Chapter 10.6, Problem 53E

Television Advertising The cost, in thousands of dollars, of a 30-second television ad during the Super Bowl in the years 1970–2010 can be approximated by the following piecewise-linear function ( $t=0$ represents1970):⁸²

$C\left(t\right)=\{\begin{array}{l}31.1t+78\text{ if }0\le t<20\\ 90t-1,100\text{ if 2}0\le t<40.\end{array}$

a. Is C a continuous function of t? Why? [HINT: See Example 4 of Section 10.3]

b. Is C a differentiable faction of t? Compute ${\lim }_{t\to {20}^{-}}C\text{'}\left(t\right)$ and ${\lim }_{t\to {20}^{+}}{C}^{\prime }\left(t\right)$

[HINT: See the “Before we go on “discussion after Example 5]