of concavity =

Inflection point(s) =

(d) Use the information from parts(a)-(c) to sketch the graph of f .


FR4 (8 points). A box with a square base and open top has volume 16 ft3. The material usedto make the base costs 4 dollars per square foot while the material used to make the sides costs1 dollar per square foot.

(a) What are the dimensions of the cheapest such box?

Dimensions =

(b) How much does it cost to produce?

Cost =

(This page intentionally left blank.You can use it for scratch work. Please do not remove it.)

Problem Possible PointsNumber Points Earned

MC 36FR1 5FR2 9FR3 12FR4 8

Total 70

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