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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Precalculus Review Problems}
\centerline{August 31, 2005}
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\noindent
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.
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\item{1.}  Find all values of $x$ that satisfy the given inequality
or inequalities:
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\itemitem{a)}  $-4x \ge 20$
\itemitem{b)} $x + 1 > 4$, or $x + 2 < -1$
\itemitem{c)}  $x + 3 > 1$ and $x - 2 < 1$
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\item{2.}  
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\itemitem{a)} Rewrite using positive exponents only: ${x^{-1/3} \over x^{1/2}}$
\itemitem{b)} Simplify:  $(x^2y^{-3})(x^{-5}y^3)$
\itemitem{c)} Simplify:  $\left({x^3\over -27y^{-6}}\right)^{-2/3}$
\itemitem{d)} Simplify:  $\left({x^{-3}\over y^{-2}}\right)^2 
\left({y\over x}\right)^4$
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\item{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?
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\item{4.}  The diameter $x$ in inches of a batch of ball bearings
manufactured by PAR Mfg. satisfies the inequality $|x - .1| \le .001$.
What are the largest and smallest diameters a ball bearing in 
the batch can have?
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\item{5.}  Perform the indicated operations and simplify:
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\itemitem{a)}  $x - (2x - (-x - (1 - x)))$
\itemitem{b)}  $2(t + \sqrt{t})^2 - 2t^2$
\itemitem{c)}  $(2x^2-1)(x) - x^2(x + 2)$
\itemitem{d)}  $5x^2(3x+1)^4(6x) +(3x+1)^5(2x)$
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\item{6.}  Factor out the greatest common factor:
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\itemitem{a)}  $7a^4  - 42a^2b^2 + 49 a^3 b$
\itemitem{b)}  $xe^{-2x} - x^3e^{-x}$
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\item{7.}  Factor:
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\itemitem{a.}  $9x^2 - 16y^4$
\itemitem{b.}  $3x^2 - 6x - 24$
\itemitem{c.}  $6ac+3bc-4ad-2bd$
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\item{8.}  Solve for $x$:
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\itemitem{a.}  $x^2 + x - 12 = 0$
\itemitem{b.}  $4x^3 + 2x^2 - 2x = 0$
\itemitem{c.}  $8x^2 - 8x - 3 = 0$
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\item{9.}  Simplify:
\itemitem{a.}  $${2a^2 - 2b^2\over b - a}\cdot {4a+4b\over a^2 + 2ab + b^2}$$
\itemitem{b.}  $${58\over 3(3+t^2)} + {1\over 3}$$
\itemitem{c.}  $${2x\over 2x - 1} - {3x \over 2x + 5}$$
\itemitem{d.}  $${ 1 + {1\over x}\over 1 - {1\over x^2}}$$
\itemitem{e.}  $${2x(x+1)^{-1/2}-(x+1)^{1/2}\over x^2}$$
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\item{10.} 
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\itemitem{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.
\itemitem{b.}  What is the Cartesian equation of the circle with
center $(h,k) = (4,2)$ and radius $r = 6$.
\itemitem{c.}  The equation $x^2 + y^2 + x - 6y = 0$
defines a circle in the plane.  Find its center and radius.
\itemitem{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?
\bye