3.4.1Do your points match? Do your slopes match?

Conditions for Differentiability

3-141.

Review the definition of a derivative. Explain why a limit is needed to calculate the slope of a tangent line. Be clear in your explanation and use a diagram to help demonstrate the limit.

3-142.

A line tangent to $f$ at $x = 2$ is shown at right. Does the slope of the tangent exist? Does $f ^\prime(2)$ exist? Why or why not?

First quadrant, dashed gray vertical line at x = 2, solid black curve, labeled f of x, coming from lower left, passing through x axis between 1 & 2, changing from concave up to concave down at the point (2, comma 2), rises almost vertically at, x = 2, then rising more slowly up & right.

3-143.

On your graphing calculator, graph $f ( x ) = \sqrt [ 3 ] { x }$ .

Describe what happens to the slope of $f$ at $x = 0$ .
Use the Power Rule to write an equation for $f ^\prime(x)$ and use it to calculate $f ^\prime (0)$ . Explain what happened. Does this confirm what you see in the graph?

3-144.

HELP!

Koy’s team needs your help to settle a dispute. For the function shown in bold at right, each of her team members drew a different tangent at $x = c$ .

Khi reasoned that if you follow $x$ as it approaches c from the left side, then his tangent makes the most sense. Karen says hers makes sense because hers is tangent to the right side of the curve at $x = c$ . Kirt argues that his is the best solution for both sides.

Koy does not think any of them are correct because the slope of a tangent is a limit, and a limit cannot exist if both sides do not agree.

Which team member is correct and why?

Your teacher will provide you with a model.

3-145.

If a function $f$ is continuous at $x = c$ and exactly one tangent line can be drawn in at $x = c$ (i.e. $\lim\limits _ { x \rightarrow c } f ^ { \prime } ( x )$ exists), then the function is differentiable at that point $(c, f(c))$ . The graphs below show examples of functions that are not differentiable at $x = c$ , each for a different reason.

With your team, discuss why the slope of a tangent line does not exist at $x = c$ in each case.
Sketch $y = f ^\prime (x)$ for each function. Pay close attention for what happens at $x = c$ .
In your own words, describe when a function, $f$ , is differentiable and when it is non-differentiable at a point, $x = c$ .

3-146.

ANTIDERIVATIVES

If $f ^\prime (x) = 8x^3 − 10x − 5$ , write a possible equation for $f$ . The equation for $f$ is called an antiderivative of $f^\prime$ .
Explain why there is always more than one antiderivative.

Math Notes

Antiderivatives

An antiderivative of a function $f$ is a function $F$ whose derivative is $f$ . That is $\frac { d } { d x }F(x) = f(x)$ .

The antiderivative of $f^\prime$ is $f$ . However, for the antiderivative of a function $f$ , we use a capital letter $F$ . For example, we write the antiderivative of $g$ as $G$ .

Since there are an infinite number of antiderivatives that are different only by a constant term, called a “family,” we add a constant “ $C$ ” to represent all of them. This is known as the general antiderivative (or simply the antiderivative).

For example: If $F(x) = 5x^3 + 3x^2 + C$ , then $F(x) = 5x^3 + 3x^2 + 5$ and $F(x) = 5x^3 + 3x^2 − 9$ are both antiderivatives of $f$ . This is why we write $F(x) = 5x^3 + 3x^2 + C$ .

3-147.

Write the general antiderivative $F$ for each function below. Test your solution by verifying that $F ^\prime (x) = f(x)$ . Homework Help ✎

$f(x) = −6x^5 + 12x^2$

$f(x)=3\cos(x)$

3-148.

Sketch the graph of a function that satisfies both of the following properties. Homework Help ✎

$\lim\limits _ { x \rightarrow 2 ^ { - } } f ^ { \prime } ( x ) = 0$
$\lim\limits_ { x \rightarrow 2 ^ { + } } f ^ { \prime } ( x ) = \infty$

3-149.

Review problem 3-109. Then on graph paper, graph the functions below and label their parts with increasing, decreasing, concave up, concave down, and point(s) of inflection. Then use the first and second derivatives to determine the intervals of increasing, decreasing, and concavity. Use different colors to represent concavity. Homework Help ✎

$f(x) = x^2 + 5x − 6$

$g(x) = \frac { 1 } { 3 }x^3 +3x^2 - x +5$

3-150.

Use the definition of a derivative to write an equation for $y^\prime$ if $y = \frac { 3 } { 4 }x^2 − 11x + 34$ . Confirm your answer with the Power Rule. Homework Help ✎

3-151.

State the domain and the range of each function below. Homework Help ✎

$y = \large\frac { x ^ { 2 } - x - 6 } { x + 1 }$

$y = \large\frac { x ^ { 2 } - x - 6 } { x + 2 }$

3-152.

Write an equation for $z^\prime$ if $z(x) = 3x^2 + 5x + 1$ . Then, write the equation of the tangent line in point slope form at $x = -2$ . Homework Help ✎

3-153.

Sketch a continuous curve which meets all of the following criteria: Homework Help ✎

$f ^\prime (x) > 0$ for all $x$

$f$ is concave down.

$f(2) = 1$

$f ^\prime(2) =\frac { 1 } { 2 }$

How many roots does $f$ have?
What can you say about the location of the root(s)?
What is $\lim\limits _ { x \rightarrow - \infty } f ( x )$ ?
Is it possible that $f^\prime (1) = 1$ ? $f ^\prime (1) =\frac { 1 } { 4 }$ ? For each case, explain why or why not.

3-154.

Sketch a graph and write the equation of the line tangent to the curve $y = 3 \sqrt [ 3 ] { x - 2 }$ at $x = 3$ . Then, locate another point on the curve with the same slope as the tangent 3-154 HW eToo. Homework Help ✎

3-155.

Re-examine your graph of $y = 3 \sqrt [ 3 ] { x - 2 }$ from problem 3-154. Homework Help ✎

As accurately as possible, draw a tangent to $y$ at $x = 2$ . Estimate its slope.
According to your derivative, $y^\prime$ what is the slope of the tangent at $x = 2$ ?

Calculus Third Edition

Antiderivatives