Mathematics 133 – Intensive Calculus for Science 1 Precalculus Review Problems August 31, 2005

The following problems will not be collected or graded. They are similar to questions on the first diagnostic quiz to be given in class on Friday, September 2.

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}}{−27y^{−6}} )^{−2/3}

d) 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 monthly sales figure should he or she attain?

4. The diameter x in inches of a batch of ball bearings manufactured by PAR Mfg. satisfies the inequality |x − .1| ≤ .001. What are the largest and smallest diameters a ball bearing in the batch can have?

5. Perform the indicated operations and simplify:

a) x − (2x − (−x − (1 − x)))

b) 2(t + \sqrt{t})^{2} − 2t^{2}

c) (2x^{2} − 1)(x) − x^{2}(x + 2)

d) 5x^{2}(3x + 1)^{4}(6x) + (3x + 1)^{5}(2x)

6. Factor out the greatest common factor:

a) 7a^{4} − 42a^{2}b^{2} + 49a^{3}b

b) xe^{−2x} − x^{3}e^{−x}

7. Factor:

a. 9x^{2} − 16y^{4} b. 3x^{2} − 6x − 24 c. 6ac + 3bc − 4ad − 2bd 8. Solve for x:

a. x^{2} + x − 12 = 0 b. 4x^{3} + 2x^{2} − 2x = 0 c. 8x^{2} − 8x − 3 = 0

9. 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}

d.

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

e.

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

10.

a. On a set of coordinate axes, plot the points P = (1, 3) and Q = (4, 7). Determine

the distance d(P, Q) between them. b. What is the Cartesian equation of the circle with center (h, k) = (4, 2) and radius

r = 6. c. The equation x^{2} + y^{2} + x − 6y = 0 defines a circle in the plane. Find its center

and radius. d. Do the points (3, 4), (−3, 7), (−6, 1), (0, −2) form the vertices of a square in the

plane? Why or why not?