Math 243: Calculus II


Unit II Exam: Review Guide


This page contains problems similar to those that will appear on the Unit II Exam. These questions are all similar to ones that you encountered while doing your WebAssign homework.


1. Derive the formulas for \(\tfrac{d}{dx}\big[ \arcsin(x)\big]\) and \(\tfrac{d}{dx}\big[ \arctan(x)\big]\).

2. Derive the formulas for \(y = \sinh^{-1}(x)\) and \(\tfrac{d}{dx}\big[\sinh^{-1}(x)\big]\).

3. Evaluate the limits.

a.) \(\displaystyle \lim_{x \to \tfrac{\pi}{2}^+} \frac{\cos x}{1 - \sin x} \)

b.) \(\displaystyle \lim_{t \to 0^+} \frac{\ln(t)}{t} \)

c.) \(\displaystyle \lim_{x \to 0} \frac{\sqrt{1 - 3x} - \sqrt{1 + 8x}}{x} \)

d.) \(\displaystyle \lim_{x\to\infty} \left( 1 + \frac{2}{x}\right)^x \)

e.) \(\displaystyle \lim_{x \to \tfrac{\pi}{2}} \left(\sec(x) - \tan(x)\right) \)

4. Evaluate the definite and indefinite integrals.

a.) \(\displaystyle \int x^2e^{2x}\,dx\)

h.) \(\displaystyle \int_0^\pi e^{\cos t} \sin(2 t)\, dt \)

b.) \(\displaystyle \int \arccos(x) \,dx\)

i.) \(\displaystyle \int \tan^3\theta \sec^6\theta\, d\theta \)

c.) \(\displaystyle \int \ln(\sqrt{x}) \,dx\)

j.) \(\displaystyle \int \sin^2 x\cos^2 x\, dx \)

d.) \(\displaystyle \int e^\theta\sin\theta\, d\theta \)

k.) \(\displaystyle \int \frac{x^2}{\sqrt{81 - x^2}} \, dx \)

e.) \(\displaystyle \int_0^{2\pi} t\sin(t)\cos(t)\, dt\)

l.) \(\displaystyle \int_0^7 \sqrt{x^2 + 49}\, dx \)

f.) \(\displaystyle \int \frac{z}{\sqrt{z^2 - 9}} \, dz\)

m.) \(\displaystyle \int \frac{7t - 5}{t + 5} \, dt\)

g.) \(\displaystyle \int \frac{17}{(x - 1)(x + 1)^2}\, dx\)

n.) \(\displaystyle \int \frac{25}{v^3 - 8} \,dv\)

5. Use Simpson's Rule with 6 boxes to approximate the value of the integral \(\displaystyle \int_1^4 \frac{1}{x^2}\, dx\). Compare your answer to the exact value.

6. Determine whether the improper integrals are convergent or divergent. If convergent, evaluate the integral.

a.) \(\displaystyle \int_8^9 \frac{x}{x-9} \,dx\)

c.) \(\displaystyle \int_1^\infty \frac{1}{t^2} \, dt \)

b.) \(\displaystyle \int_1^\infty \frac{\ln x}{x} \,dx\)

d.) \(\displaystyle \int_0^1 \frac{1}{t^2} \,dt \)

7. Find the area of the region bounded by the hyperbola \(25x^2 - 4y^2 = 100\) and the line \(x = 3\).

8. Compute the area of a lune: the shaded region in the image below.


9. Find the area of the region bounded between the curves \(y_1 = \sin^2 x\) and \(y_2 = \sin^3 x\), \(0 \leq x \leq \pi\).

10. You wish to estimate the integral \(\displaystyle \int_0^2 e^{-2x^2}\, dx\) with maximum error of \(\vert E \vert \leq \tfrac{1}{1000}\). What is the minimum number of boxes you should use for: a.) the Trapezoidal Rule, and b.) Simpson's Rule?




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