3.4.2What's up with a cusp?

Curve Constructor: Part Two

3-156.

Explain why a function must be continuous at $x = c$ to be differentiable at $x = c$ . The graph at right may help you.

Your teacher will provide you with a model.

Math Notes

Differentiability

A function, $f$ , is differentiable at $x = c$ if:

$f$ is continuous at $x = c$ and
$\lim\limits _ { x \rightarrow c } f ^ { \prime } ( x )$ exists

When a function is differentiable at all values of $x$ in its domain, then it is said to be differentiable everywhere. A function that is twice differentiable is both differentiable everywhere and $\lim\limits_ { x \rightarrow c } f ^ { \prime \prime } ( x )$ exists everywhere.

The graphs below illustrate functions that are not differentiable at $x = c$ . They each fail at least one of the conditions of differentiability listed above.

First quadrant, labeled Graph A, point on x axis labeled, c, increasing curve, opening up, stopping at closed point at, x = c.

The “sharp point” in Graph D is often called a cusp.

3-157.

FUNKY FUNCTIONS, Part One

Graph $f (x) = 2 + (0.1 − \left|x\right|)^2$ and rewrite $f$ as a piecewise-defined function.
Zoom in at $x = 0$ on your graphing calculator and carefully examine the shape of the graph at $x = 0$ . Does $f$ appear differentiable at $x = 0$ ? Why or why not?
To confirm whether or not $f (x) = 2 + (0.1 − \left|x\right|)^2$ is differentiable at $x = 0$ , we need to examine $f^\prime$ . Use your piecewise-defined function from part (a) to demonstrate which condition of differentiability fails at $x = 0$ .
Analyze $\lim\limits _ { h \rightarrow 0 ^ { - } } \frac { f ( x + h ) - f ( x ) } { h }$ and $\lim\limits _ { h \rightarrow 0 ^ { + } } \frac { f ( x + h ) - f ( x ) } { h }$ . Do they agree? Explain.

3-158.

Use the definition of the derivative as a limit to write the slope function for $f(x) = 4x^2 − 3$ . Then use your slope function to calculate $f ^\prime (11)$ and $f ^\prime (1000)$ .

3-159.

THE ABSOLUTE VALUE FUNCTION

Graph $f(x) =\left|x\right|$ on graph paper and without a calculator, sketch $y = f ^\prime (x)$ .
What happens to $y = f ^\prime (x)$ at the vertex of $f(x) =\left|x\right|$ ? Verify your observations by examining the slopes on both sides of the vertex.
Use your graphing calculator to determine the slope of $f(x) =\left|x\right|$ at the vertex. What happened?
Part of the reason most graphing calculators incorrectly determine slopes at the vertex of an absolute value graph, as well as other cusps, is because they use the symmetric difference quotient (Hanah’s Method) to calculate the slope of a tangent.

For $f(x) =\left|x\right|$ , use $f ^ { \prime } ( x ) = \frac { f ( x + h ) - f ( x - h ) } { 2 h }$ to calculate $f ^\prime (0)$ for $h = 0.1, -0.1,$ and $0.01$ . What do you notice? For functions like of $f(x) =\left|x\right|$ , some calculators falsely calculate the derivative at the cusp as 0. Why do you think this happens?

3-160.

CURVE CONSTRUCTOR, Part Two

Revisit our computer graphics program from problem 3-82. By using your arc tool, you can make four different types of arcs, shown at right.

The software user can use the arc tool twice and then connect the curves to make a continuous curve, such as the one shown at right. Draw every possible combination of two of these arcs together.

4 curves as follows: Increasing opening up, decreasing opening up, increasing opening down, decreasing opening down.

2 continuous curves as follows: left curve, decreasing opening up right. curve, increasing opening down.

Which of the combinations you made in part (a) must create a curve with a cusp?
Which of the combinations you made in part (a) have a point of inflection?
Place the combinations appropriately in the Venn diagram at right.

Venn diagram, with 2 overlapping circles, dividing the rectangle into 4 sections. The section contained in only the left circle, is labeled points of inflection,. The section contained in only the right circle, is labeled cusps. The overlapping section, is blank. The section outside of the circles, is labeled neither.

3-161.

Write the equations of the lines tangent to the curve $y = x^3 − 4x$ at both $x = 0$ and $x = 2$ . Then, determine the point of intersection for these two tangent lines. 3-161 HW eTool. Homework Help ✎

3-162.

For each graph below: Homework Help ✎

Trace the graph onto your paper and write a slope statement for $f$ .
Sketch the graph of $y = f ^\prime (x)$ using a different color.

3-163.

Write and evaluate a Riemann sum to estimate the area under the curve for $1 ≤ x ≤ 8$ using $10$ left endpoint rectangles of equal width given $y = \frac { 4 } { x }$ . Homework Help ✎

3-164.

Sketch the graph of a function $f$ given the following information about its slope function. Homework Help ✎

$f^\prime (x)>0$ for $1<x<4$
$f ^\prime (x) < 0$ for $x<1$ and $x>4$
$f ^\prime(x) = 0$ for $x=1$ , $x=4$

3-165.

What is $\frac { d y } { d x }$ for each of the following functions? You will need to rewrite each equation first. Homework Help ✎

$y = \sqrt [ 3 ] { \frac { 1 } { x ^ { 2 } } }$

$y = x\sqrt { x }$

$y = \sin^2(x) + \cos^2(x)$

$y = \large\frac { x + 2 } { x }$

3-166.

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

$f(x) = 3x^{1/2} − 7x$

$f(x) = \cos(x) + 2\sin(x)$

3-167.

Define $f$ and $g$ so that $h(x) = f(g(x))$ , for the following functions where $f(x)\ne x$ and $g(x)\ne x$ . Homework Help ✎

$h(x)=\cos(3x−11)$

$h ( x ) = \large\sqrt [ 3 ] { 2 ^ { x } - 1 }$

$h(x) = 3^{5−2x}$

3-168.

Evaluate each limit. If the limit does not exist due to a vertical asymptote, then add an approach statement stating if $y$ is approaching negative or positive infinity. Homework Help ✎

$\lim\limits _ { x \rightarrow 0 } \large\frac { \sqrt { 3 - x } - \sqrt { 3 } } { x }$

$\lim\limits _ { x \rightarrow 2 } \large\frac { x + 3 } { x - 4 }$

$\lim\limits _ { x \rightarrow 2 ^ { + } } \large\frac { x ^ { 2 } | x - 2 | } { x - 2 }$

$\lim\limits _ { x \rightarrow \infty } ( e ^ { - x } + 1 )$

3-169.

Use the definition of a derivative as a limit to write an equation for $f ^\prime$ if $f(x) = 2x + 9$ . Use the Power Rule to confirm your answer. Homework Help ✎

Calculus Third Edition

Differentiability