3.4.3How do I apply differentiability rules to functions?

Differentiability of Specific Functions

3-170.

FUNKY FUNCTIONS, Part Two

One of the reasons we need to analyze functions analytically is because graphs can be misleading. When viewed with a standard window, the graph of $f (x) = 2 + (0.1 − \left|x\right|)^2$ can look differentiable at $x = 0$ when it is not! Examine the graphs of the following “funky functions” and their equations to determine if they are differentiable at $x = c$ .

$f ( x ) = \frac { \operatorname { sin } ( x ) } { x }$ , $c = 0$
$f ( x ) = \left\{ \begin{array} { c l } { | x | ^ { x } } & { \text { for } x \neq 0 } \\ { 1 } & { \text { for } x = 0 } \end{array} \right.$ , $c = 0$
$f ( x ) = \left| x ^ { 3 } + 0.125 \right|$ , $c = −0.5$

3-171.

Graph the function defined by $g ( r ) = \left\{ \begin{array} { l l } { 0 } & { \text { for } - 2 \leq r \leq 2 } \\ { r ^ { 3 } + 2 r ^ { 2 } - 4 r - 8 } & { \text { otherwise } } \end{array} \right.$ .

Is $g$ continuous at $r = 2$ ? Explain your answer.
Is $g$ is differentiable at $r = 2$ ? Explain your answer.

3-172.

Sketch the graph of a function defined for all real numbers that satisfies all of the following properties. (There are many possible answers.)

$f (0) = −1$
$f (x)$ is not differentiable at $x = 2$ .
$f (x)$ is decreasing for all $x\ne2$ .

3-173.

Compare how distance and velocity are related with the scenarios in parts (a) and (b).

A ball is rolled down a ramp so that the distance it travels in feet at time $t$ is $d(t) = 6t^2 + 2t$ . Without your calculator, determine the velocity, $d ^\prime (t)$ , at $t = 1, 3,$ and $10$ seconds. Explain what concepts of calculus you applied in order to solve this problem.
When a football is kicked from the ground straight up into the air its velocity, measured in feet per second, is $v(t) = −32t + 80$ . On one set of axes, sketch a graph of the height function and a graph of the velocity function. Calculate the maximum height obtained the ball. Explain what calculus concepts you applied to solve this problem.
Both (a) and (b) involve distance and velocity. However, each part required a different solution method or approach. Describe the relationship between distance and velocity, as well as the derivative and area under a curve.

3-174.

Sketch the graph of $f(x) = x(x − 1)$ , and use the graph to sketch the graph of the slope function. 3-174 HW eTool. Homework Help ✎

3-175.

Use the definition of a derivative as a limit to write the slope function of $f(x) = 2x^2 − 3x + 4$ . Confirm your slope function with the Power Rule. Then use your slope function to calculate $f ^\prime (3)$ and $f ^\prime (−2)$ . Homework Help ✎

3-176.

Evaluate the following limits. (Hint: Review your solution for problem 3-92 first!) Homework Help ✎

$\lim\limits_ { h \rightarrow 0 } \large\frac { ( x + h ) ^ { 5 } - x ^ { 5 } } { h }$

$\lim\limits_ { h \rightarrow 0 } \large\frac { 2 \sqrt { x + h } - 2 \sqrt { x } } { h }$

3-177.

Write a Riemann sum to estimate the area under the curve for $0 ≤ x ≤ 2$ using $n$ left endpoint rectangles given $g(x) = −x^2 + 4$ . Homework Help ✎

Calculate the sum for $n = 20$ .
How can you use your result to estimate the area under the curve for $−2\le x\le0$ ? What about the area under the curve for $−2\le x\le2$ ?

3-178.

What is the end-behavior function for each of the following functions? Homework Help ✎

$f ( x ) = \large\frac { 2 x ^ { 2 } - 3 x + 1 } { x + 2 }$

$g(x) = \frac { 1 } { x }+ \sin(x)$

$h ( x ) = \large\frac { \operatorname { sin } ( x ) } { x }$

3-179.

For each function below, solve for $x$ . Homework Help ✎

$y = -10x + 8$

$y = (x + 4)^2$

$y = x^3 + 2$

$y=3\sin(x)$

Assuming that no domains are restricted, which of the functions above has an inverse that is also a function? Give a reason for your answer.

3-180.

Let $f(x) = x^3 − 3x^2 − 24x + k$ where $k$ is a constant. Over what intervals is $f$ increasing? Homework Help ✎

3-181.

Calculate the volume of the solid created when the semi-circular flag at right is rotated about the pole. Describe the rotated solid. Homework Help ✎

Horizontal segment, with diameter of semicircle on the right 3 fourths of the segment, rectangle inscribed in semicircle, top edge labeled 8, right edge labeled 6, regions outside of rectangle, but inside semicircle are shaded.

Calculus Third Edition