# 009B Sample Midterm 3

This is a sample, and is meant to represent the material usually covered in Math 9B 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

Divide the interval  ${\displaystyle [0,\pi ]}$  into four subintervals of equal length   ${\displaystyle {\frac {\pi }{4}}}$  and compute the right-endpoint Riemann sum of  ${\displaystyle y=\sin(x).}$

## Problem 2

State the fundamental theorem of calculus, and use this theorem to find the derivative of

${\displaystyle F(x)=\int _{\cos(x)}^{5}{\frac {1}{1+u^{10}}}~du.}$

## Problem 3

Find a curve  ${\displaystyle y=f(x)}$  with the following properties:

(i)   ${\displaystyle f''(x)=6x}$

(ii)   Its graph passes through the point  ${\displaystyle (0,1)}$  and has a horizontal tangent there.

## Problem 4

Compute the following integrals:

(a)   ${\displaystyle \int x^{2}\sin(x^{3})~dx}$

(b)   ${\displaystyle \int _{-{\frac {\pi }{4}}}^{\frac {\pi }{4}}\cos ^{2}(x)\sin(x)~dx}$

## Problem 5

Evaluate the indefinite and definite integrals.

(a)   ${\displaystyle \int x\ln x~dx}$

(b)   ${\displaystyle \int _{0}^{\pi }\sin ^{2}x~dx}$