I am currently pursuing a Bachelor's degree in Applied Math and Computational Sciences. My dad instilled a love of math in me from a very young age by explaining things in a way that made them fun. When I was younger I was always begging for him to give me long division problems to solve in my head.
Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$.(GRAPH CAN'T COPY)
Use the graph shown in the figure. (GRAPH CAN'T COPY)Identify or sketch each of the quantities on the figure.(a) $f(1)$ and $f(4)$(b) $f(4)-f(1)$(c) $y=\frac{f(4)-f(1)}{4-1}(x-1)+f(1)$
Use the graph shown in the figure. (GRAPH CAN'T COPY)Insert the proper inequality symbol $(<\text { or }>)$ between the given quantities.(a) $\frac{f(4)-f(1)}{4-1} \quad \frac{f(4)-f(3)}{4-3}$(b) $\frac{f(4)-f(1)}{4-1} \quad f^{\prime}(1)$
Show that the slopes of the graphs of $f$ and $f^{-1}$ are reciprocals at the given points.$$\begin{aligned}&f(x)=x^{3} \quad \quad \left(\frac{1}{2}, \frac{1}{8}\right) \\&f^{-1}(x)=\sqrt[3]{x} \quad \quad \left(\frac{1}{8}, \frac{1}{2}\right)\end{aligned}$$
Show that the slopes of the graphs of $f$ and $f^{-1}$ are reciprocals at the given points.$$\begin{aligned}&f(x)=3-4 x \quad \quad (1,-1)\\&f^{-1}(x)=\frac{3-x}{4} \quad \quad (-1,1)\end{aligned}$$
