008A 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

Find $\displaystyle f^{-1}(x)$ for $\displaystyle f(x) = \log_3(x+3)-1$

Question 2

Find f(5) for f(x) given in problem 1.

Question 3

a) Find the vertex, standard graphing form, and X-intercept for $\displaystyle x = -3y^2-6y+2$
b) Sketch the graph. Provide the focus and directrix.

Question 4

Solve. Provide your solution in interval notation. $\displaystyle (x-4)(2x+1)(x-1)<0$

Question 5

Graph the system of inequalities $\displaystyle y < \vert x\vert +1$           $\displaystyle x^2 + y^2 \le 9$

Question 6

Sketch $\displaystyle 4x^2 + 9(y + 1)^2 = 36$ . Give coordinates of each of the 4 vertices of the graph.

Question 7

Solve $\displaystyle 2\vert 3x-4\vert -7 = 7$

Question 8

Given a sequence $\displaystyle 27, 23, 19, 15, \ldots$ use formulae to compute $\displaystyle S_{10}$ and $\displaystyle A_{15}$ .

Question 9

a) List all the possible rational zeros of the function $\displaystyle f(x) = x^4 + 5x^3 - 27x^2 +31x -10$
b) Find all the zeros, that is, solve f(x) = 0

Question 10

Graph the function. Give equations of any asymptotes, and list any intercepts $\displaystyle y = \frac{x-1}{2x+2}$

Question 11

Decompose into separate partial fractions $\displaystyle \frac{3x^2+6x+7}{(x+3)^2(x-1)}$

Question 12

Find and simplify the difference quotient $\displaystyle \frac{f(x+h)-f(x)}{h}$ for f(x) = $\displaystyle \frac{2}{3x+1}$

Question 13

Set up, but do not solve, the following word problem.
One evening 1500 concert tickets were sold for the Riverside Jazz Festival. Tickets cost $25 for a covered pavilion seat and$15 for a lawn seat. Total receipts were \$28,500. How many of each type of ticket were sold?

Question 14

Compute $\displaystyle \displaystyle{\sum_{n=1}^\infty 5\left(\frac{3}{5}\right)^n}$

Question 15

a) Find the equation of the line passing through (3, -2) and (5, 6).
b) Find the slope of any line perpendicular to your answer from a).

Question 16

Solve. $\displaystyle \log_6(x+2)+\log_6(x-3) = 1$

Question 17

Compute the following trig ratios: a) $\displaystyle \sec \frac{3\pi}{4}$       b) $\displaystyle \tan \frac{11\pi}{6}$       c) $\displaystyle \sin(-120)$

Question 18

Compute $\displaystyle \cos(\arctan\frac{5}{3})$

Question 19

Compute $\displaystyle \arcsin -\frac{\sqrt{3}}{2}$ . Provide your answer in radians.