1. f(x) = x for all x. Find f ′(f ′(f ′(f ′(2222)))).

A. 1 B. 0 C. 2 D. 2222 E. NOTA

2. Find the slope of the tangent line to the function f(x) = x2 sinx cosx at x = π.

A. π B. 0.5 C. π2 D.π

2E. NOTA

3. A scientist is trying to find a mathematical model for the position of a particle at time t, where t ismeasured in seconds. He eventually defines the position function p(t) in terms of itself as p(t) = (p(t))2 + t.However, this definition of p(t) fails to provide a result for the position of the particle after E seconds (that is,the function is undefined for any t > E). Which of the following is closest to E?

A. 0.25 B. 0.33 C. 0.50 D. 1.00 E. NOTA

4. For some continuous function f(x), f(1) = 4 and f ′(1) = 0. The function g(x) is defined as g(x) = f ′′(x).If g(1) = 3, then f(x) has which of the following at x = 1?

A. local maximum B. local minimum C. absolute maximum D. inflection point E. NOTA

5. Two tangent lines to the function f(x) = x2 + 2 meet at (0,−3). What is the length of the chord formedby the points of tangency to f(x)?

A. 2√

2 B. 2√

3 C. 4 D. 2√

5 E. NOTA

6. Aaron attempts to approximate e1.1 with differentials, using x = 1 as a starting point. However, hedifferentiates f(x) = ex incorrectly and works under the assumption that f ′(x) = 2x. What is the error in hisfinal approximation?

A.e

10B.

e− 2

2C.

e− 2

10D.

e

4E. NOTA

7. Evaluate

∫ 5

0

bxcx

dx.

A. 15 B. ln

(625

24

)C. ln

(64

27

)D. 60 E. NOTA

8. The cubic function f(x) = ax3 + bx2 + cx+ d has only one critical point on its entire domain. If ac = 2,what is |b|?

A.√

2 B.√

3 C.√

5 D.√

6 E. NOTA

9. f(x) =1

1− x. Define f 1(x) as f(x), fn(x) as f(fn−1(x)) and f

′n(x) asd

dxfn(x). Find

100∑n=1

f′n(2).

A.297

4B.

301

4C.

305

4D.

309

4E. NOTA

1

10. The function y is implicitly defined as 2 sin(x + y) = 1. What is the slope of the tangent line to y at

x = arcsin24

25?

A.−7

25B. −1 C.

−1

2D.

24

25E. NOTA

11. f(x) =n∏k=0

(x+ k)k. Findf ′(0)

f(0)in terms of n.

A. n! B.n∑k=0

1

k + 1C. n D.

ln(n+ 1)

n!E. NOTA

12. A particularly confusing clock is built with hour and minute hands both of length 1 unit. At exactly4:20 PM, at what rate is the distance between the two hands of the clock changing?

A. cosπ

72B. cos

π

36C. cos

π

18D. cos

π

9E. NOTA

13. Evaluate limx→0

cos(sin(cos(x)))

cos(sin(cos(sin(x)))).

A. 1 B. π C. 0 D. -1 E. NOTA

14. Define Si(x) as

∫ x

0

sin t

tdt. Given that

∫ 1

0

sin(ex)dx = Si(m) − Si(n), findm

n.

A. 1 B. e C. e2 D. e3 E. NOTA

15. Evaluate

∫ π/2

−π/2arccos (sinx)dx.

A.π2

4B.

π2

2C.

π

4D.

π

2E. NOTA

16. Express

∫ 1

0

sinx

xdx as a series.

A.∞∑n=1

(−1)n+1

(n)(n)!B.

∞∑n=1

(−1)n+1

(2n− 1)(2n− 1)!C.

∞∑n=1

(−1)n+1

(2n− 1)(n)!D.

∞∑n=1

(−1)n+1

(n)(2n− 1)!E. NOTA

17. Evaluate the improper integral

∫ 1

0

ln(x+ 1)

xdx.

A.π2

6B.

π2

12C. ln 2 D.

π ln 2

8E. NOTA

2

18. The region of the plane bounded by the graph of y = sin x and the x-axis for 0 ≤ x ≤ 2π is rotatedabout the line y = x− π. What is the volume of the object formed?

A.5√

2π2

4B.

√2

2C.

√5π2

3D.

5√

2π2

2E. NOTA

19. A line through the origin divides the region bounded by y = ex, y = 0, x = 0, and x = 1 into two equalpieces. Find the slope of this line.

A.e

2B. e C. e− 1 D. e− 2 E. NOTA

20. A dartboard is built in the shape of the parabolic sector bounded by y = −x2 + c and y = 0. In thisparabolic sector, a circular bullseye of maximum possible area is inscribed. Given that c ≥ 1, what is theradius of the bullseye region in terms of c?

A.√c B. c− 1

2C.√c− 1

2D.

√c

2E. NOTA

21. Evaluate

∫ 2π

π

1

1− 2 sin θ + sin2 θdθ

A.4

3B.

2

3C. 1 D.

5

3E. NOTA

22. Two real numbers x and y are randomly chosen from the interval (0, 1) with respect to uniform

distribution. What is the probability that x− y2 < 1

2?

A.

√3

2B.

√2

2C.

√2

3D.

√2

6E. NOTA

23. The function f(x) =xk

exis maximized at x = 3. Find the value of x for which g(x) =

xe

kxis maximized.

A. x =e

ln 3B. x =

3

eC. x =

e

3D. x =

ln 3

eE. NOTA

24. If y′ = xy and y(0) = 1, what is the smallest positive x for which y(x) is an integer?

A.√

ln 2 B.√

2 C.√

ln 4 D. 2 E. NOTA

25. Let S be the set of all right triangles with hypotenuse of length 4. Let A(S) be the set of the areas ofall the triangles in S. Find the average value of the elements in A(S).

A.8

3B.

2

3C.

3

4D.

9

4E. NOTA

26. Evaluate∞∑k=0

cos k

k!.

A. 0.5(ee−i

+ eei) B. (ee

−i+ ee

i) C. 0.5i(ee

−i+ ee

i) D. i(ee

−i+ ee

i) E. NOTA

3

27. Batra is making cups for his tea party. He makes cups by rotating the region bounded by the linesx = 0, x = a, y = a2 and the function y = x2 about the y-axis. What is the smallest integral value of a forwhich each of his cups holds more than 300π cubic units of tea?

A. 3 B. 4 C. 5 D. 6 E. NOTA

28. A nonzero polynomial p(x) with real coefficients satisfies p(x) = p′(x)p′′(x). What is the leadingcoefficient of p(x)?

A.1

3B.

1

6C.

1

9D.

1

18E. NOTA

29. Evaluate

∫ ∞0

1

4 + x4dx.

A.π

2√

2B.

π

4C.

π

8D.

π

2E. NOTA

30. Evaluate limx→∞

1

xln

(x!

xx

).

A. 0 B.1

2C. 1 D. ln 2 E. NOTA

31. f(x) is a continuous, differentiable function satisfying f ′(x) ≤ f(x) for all x in the domain of f . Whichof the following statements must be true?

I.f(2012)

f(2011)> e

There exists some c ∈ [a, b] for which f(b)− f(a) < f(c)(b− a).

A. B. C. D. E. NOTA

4