§1.7 Limits, continuity, and differentiability (continued)

Summary

Consider the graph of the function $y = p(x)$ that is provided in Figure 1.7. Assume that each portion of the graph of $p$ is a straight line, as pictured.

• State all values of $a$ for which $\lim_{x \to a} p(x)$ does not exist. - State all values of $a$ for which $p$ is not continuous at $a$. - State all values of $a$ for which $p$ is not differentiable at $x = a$. - On the axes provided in Figure 1.7, sketch an accurate graph of $y = p'(x)$.

For each of the following prompts, give an example of a function that satisfies the stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why. - A function $f$ that is continuous at $a = 2$ but not differentiable at $a = 2$. - A function $g$ that is differentiable at $a = 3$ but does not have a limit at $a=3$. - A function $h$ that has a limit at $a = -2$, is defined at $a = -2$, but is not continuous at $a = -2$. - A function $p$ that satisfies all of the following: -$p(-1) = 3$ and $\lim_{x \to -1} p(x) = 2$-$p(0) = 1$ and $p'(0) = 0$-$\lim_{x \to 1} p(x) = p(1)$ and $p'(1)$ does not exist

Let $h(x)$ be a function whose derivative $y= h'(x)$ is given by the graph on the right in Figure 1.7. - Based on the graph of $y = h'(x)$, what can you say about the behavior of the function $y = h(x)$? - At which values of $x$ is $y = h'(x)$ not defined? What behavior does this lead you to expect to see in the graph of $y=h(x)$? - Is it possible for $y = h(x)$ to have points where $h$ is not continuous? Explain your answer. - On the axes provided at left, sketch at least two distinct graphs that are possible functions $y = h(x)$ that each have a derivative $y = h'(x)$ that matches the provided graph at right. Explain why there are multiple possibilities for $y = h(x)$.

Consider the function $g(x) = \sqrt{|x|}$. - Use a graph to explain visually why $g$ is not differentiable at $x = 0$. - Use the limit definition of the derivative to show that

. - Investigate the value of $g'(0)$ by estimating the limit in (b) using small positive and negative values of $h$. For instance, you might compute $\frac{\sqrt{|-0.01|}}{0.01}$. Be sure to use several different values of $h$ (both positive and negative), including ones closer to 0 than 0.01. What do your results tell you about $g'(0)$? - Use your graph in (a) to sketch an approximate graph of $y = g'(x)$.