Sample Math 212 Final Exam

(a)

Evaluate $\displaystyle \int 5^x \tan(5^x)~dx$.

(b)

Find $\displaystyle \frac{d}{dx}\big[\sqrt{\log_3 x}\big]$.

(c)

Does the graph of $x + x^3 + y^2 + y^4 + z^4 + z^5 = 1$ have symmetry about

1. the $x$-axis?
2. the $xz$-plane?
3. the origin?

(d)

What is the value of $\displaystyle \int_{-\sqrt{\pi}}^\sqrt{\pi} \sin(x^3)~dx$? Explain why.

(a)

Let the function $y(x)$ be the solution to the differential equation $$y' = \frac{12 x^3 + 12 x^2}{y^2 e^{y^3}}$$ for which $y(1)=0$. Find $y(x)$ explicitly as a function of $x$.

(b)

Find $\displaystyle \int 4x\big[\sec(x^2+2)\big]~dx$.

(a)

A package initially with a temperature of $150^\circ$F is placed in a room maintained at $70^\circ$F. Assume the temperature difference between package and the room $t$ hours after the package was initially placed is $(\Delta T)(t)=T(t)-70$, where $T(t)$ is the temperature of the package $t$ hours after being placed in the room, satisfies an exponential decay law. The temperature of the package $2$ hours after being placed in the room is $86^\circ$F.

1. Find the function $(\Delta T)(t)$.
2. Find the temperature, expressed as a rational number (i.e., a quotient of integers), of the package after $4$ hours.
3. How long does it take for the package to reach $71^\circ$F?

(b)

Evaluate $\displaystyle \int \sec(3x+4)~dx$.

(a)

Evaluate $\displaystyle \int_0^1 \arctan x~dx$.

(b)

Evaluate $\displaystyle \int_1^2 \frac{x^2-2x-4}{x^3+2x}~dx$.

(a)

Evaluate $\displaystyle \int (\tan x + \sin^2 x \cos x) \sin 2x~dx$.

[Suggestion: Use the double angle formula for $\sin 2x$.]

(b)

Evaluate $\displaystyle \int \sqrt{1 - 4x^2}~dx$.

(a)

Which of the following improper integrals is/are convergent? Show why.

1. $\displaystyle \int_0^1 \frac{\ln(x+2)}{x \sqrt{x}}~dx$.
2. $\displaystyle \int_1^\infty x e^{-2x}~dx$.
3. $\displaystyle \int_1^\infty \frac{x \arctan x}{\sqrt[3]{x^7+7}}~dx$.

(b)

Find $\displaystyle \lim_{x \to \infty} e^{(x+\ln x)/\sqrt{x^2+x+1}}$.

(a)

$\displaystyle \sum_{n=1}^{\infty}(-1)^{n}\left(1+\frac{1}{n}+\frac{1}{n^{2}}\right)^{n}$.

(b)

$\displaystyle \sum_{n=0}^{\infty}\frac{n^{2}3^{n}\ln(n+2)}{2^{2n+2}}$.

(c)

$\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}(n^{2}+3)}{n^{3}+4}$.

(a)

Find the interval of convergence of the power series $$\sum_{n=0}^{\infty}\frac{n(x+1)^n}{{5}^n\sqrt{n^2+4}}.$$ Remember to check the endpoints if applicable.

(b)

Sketch the graph of the polar equation $r=3+2 \sin \theta$, and find the area which is both inside the graph and above the $x$-axis.

(a)

Find the first four nonzero terms of the Maclaurin series (i.e., power series centered at $0$) for the function $$g(x)=e^{-x^{2}}.$$

(b)

Find the sum of the first four terms of the Maclaurin series for the derivative of the function $$f(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{(n+1)^{3}}.$$

(c)

Use the answer in part (b) to approximate $f'(-\frac{1}{2})$ with an error of less than $0.01$.

(a)

Graph $4x^2 + 36 y^2 + 9 z^2 + 16x - 20 = 0$, and graph the trace of the answer to (a) in the $xy$-plane.

(b)

Sketch the portion of the graph of $y = 1-x^2$ which is in the first octant.