Some extra review problems

These problems concern optimization, graphing, and l'Hospital's rule.
  1. Sketch the graphs of the following functions:
    1. $\frac{1}{x^{2}-1}$
    2. $x^{2}+2x+3$
    3. $x^{3}-3x^{2}+4x$ (this one won't have any critical points, but it still has inflection points!)
  2. Sketch graph that have the following descriptions:
    1. $f$ is concave up and increasing on $(-\infty,0)$, concave down and increasing on $(0,\infty)$, has a vertical asymptote at $x=0$ with $\lim_{x\rightarrow 0^{-}}f(x)=\infty$ and $\lim_{x\rightarrow 0^{+}}f(x)=-\infty$, and a horizontal asymptote $y=0$ (so $\lim_{x\rightarrow \pm\infty} f(x)=0$
    2. $f$ is concave down from $(-\infty,0)$ and concave up from $(0,\infty)$, increasing on $(-\infty, -2)$ and $(2,\infty)$, decreasing on $(-2,2)$. Also, $f(-2)=1$, $f(2)=-1$, and $f(0)=0$.
    3. $f$ is concave down on $(-\infty,-1)$, concave up on $(-1,1)$, concave up on $(1,\infty)$, has vertical asymptotes at $x=\pm 1$ with $\lim_{x\rightarrow -1^{-}}f(x)=-\infty$, $\lim_{x\rightarrow -1^{+}}f(x)=\lim_{x\rightarrow 1^{+}}f(x)=\lim_{x\rightarrow 1^{-}}f(x)=+\infty$, $f$ is increasing on $(-\infty,-2)\cup (0,1)\cup (1,\infty)$, decreasing on $(-2,-1)\cup (-1,0)$, and has oblique asymptote $y=x$.
  3. Find the base and height of an isociles triangle of maximum area so that its perimeter is 1.
  4. Find the rectangle of largest area enclosed in a right triangle of height 3 and base 4.
  5. Find the maximum area of a rectangle enclosed between the $x$ axis and the graph of $1-x^{2}$ on the interval $[-1,1]$.
    Compute the following limits:
    1. $\lim_{x\rightarrow 0}\frac{\cos \sin x-1}{x^{2}}$
    2. $\lim_{x\rightarrow\infty} x\sin\frac{1}{x}$
    3. $\lim_{x\rightarrow 0} \frac{1}{\sin x}-\frac{1}{x}$
    4. $\lim_{x\rightarrow 0} (\sin x)^{\frac{1}{\ln x}}$