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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=1-x^{2}} .

(a) Compute the left-hand Riemann sum approximation of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{0}^{3}f(x)~dx}   with  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=3}   boxes.

(b) Compute the right-hand Riemann sum approximation of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{0}^{3}f(x)~dx}   with  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=3}   boxes.

(c) Express  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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)   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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)   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {\frac {5x-7}{x^{2}-3x+2}}~dx}

 Problem 5 

Find the area bounded by  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\sin(x)}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\cos(x)}   from  $x=0$  to  $x={\frac {\pi }{4}}.$

Contributions to this page were made by Kayla Murray