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\centerline{Mathematics 131 -- Calculus for Physical and Life Sciences 1}
\centerline{Precalculus Review Problems}
\centerline{September 1, 2004}
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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 Tuesday, September 7.
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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:  $\left({x^{-3}\over y^{-2}}\right)^2 
\left({y\over x}\right)^4$
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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 should his/her monthly sales be?
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\item{4.}  Factor:
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\itemitem{a)}  $7a^4  - 42a^2b^2 + 49 a^3 b$
\itemitem{b)}  $xe^{-2x} - x^3e^{-x}$
\itemitem{c)}  $6ac+3bc-4ad-2bd$
\itemitem{d)}  $3x^2 - 6x - 24$
\itemitem{e)}  $9x^2 - 16y^4$
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\item{5.}  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{6.}  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{7.} Let $f(x) = (x+1)^3$ and $g(x) = x^2 - 1$.  
\itemitem{a)} What is the 
function $f(g(x))$?  
\itemitem{b)} What is the function $g(f(x))$? 
\itemitem{c)} What is the function $f(x)g(x)$?  
\itemitem{d)} What is the domain of the function ${f(x)\over g(x)}$?
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\item{8.} 
\itemitem{a)} Let  $f(x) = x^3-2x^2+3$.  Simplify as far as possible:
${f(a+h) - f(a) \over h}$.
\itemitem{b)}  Same question for $f(x) = {1\over \sqrt{x}}$.  
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\item{9.}  Express in terms of the sine and cosine functions
and simplify:
$$g(x) = \csc^2(x) + \sec^2(x)$$
\bye