009C Sample Final 1

This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Compute

a) $\lim _{n\rightarrow \infty }{\frac {3-2n^{2}}{5n^{2}+n+1}}$

b) $\lim _{n\rightarrow \infty }{\frac {\ln n}{\ln 3n}}$

 Problem 2 

We would like to evaluate

${\frac {d}{dx}}{\bigg (}\int _{-1}^{x}\sin(t^{2})2tdt{\bigg )}$.

a) Compute $f(x)=\int _{-1}^{x}\sin(t^{2})2tdt$.

b) Find $f'(x)$.

c) State the fundamental theorem of calculus.

d) Use the fundamental theorem of calculus to compute ${\frac {d}{dx}}{\bigg (}\int _{-1}^{x}\sin(t^{2})2tdt{\bigg )}$ without first computing the integral.

 Problem 3 

Consider the area bounded by the following two functions:

$y=\sin x$ and $y={\frac {2}{\pi }}x$

a) Find the three intersection points of the two given functions. (Drawing may be helpful.)

b) Find the area bounded by the two functions.

 Problem 4 

Compute the following integrals.

a) $\int e^{x}(x+\sin(e^{x}))~dx$

b) $\int {\frac {2x^{2}+1}{2x^{2}+x}}~dx$

c) $\int \sin ^{3}x~dx$

 Problem 5 

Consider the solid obtained by rotating the area bounded by the following three functions about the $y$-axis:

$x=0$, $y=e^{x}$, and $y=ex$.

a) Sketch the region bounded by the given three functions. Find the intersection point of the two functions:

$y=e^{x}$ and $y=ex$. (There is only one.)

b) Set up the integral for the volume of the solid.

c) Find the volume of the solid by computing the integral.

 Problem 6 

Evaluate the improper integrals:

a) $\int _{0}^{\infty }xe^{-x}~dx$

b) $\int _{1}^{4}{\frac {dx}{\sqrt {4-x}}}$

 Problem 7 

a) Find the length of the curve

$y=\ln(\cos x),~~~0\leq x\leq {\frac {\pi }{3}}$.

b) The curve

$y=1-x^{2},~~~0\leq x\leq 1$

is rotated about the $y$-axis. Find the area of the resulting surface.