M119 Midterm In-class Review

\hsize=6truein \newcount\qno \def\qn {\par\advance\qno by1\noindent\strut \hbox to\parindent{\the\qno.\hfil}\hangindent\parindent\ignorespaces} \font\bg=cmbx10 scaled \magstep2 \font\hg=cmr7 scaled 2985 \vglue 0pt \rm \qn Differentiate $\displaystyle f(x)=3\sqrt{x}+{1\over x^2}$.
\qn Differentiate $\displaystyle g(x)={2x+5\over x^2+9}$.
\qn Find $h''(x)$ where $h(x)=(6x^3-27)^5$.
\qn State the definition of derivative in terms of limits.
\qn Find $\displaystyle\lim_{x\to 10}{3x-1\over x-3}$.
\qn Find $\displaystyle\lim_{x\to\infty}{x^2+3x-6\over 4x^2-1}$
\qn Find $\displaystyle\lim_{x\to -1}{2x^2-x-3\over x+1}$
\qn Find the equation of the tangent line to the curve $y=\sqrt x$ at the point $(1,1)$.
\qn Let $f(x)=x^4-4x^3-3$. Where is $f$ increasing? Decreasing? Where is it concave up? Concave down?
\qn Let $$ f(x)=\cases{{\displaystyle {1\over x+1}}& for $x<1$\cr x& for $x\ge 1$\cr}$$ Determine all values of $x$ where the function $f(x)$ is discontinuous.
\qn Find the maximum and minimum values of $\displaystyle f(x)={2x\over x^2+4}$ on the interval $[-2,3]$.
\qn Sketch a graph of $f'(x)$ where $f(x)$ defined by the graph
A graph of f(x) is here.

\qn The demand equation for a monopolist is $p=200-3x$ and the cost function is $C(x)=75+80x-x^2$ where $0\le x\le 40$. Determine the value of $x$ and the corresponding price that maximize the profit.
\qn Given that $f(x)=\sqrt x$, use the general power rule to calculate the limit $\displaystyle\lim_{h\to 0} {f(4+h)-f(4)\over h}$.
\qn Suppose $f$ and $g$ are differentiable functions such that $f(3)=5$, $f'(3)=2$, $g(7)=3$, and $g'(7)=4$. What is $(f\circ g)'(7)$?