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2025 · Spring · Calculus

2025 Spring · Calculus

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Problem 1
Welcome to the Excel Academe Math Competition! This is our first time doing a calculus test ever, so congratulations on being a part of this historical moment! We hope everyone did good on their AP exams! The following equates to a simplified fraction of the form mk\frac{m}{k}. What is m+km+k?
d(5x2+18x+2008)d(e2025x)∣x=0\frac{d(5x^2+18x+2008)}{d(e^{2025x})}\bigg|_{x=0}
(A) 225(B) 227(C) 229(D) 231(E) 233\text{(A) } 225 \quad \text{(B) } 227 \quad \text{(C) } 229 \quad \text{(D) } 231 \quad \text{(E) } 233

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Problem 2
Find the slope of the normal line of x3+2x+5x^3 + 2x + 5 at x=1x = 1.
(A) −15(B) 15(C) −5(D) 5(E) 0\text{(A) } -\tfrac{1}{5} \quad \text{(B) } \tfrac{1}{5} \quad \text{(C) } -5 \quad \text{(D) } 5 \quad \text{(E) } 0

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Problem 3
David is running on 30 minutes of sleep again and has Calculus homework due at 11:59PM (it is currently 11:50PM). Solve this integral so David can get a 100% on his homework.
∫01ln⁡(x) dx\int_0^1 \ln(x)\,dx
(A) e(B) −1(C) 0(D) 1(E) 1e\text{(A) } e \quad \text{(B) } -1 \quad \text{(C) } 0 \quad \text{(D) } 1 \quad \text{(E) } \tfrac{1}{e}

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Problem 4
If f(x)=e3x−3+5x2−6f(x) = e^{3x-3} + 5x^2 - 6, find the derivative of f−1(x)f^{-1}(x) at x=0x = 0.
(A) −113(B) 113(C) 13(D) −13(E) 0\text{(A) } -\tfrac{1}{13} \quad \text{(B) } \tfrac{1}{13} \quad \text{(C) } 13 \quad \text{(D) } -13 \quad \text{(E) } 0

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Problem 5
∫x2+16x4 dx=−(xα+β)γλ⋅xρ+C\int \frac{\sqrt{x^2+16}}{x^4}\,dx = -\frac{\left(x^{\alpha}+\beta\right)^{\gamma}}{\lambda \cdot x^{\rho}} + C
 What is α+β+2γ+λ+ρ\alpha + \beta + 2\gamma + \lambda + \rho?
(A) 72(B) 81(C) 90(D) 99(E) 108\text{(A) } 72 \quad \text{(B) } 81 \quad \text{(C) } 90 \quad \text{(D) } 99 \quad \text{(E) } 108

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Problem 6
lim⁡n→∞  n2009∫01/nx 5x2+18x+2008 dx\lim_{n\to\infty}\; n^{2009}\int_0^{1/n} x^{\,5x^2+18x+2008}\,dx
(A) 0(B) 12007(C) 12008(D) 12009(E) DNE\text{(A) } 0 \quad \text{(B) } \tfrac{1}{2007} \quad \text{(C) } \tfrac{1}{2008} \quad \text{(D) } \tfrac{1}{2009} \quad \text{(E) DNE}

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Problem 7
An important concept in classical physics is the center of mass of an object with non-uniform density. Given a density along the xx-direction, the center of mass is
xcm=∫r dm∫dm.x_{cm} = \frac{\int r\,dm}{\int dm}.
 A one-dimensional rod of length 55 (with x=0x=0 at its left end) has linear density λ(x)=x2+3x\lambda(x) = x^2 + 3\sqrt{x}. Let MM be its total mass. If ∣xcm∣−M=b5+mn|x_{cm}| - M = b\sqrt{5} + \dfrac{m}{n}, what is b+m+nb + m + n?
(A) 1407(B) 1537(C) 1497(D) 1337(E) 1607\text{(A) } 1407 \quad \text{(B) } 1537 \quad \text{(C) } 1497 \quad \text{(D) } 1337 \quad \text{(E) } 1607

