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\centerline{Mathematics 135 -- Calculus 1}
\centerline{Precalculus Review Problems}
\centerline{August, 2016}
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\noindent
The following questions indicate the sorts of topics and questions that 
will appear on the 
precalculus diagnostic quiz that will be given at the start of 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:  $\left({x^{-3}\over y^{-2}}\right)^2 
\left({y\over x}\right)^4$
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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
