# 007B Sample Midterm 1

This is a sample, and is meant to represent the material usually covered in Math 7B for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

## Problem 1

Let  $f(x)=1-x^{2}$ .

(a) Compute the left-hand Riemann sum approximation of  $\int _{0}^{3}f(x)~dx$ with  $n=3$ boxes.

(b) Compute the right-hand Riemann sum approximation of  $\int _{0}^{3}f(x)~dx$ with  $n=3$ boxes.

(c) Express  $\int _{0}^{3}f(x)~dx$ as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.

## Problem 2

A population grows at a rate

$P'(t)=500e^{-t}$ where  $P(t)$ is the population after  $t$ months.

(a)   Find a formula for the population size after  $t$ months, given that the population is  $2000$ at  $t=0.$ (b)   Use your answer to part (a) to find the size of the population after one month.

## Problem 3

Evaluate the following integrals.

(a)   $\int x^{2}{\sqrt {1+x^{3}}}~dx$ (b)   $\int _{\frac {\pi }{4}}^{\frac {\pi }{2}}{\frac {\cos(x)}{\sin ^{2}(x)}}~dx$ ## Problem 4

Evaluate the following integrals.

(a)   $\int x^{2}e^{x}~dx$ (b)   $\int {\frac {5x-7}{x^{2}-3x+2}}~dx$ ## Problem 5

Find the area bounded by  $y=\sin(x)$ and  $y=\cos(x)$ from  $x=0$ to  $x={\frac {\pi }{4}}.$ 