Sample Final Exams

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Sample Final Exams

1 Sample Final, version A

(problem 1) Find the area bounded by the graphs of the parabola $y = x^2 + x - 3$ and the line $y = 2x - 1$ .

(problem 2) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = \tan (x)$ and $y = 0$ from $x = 0$ to $x = \pi /4$ about the $x$ -axis.
answer: $\displaystyle \pi \left (1-\frac {\pi }{4}\right )$

(problem 3) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = \sin (x)$ and $y = 0$ from $x = 0$ to $x = \pi$ about the $y$ -axis.
answer: $\displaystyle 2\pi ^2$

(problem 4) Find the length of the graph of $y = \frac {x^3}{12} + \frac {1}{x}$ from $x = 1$ to $x = 2$ .

(problem 5) Find the average value of the function $f(x) = xe^{2x}$ over the interval $(0,2)$ .
answer: $\displaystyle \frac 38 e^4 + \frac 18$

(problem 6) Solve the initial value problem $\displaystyle (1+t^2)\frac {dy}{dt} = 2ty, \; y(1) = 8$ .

(problem 7) Compute $\displaystyle \int x^3 \sqrt {1-x^2} \, dx$

(problem 8) Compute $\displaystyle \int \frac {3x-4}{x^3 +2x^2 +x} \, dx$

(problem 9) Find the sum of the infinite series, if it converges: $\displaystyle \sum _{n=1}^\infty \frac {1}{n^2 +n}$

(problem 10) Use the Integral Test to determine whether the infinite series converges or diverges: $\displaystyle \sum _{n=0}^\infty \frac {1}{1+n^2}$

(problem 11) Determine whether the infinite series converges or diverges: $\displaystyle \sum _{n=1}^\infty \frac {n+1}{2n+3}$

(problem 12) Determine whether the infinite series converges or diverges: $\displaystyle \sum _{n=1}^\infty \frac {n^2 + 5n + 12}{n^3 + 6n^2 + 2n}$

(problem 13) Find the interval of convergence of the power series $\displaystyle {\sum _{n=1}^\infty \frac {(x+2)^n}{(2n+1)3^n}}$

(problem 14) Find a power series representation for the function $\displaystyle f(x) = \frac {1}{2-x}$ . Be sure to include the interval of convergence in your answer.

(problem 15) Find the first four terms of the Taylor Series $f(x) = \sqrt x$ centered at $x = 4$ .

(problem 16) Find the Maclaurin series representation of the function $f(x) = x^2 e^{2x}$ .

2 Sample Final, version B

(problem 1) Find the area between the curves $y = \cos (x)$ and $y=\sin (x)$ over the interval $[0, \pi ]$ .

(problem 2) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = \sin (x)$ and $y = 0$ from $x = 0$ to $x = \pi /2$ about the $x$ -axis.
answer: $\displaystyle \frac {\pi ^2}{4}$

(problem 3) Find the volume of the solid obtained by revolving the region bounded by the graphs of $y = e^x$ and $y = 0$ from $x = 0$ to $x = 1$ about the $y$ -axis.
answer: $\displaystyle 2\pi$

(problem 4) Find the length of the graph of $y = 1+ x^{3/2}$ from $x = 0$ to $x = 4$ .

(problem 5) Find the average value of the function $f(x) = \frac {\ln (x)}{x}$ over the interval $(1,e)$ .
answer: $\displaystyle \frac {1}{2e-2}$

(problem 6) Solve the differential equation $y' = xy^2$ .

(problem 7) Compute $\displaystyle \int \frac {x^3}{ \sqrt {9+x^2}} \, dx$

(problem 8) Compute $\displaystyle \int \frac {x^2 + 5x + 2}{x^3 + 4x} \, dx$

(problem 9) Find the sum of the infinite series, if it converges: $\displaystyle \sum _{n=0}^\infty \frac {2^{2n}}{3^n}$

(problem 10) Use the integral test to determine whether the infinite series $\displaystyle \sum _{n=1}^\infty \frac {n}{e^n}$ converges or diverges.

(problem 11) Determine whether the infinite series converges or diverges: $\displaystyle \sum _{n=1}^\infty \frac {\sin ^2(n)}{n^3}$

(problem 12) Determine whether the infinite series converges absolutely, converges conditionally or diverges: $\displaystyle \sum _{n=0}^\infty (-1)^n \frac {n^2 + 2}{n^3 + 3}$

(problem 13) Find the interval of convergence of the power series $\displaystyle \sum _{n=1}^\infty \frac {(x-4)^n}{n2^n}$

(problem 14) Find a power series representation for the function $\displaystyle f(x) = \frac {3x^3}{4 + x^2}$ . Be sure to include the interval of convergence in your answer.

(problem 15) Find the first four terms of the Taylor Series $f(x) = \ln (x)$ centered at $x = 1$ .

(problem 16) Find the Maclaurin series representation for the function $f(x) = x\sin (3x)$ .

2024-09-27 14:08:14