Math 243: Calculus II

Midterm Exam

Review Guide

This page contains questions similar in spirit to those that will appear on the Midterm Exam. The Midterm Exam will cover sections 12.1--6, 6.5--8, and 7.1--6 of the book. Students should also study the WebAssign problems for these sections.

The Midterm Exam is scheduled for 8 March 2019.

1. You must be able to complete all quiz problems. While some of these problems take more time than an acceptable test question should, you will need to know all of the ideas necessary to complete the problems.

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

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

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

5. Find an equation of the plane through the points \(P(1,2,3)\), \(Q(4,0,-1)\), and \(R(2,-4,-2)\).

6. Find the distance between the point and the given plane. \[\begin{cases} P(1,-3,2) \\ \Pi:\ \ 3x + 2y + 6z = 5 \end{cases}\]
7. Reduce the equation to one of the standard forms and classify the surface. \[x^2 - y^2 - z^2 - 4x - 2z +3 = 0\]
8. You must be able to match the graph of a surface in \(\mathbb{R}^3\) to its equation. (This would be a multiple choice question.)

9. Compute the area of the lune in the figure below assuming that the radius of the larger circle is \(R = 4\) and the radius of the smaller circle is \(r = 1\).

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

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