005 Sample Final A

This is a sample final, and is meant to represent the material usually covered in Math 8A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

 Question 1  Please circle either true or false,
    a. (True/False)In a geometric sequence, the common ratio is always positive.
    b. (True/False) A linear system of equations always has a solution.
    c. (True/False) Every function has an inverse.
    d. (True/False) Trigonometric equations do not always have unique solutions.
    e. (True/False) The domain of $f(x)=\tan ^{-1}(x)$ is all real numbers.
    f. (True/False) The function $\log _{a}(x)$ is defined for all real numbers.

 Question 2  Find the domain of the following function. Your answer should be in interval notation $f(x)={\frac {1}{\sqrt {x^{2}-x-2}}}$

 Question 3  Find f $\circ$ g and its domain if $f(x)=x^{2}+1\qquad g(x)={\sqrt {x-1}}$

 Question 4  Find the inverse of the following function $f(x)={\frac {3x}{2x-1}}$

 Question 5  Solve the following inequality. Your answer should be in interval notation. ${\frac {3x+5}{x+2}}\geq 2$

 Question 6  Factor the following polynomial completely,     $p(x)=x^{4}+x^{3}+2x-4$

 Question 7  Solve the following equation,      $2\log _{5}(x)=3\log _{5}(4)$

 Question 8  Solve the following equation,      $3^{2x}+3^{x}-2=0$

 Question 9  Solve the following system of equations

${\begin{aligned}2x+3y&=&1\\-x+y&=&-3\end{aligned}}$

 Question 10  Write the partial fraction decomposition of the following,

${\frac {x+2}{x^{3}-2x^{2}+x}}$

 Question 11  Solve the following equation in the interval $[0,2\pi )$

$\sin ^{2}(\theta )-\cos ^{2}(\theta )=1+\cos(\theta )$

 Question 12  Given that $\sec(\theta )=-2$ and $\tan(\theta )>0$, find the exact values of the remaining trig functions.

 Question 13  Give the exact value of the following if its defined, otherwise, write undefined.
$(a)\sin ^{-1}(2)\qquad \qquad (b)\sin \left({\frac {-32\pi }{3}}\right)\qquad \qquad (c)\sec \left({\frac {-17\pi }{6}}\right)$

 Question 14 Prove the following identity,

${\frac {1-\sin(\theta )}{\cos(\theta )}}={\frac {\cos(\theta )}{1+\sin(\theta )}}$

 Question 15  Find an equivalent algebraic expression for the following,

$\cos(\tan ^{-1}(x))$

 Question 16  Graph the following,

$-x^{2}+4y^{2}-2x-16y+11=0$

 Question 17  Graph the following function,

$f(x)=\log _{2}(x+1)+2$

Make sure to label any asymptotes, and at least two points on the graph.

 Question 18  Graph the following function,

$f(x)=\left({\frac {1}{3}}\right)^{x+1}+1$

Make sure to label any asymptotes, and at least two points on the graph.

 Question 19  Consider the following function,

$f(x)=-\sin \left(3x+{\frac {\pi }{2}}\right)+1$

     a. What is the amplitude?
     b. What is the period?
     c. What is the phase shift?
     d. What is the vertical shift?
     e. Graph one cycle of f(x). Make sure to label five key points.

 Question 20  Consider the following rational function,

$f(x)={\frac {x^{2}+x-2}{x^{2}-1}}$

     a. What is the domain of f?
     b. What are the x and y-intercepts of f?
     c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes?
     d. Graph f(x). Make sure to include the information you found above.

 Question 21  Find the sum

$5+9+13+\cdots +49$

 Question 22  Consider the following sequence,

$-3,1,-{\frac {1}{3}},{\frac {1}{9}},-{\frac {1}{27}},\cdots$

     a. Determine a formula for $a_{n}$, the n-th term of the sequence.
     b. Find the sum $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$