Mathematics 135 – Calculus 1 Answers for Precalculus Review Problems August 28, 2013

1. Find all values of x that satisfy the given inequality or inequalities:

a) −4x ≥ 20

b) x + 1 > 4, or x + 2 < −1

c) x + 3 > 1 and x − 2 < 1

2.

a) Rewrite using positive exponents only: \frac{x^{−1/3}}{x^{1/2}}

b) Simplify: (x^{2}y^{−3})(x^{−5}y^{3})

c) Simplify: (\frac{x^{−3}}{y^{−2}} )^{2} (\frac{y}{x})^{4}

3. A salesperson’s monthly commission is 15% on all sales over $12000. If the goal is to make a commission of at least $3000 per month, what should his/her monthly sales be?

4. Factor:

a) 7a^{4} − 42a^{2}b^{2} + 49a^{3}b

b) xe^{−2x} − x^{3}e^{−x}

c) 6ac + 3bc − 4ad − 2bd

d) 3x^{2} − 6x − 24

e) 9x^{2} − 16y^{4}

5. Solve for x:

a) x^{2} + x − 12 = 0

b) 4x^{3} + 2x^{2} − 2x = 0

c) 8x^{2} − 8x − 3 = 0

6. Simplify:

a)

\frac{2a^{2} − 2b^{2}}{b − a} · \frac{4a + 4b}{a^{2} + 2ab + b^{2}}

b)

\frac{58}{3(3 + t^{2})} + \frac{1}{3}

c)

\frac{2x}{2x − 1} − \frac{3x}{2x + 5}

\frac{1 + \frac{1}{x}}{1 − \frac{1}{x^{2}}}

e)

\frac{2x(x + 1)^{−1/2} − (x + 1)^{1/2}}{x^{2}}

7. Let f(x) = (x + 1)^{3} and g(x) = x^{2} − 1.

a) What is the function f(g(x))?

b) What is the function g(f(x))?

c) What is the function f(x)g(x)?

d) What is the domain of the function \frac{f(x)}{g(x)}?

8.

a) Let f(x) = x^{3} − 2x^{2} + 3. Simplify as far as possible: \frac{f(a+h)−f(a)}{h}.

b) Same question for f(x) = \frac{1}{\sqrt{x}}.

9. Express in terms of the sine and cosine functions and simplify:

g(x) = csc^{2}(x) + sec^{2}(x)