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Chapter 2: Q. 2 (page 183)

2. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) The graph of a function that is continuous, but not differentiable, at $x=2$.
(b) The graph of a function that is left and right differentiable, but not differentiable, at $x=3$.
(c) The graph of a function that is differentiable on the interval $[-1,1]$ but not differentiable at the point $x=1$.

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Most popular questions from this chapter

A tomato plant given $x$ ounces of fertilizer will successfully bear $T (x)$ pounds of tomatoes in a growing season.

(a) In real-world terms, what does $T (5)$ represent and what are its units? What does $T' (5)$ represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than $20$ ounces are used, but inhibits production when more than $20$ ounces are used. When is $T (x)$ positive and when is $T (x)$ negative? When is $T' (x)$ positive and when is $T' (x)$ negative?

Stuart left his house at noon and walked north on Pine Street for $20$ minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was $10$ minutes early for his $12 : 40$ appointment. Stuart waited at the office for $10$ minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

For each function graphed in Exercises 65-68, determine the values of $x$ at which $f$ fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f (x) = {(1 - 4 x)}^{2} {(3 x^{2} + 1)}^{9}$

Use (a) the $h \to 0$ definition of the derivative and then

(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.

25. $f (x) = \frac{1}{x}, x = - 1$

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