2025 Fall · Calculus · Excel Academe

Problem 1

Its a known fact that James Doakes is the Bay Harbor Butcher. Doakes is taking care of business on the Miami coastline on his boat which is traveling at a velocity of

f(t)\left[\frac{m}{s}\right]

eastward. What is his acceleration at time

t = 5

minutes if

f(t) = t^2 - 4t + 57

?

\text{(A) } 596\tfrac{m}{s^2}\text{ west} \quad \text{(B) } 600\tfrac{m}{s^2}\text{ east} \quad \text{(C) } 604\tfrac{m}{s^2}\text{ east} \quad \text{(D) } 608\tfrac{m}{s^2}\text{ east} \quad \text{(E) } 596\tfrac{m}{s^2}\text{ east}

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Problem 2

L = \int_1^5 \frac{1}{x^2 - 5x + 6}\,dx

What is

e^L

?

\text{(A) } \tfrac{1}{3} \quad \text{(B) } \tfrac{1}{5} \quad \text{(C) } \tfrac{1}{7} \quad \text{(D) } \tfrac{1}{9} \quad \text{(E) None of the Above}

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Problem 3

Lil Mike (a friend of the test writer) is obsessed with One Piece. Unfortunately, he is not a fan of the pacing of One Piece so he must watch One Pace (a version of One Piece with filler cut out of it). If

f(x)

represents the amount of filler cut out of the series in hours at an instance of time and

x

is the number of years since the One Piece anime has released, then how much content has been cut in total for the entire show given it has been running for approximately

25

years?

f(x) = |x - 4|

\text{(A) } \tfrac{451}{2}\text{ Hours} \quad \text{(B) } \tfrac{453}{2}\text{ Hours} \quad \text{(C) } \tfrac{455}{2}\text{ Hours} \quad \text{(D) } \tfrac{457}{2}\text{ Hours} \quad \text{(E) } \tfrac{459}{3}\text{ Hours}

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Problem 4

\int_1^2 \frac{3x^2 - 2 + \frac{1}{2\sqrt{x}}}{x^3 - 2x + 7 + \sqrt{x}}\,dx = D

If

D

is of the form

\ln\!\left(\dfrac{A + \sqrt{B}}{C}\right)

, what is

A + B + C

?

\text{(A) } 18 \quad \text{(B) } 20 \quad \text{(C) } 22 \quad \text{(D) } 24 \quad \text{(E) } 26

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Problem 5

What can be said about the behavior of

\sum_{n=0}^{\infty} \frac{\sin(n)}{n}\,?

\text{(A) Converges Absolutely} \quad \text{(B) Converges Conditionally} \quad \text{(C) Diverges} \quad \text{(D) Bifurcates} \quad \text{(E) None of the Above}

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Problem 6

Congratulations! You have opened your own pizzaria! You are currently making your own cheese pizza with the radius of only the cheese portion being

r

and the radius of the whole pizza being

R

(including the crust and the cheese). The cheesy part of the pizza costs \

0.05 per square inch and the crust costs \

0.02 per square inch. You sell to the customer the cheese pizza for \

0.10 per square inch of the cheesy part of the pizza and \

0.05 per square inch of the crusty part of the pizza. The radius

R

of your entire pizza is

8

inches and you can only spend a total of

2.75\pi

dollars creating the pizza. What is the most optimal thickness of the crust that maximizes profit for you?

\text{(A) } 0\text{ inches} \quad \text{(B) } 1\text{ inch} \quad \text{(C) } 3\text{ inches} \quad \text{(D) } 5\text{ inches} \quad \text{(E) } 7\text{ inches}

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Problem 7

Let

\lfloor x \rfloor

denote the greatest integer less than

x

and

\lceil x \rceil

the least integer greater than

x

. Evaluate

\int_1^{\infty} \frac{\lfloor x \rfloor}{x^2\,\lceil x \rceil}\,dx.

\text{(A) } \tfrac{\pi^2}{6} - \tfrac{1}{2} \quad \text{(B) } \tfrac{\pi^2}{2} - 1 \quad \text{(C) } \tfrac{\pi^2}{3} - 1 \quad \text{(D) } \tfrac{\pi^2}{6} - 1 \quad \text{(E) } \tfrac{\pi^2}{2} - \tfrac{1}{2}

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Problem 8

y = b^{x}\cdot\frac{d}{dx}\!\left[b^{x}\cdot\frac{d}{dx}\!\left[b^{x}\cdot\frac{d}{dx}[\,\cdots\,]\right]\right]

If

y(0) = 1

, find

\ln\!\left(y\!\left(\log_b \tfrac{3}{2}\right)\right)

.

\text{(A) } \tfrac{1}{2\ln b} \quad \text{(B) } \tfrac{1}{3\ln b} \quad \text{(C) } \tfrac{1}{4\ln b} \quad \text{(D) } \tfrac{1}{9\ln b} \quad \text{(E) } \tfrac{1}{16\ln b}

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Problem 9

L = \lim_{n\to\infty}\left[\prod_{k=0}^{n}\left(1 + \cos\tfrac{k}{n}\right)\right]^{1/n}

What is

\ln(L)

? (Use

\ln(1+x) \approx x

.)

\text{(A) } \tan(1) \quad \text{(B) } \cos(1) \quad \text{(C) } \sin(1) \quad \text{(D) } \sec(1) \quad \text{(E) } \csc(1)

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Problem 10

L = \lim_{n\to\infty}\left(\frac{\dbinom{5n}{n}}{\dbinom{5n}{2n}}\right)^{1/n}

\text{(A) } 0 \quad \text{(B) } \tfrac{27}{64} \quad \text{(C) } \tfrac{64}{27} \quad \text{(D) } 1 \quad \text{(E) } \tfrac{3}{2}

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Problem 11

\int \sin(6x)\cos(7x)\,dx = -\frac{\cos(Ax)}{B} + \frac{\cos(Ex)}{F} + C

What is

2A + \dfrac{B}{2} + (EF)^4

?

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Problem 12

L = \int_1^3 -\frac{5x + 6}{x^7 + x^2 + x}\,dx

What is

e^{L}

?

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Problem 13

Find the area of the following region as

n \to \infty

:

\left(x^2 + y^2\right)^{n} = 67\,x\,y^{n}.

If

A

is the area, your answer should be

\dfrac{3\pi}{A}

.

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Problem 14

Here

n!! = n\cdot(n-2)\cdot(n-4)\cdots

and

k

is an arbitrary constant. If

\lim_{n\to\infty} \frac{c^{n}\,(3n)!}{(2n)!\,(4n-1)!!} = k

and

c = \dfrac{a}{b}

, what is

a + b

?

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Problem 15

L = \int_0^{\pi/2} \sqrt{\tan x}\,dx

Find

\dfrac{12\pi^4}{L^4}

.

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