007B Sample Final 2

This is a sample, and is meant to represent the material usually covered in Math 7B 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 

(a) State both parts of the Fundamental Theorem of Calculus.

(b) Evaluate the integral

$\int _{0}^{1}{\frac {d}{dx}}{\bigg (}e^{\tan ^{-1}(x)}{\bigg )}dx$

(c) Compute

${\frac {d}{dx}}\int _{1}^{\frac {1}{x}}\sin t~dt$

 Problem 2 

Consider the area bounded by the following two functions:

$y=\cos x{\text{ and }}y=2-\cos x,~0\leq x\leq 2\pi .$

(a) Sketch the graphs and find their points of intersection.

(b) Find the area bounded by the two functions.

 Problem 3 

Find the volume of the solid obtained by rotating the region bounded by the curves  $y=x$  and  $y=x^{2}$  about the line  $y=2.$

 Problem 4 

Evaluate  $\int _{0}^{5}|x-1|~dx.$  (Suggestion: Sketch the graph.)

 Problem 5 

Evaluate the following integrals:

(a)  $\int {\frac {dx}{x^{2}{\sqrt {x^{2}-16}}}}$

(b)  $\int _{-\pi }^{\pi }\sin ^{3}x\cos ^{3}x~dx$

(c)  $\int _{0}^{1}{\frac {x-3}{x^{2}+6x+5}}~dx$

 Problem 6 

Evaluate the following integrals or show that they are divergent:

(a)  $\int _{1}^{\infty }{\frac {\ln x}{x^{4}}}~dx$

(b)  $\int _{0}^{1}{\frac {3\ln x}{\sqrt {x}}}~dx$

 Problem 7 

Suppose the size of a population evolves according to the logistic equation:

${\frac {dN}{dt}}=1.5N{\bigg (}1-{\frac {N}{100}}{\bigg )}.$

(a) Find all equilibria, and by using the graphical approach, discuss the stability of the equilibria.

(b) Find the eigenvalues associated with the equilibria, and use the eigenvalues to determine the stability of the equilibria.