Question 1

Given the equation $$9x^2$$ + $$y^2 - 18x = 0$, what is the standard form of the ellipse?

Accepted Answer

$$\frac{(x-1)^2$}{1} + \frac{y^2$}{9} = 1$$

Question 2

For the ellipse $$\frac{(x-1)^2$}{1} + \frac{y^2$}{9} = 1$$, what are the coordinates of the foci?

Accepted Answer

$$(1, \pm 2\sqrt{2})$$

Question 3

Find the domain of $$g(x) = \frac{x+4}{x^3$ - 4x}.$$

Accepted Answer

$$(-\infty, -2) \cup (-2, 0) \cup (0, 2) \cup (2, \infty)$$

Question 4

Given $$f(x) = 2x^2$ - 3x + 1$$, find and simplify the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$4x + 2h - 3$$

Question 5

Simplify the expression $$\frac{(x+4$$)^{1/3}$$ - 2x(x+4$$)^{-2/3}$$}{x+4} so $$that only positive exponents appear.

Accepted Answer

$$\frac{-x+4}{(x+4$$)^{4/3}$$}$$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is 100, how many bacteria will there be after 9 hours?

Accepted Answer

$$800$$

Question 7

The half-life of radium is 1690 years. If you start with 10 grams, how much will remain after 50 years? (Use $$A(t) = A_0$ e$$^{kt}$$ with k$$ = \frac{\ln(1/2)}{1690}$$)

Accepted Answer

$$9.8$$ grams

Question 8

Which of the following is the correct factored form of $$f(x) = x^3$ + 13x^2$ + 57x + 85$$?

Accepted Answer

$$(x+5)(x^2 + 8x + 17$$)$$

Question 9

Given the zeros $$-1 ($$multiplicity 1) and $$3 ($$multiplicity 2), form a degree 3 polynomial with leading coefficient 1.

Accepted Answer

$$f(x) = (x+1)(x-3)^2$

Question 10

For the function $$f(x) = 4$$^{x-1}$$ + 2$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 11

A box with a square base and no top is to be made from a 24-inch by 24-inch piece of cardboard by cutting equal squares from each corner and folding up the sides. What is the volume $$V(x) as a $$function of $$x$$, the side of the square cut out?

Accepted Answer

$$V(x) = x(24-2x)^2$

Question 12

Given the function $$R(p) = -4p^2$ + 4000p$$, what price $$p$$ maximizes the revenue?

Accepted Answer

$$p = 500$$

Question 13

Which interval describes where the function $$f(x) is $$increasing on $$(-2,0)$$, given the graph in the study material?

Accepted Answer

$$f(x) is $$increasing on $$(-2,0)$$

Question 14

What is the center of the ellipse $$\frac{(x-1)^2$}{1} + \frac{y^2$}{9} = 1$$?

Accepted Answer

$$(1,0)$$

Question 15

If $$f(x) = x^3$ + 13x^2$ + 57x + 85$$ has a zero at $$x = -5$$, what are the other zeros?

Accepted Answer

$$-4 \pm i$$

Question 16

Which quadrant does the angle $$\frac{25\pi}{3}$$ lie in?

Accepted Answer

Quadrant I

Precalculus Final Exam Review: Key Concepts and Problem Types

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