007B Sample Midterm 2

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 

This problem has three parts:

(a) State both parts of the fundamental theorem of calculus.

(b) Compute   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {d}{dx}}\int _{2}^{\cos(x)}\sin(t)~dt} .

(c) Evaluate  $\int _{0}^{\pi /4}\sec ^{2}x~dx$.

 Problem 2 

Evaluate

(a)   $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

(b)   $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

 Problem 3 

The population density of a plant species is  $f(x)$  individual per square meter, where  $x$  is the distance from the river, with  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)\geq 0}   for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\leq 100}   and  $f(x)=0$  for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\geq 100.} Construct a definite integral to calculate the number of plants along a section of the river of length  $L.$

 Problem 4 

Find the area of the region bounded by  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\ln x,~y=0,~x=1,}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=e.}

 Problem 5 

Evaluate the integral:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {\frac {4x}{(x+1)(x^{2}+1)}}~dx}

Contributions to this page were made by Kayla Murray