Allan H.

Graph the curve $y=e^{x}$ in the viewing rectangles $[-1,1]$ by $[0,2],[-0.5,0.5]$ by $[0.5,1.5],$ and $[-0.1,0.1]$ by $[0.9,1.1] .$ What do you notice about the curve as you zoom in toward the point $(0,1) ?$

5-8 Find an equation of the tangent line to the curve at the given point.
$y=x^{3}-3 x+1,(2,3)$

5-8 Find an equation of the tangent line to the curve at the given point.
$y=\frac{2 x+1}{x+2}, \quad(1,1)$

Find an equation of the tangent line to the graph of
$y=g(x)$ at $x=5$ if $g(5)=-3$ and $g^{\prime}(5)=4$

If the tangent line to $y=f(x)$ at $(4,3)$ passes through the point $(0,2),$ find $f(4)$ and $f^{\prime}(4)$

Sketch the graph of a function $g$ for which
$$\begin{array}{l}{g(0)=g(2)=g(4)=0, g^{\prime}(1)=g^{\prime}(3)=0} \\ {g^{\prime}(0)=g^{\prime}(4)=1, g^{\prime}(2)=-1, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and }} \\ {\lim _{x \rightarrow-\infty} g(x)=-\infty}\end{array}$$