005 Sample Final A

This is a sample final, and is meant to represent the material usually covered in Math 5. 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 $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$

Contributions to this page were made by Matthew Lee