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Problem 8
The probability density function of the standard normal distribution is 12π e−x2/2\dfrac{1}{\sqrt{2\pi}}\,e^{-x^2/2}, and it integrates to 11 over all of R\mathbb{R}. Using this, evaluate
∫−∞∞e−x2 dx.\int_{-\infty}^{\infty} e^{-x^2}\,dx.
(A) π2(B) 3π2(C) 2π(D) e(E) π\text{(A) } \sqrt{\tfrac{\pi}{2}} \quad \text{(B) } \sqrt{\tfrac{3\pi}{2}} \quad \text{(C) } \sqrt{2\pi} \quad \text{(D) } \sqrt{e} \quad \text{(E) } \sqrt{\pi}

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Problem 9
∫01e−x2 dx  +  ∫−eesin⁡x1+x2+cos⁡x dx  +  ∫11/eln⁡ ⁣(1x) dx  =  ek\int_0^1 e^{-x^2}\,dx \;+\; \int_{-\sqrt{e}}^{\sqrt{e}} \frac{\sin x}{1+x^2+\cos x}\,dx \;+\; \int_1^{1/e} \sqrt{\ln\!\left(\tfrac{1}{x}\right)}\,dx \;=\; e^{k}
 What is kk?
(A) −2(B) −1(C) 0(D) 1(E) 2\text{(A) } -2 \quad \text{(B) } -1 \quad \text{(C) } 0 \quad \text{(D) } 1 \quad \text{(E) } 2

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Problem 10
∫02x⋅ x⋅ x⋅ x⋯43   dx\int_0^2 x\cdot\sqrt{\,x\cdot\sqrt[3]{\,x\cdot\sqrt[4]{\,x\cdots}}}\;\,dx
(A) 2e+2e+2(B) 2ee(C) 2e+1e(D) 2e+1e−2(E) 2e+2e\text{(A) } \tfrac{2^{e+2}}{e+2} \quad \text{(B) } \tfrac{2^{e}}{e} \quad \text{(C) } \tfrac{2^{e+1}}{e} \quad \text{(D) } \tfrac{2^{e+1}}{e-2} \quad \text{(E) } \tfrac{2^{e+2}}{e}

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Problem 11
If the area of the inner loop of the polar curve r=1+2cos⁡θr = 1 + 2\cos\theta equals π−ABC\pi - \dfrac{A\sqrt{B}}{C}, where the fraction is in simplest form and properly rationalized, what is A⋅B⋅CA\cdot B\cdot C?

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Problem 12
If In=∫n/2∞(nx) dxI_n = \displaystyle\int_{n/2}^{\infty} \binom{n}{x}\,dx, then find ∑n=0∞1In\displaystyle\sum_{n=0}^{\infty}\dfrac{1}{I_n}.

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Problem 13
∫123x4+4x+6x−1x6+3x5+x2+1 dx=ln⁡ ⁣(AB)\int_1^2 \frac{3x^4 + 4x + 6x^{-1}}{x^6 + 3x^5 + x^2 + 1}\,dx = \ln\!\left(\frac{A}{B}\right)
 What is A+BA + B, given that AB\dfrac{A}{B} is a fraction in simplest form?

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Problem 14
S=∑n=1∞cos⁡(n)nS = \sum_{n=1}^{\infty} \frac{\cos(n)}{n}
 What is ⌊100S⌋\lfloor 100S \rfloor?

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Problem 15
L=∫−111−x1+x⋅arccos⁡(x) dxL = \int_{-1}^{1} \sqrt{\frac{1-x}{1+x}}\cdot \arccos(x)\,dx
 If L=πab+cL = \dfrac{\pi^{a}}{b} + c, what is a⋅b⋅ca\cdot b\cdot c?

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