3.3.2Did you notice the curve on that function?

The Shape of a Curve

3-96.

For each graph below, answer the following two questions using complete sentences. Remember that for some graphs, the conditions change as $x$ increases.

As $x$ increases, does $f$ increase or decrease?
As $x$ increases, does the slope of $f$ increase or decrease?

3-97.

Consider the four curves of calculus that you created in problem 3-86. Sketch one of the four curves for each part below.

The entire curve has a positive, increasing slope.
The entire curve has a positive, decreasing slope.
The entire curve has a negative, increasing slope.
The entire curve has a negative, decreasing slope.

Math Notes

Curve Analysis

A function is increasing on an interval if $y$ increases as $x$ increases. Likewise, a function is decreasing on an interval if $y$ decreases as $x$ increases.

One way to determine if a function is increasing or decreasing is to determine the domain on which the slope of its tangent lines, $\frac { d y } { d x }$ , is positive or negative.

For example give $y = f(x)$ shown at right, $f$ is increasing for $(a, c)\cup(e, \infty)$ because $\frac { d y } { d x }> 0$ and decreasing for $(−\infty, a)\cup(c, e)$ because $\frac { d y } { d x }< 0$ .

The coordinate point where a function changes from increasing to decreasing or vice versa is called an extrema. This extreme point is either a maximum or a minimum. The function graphed at right has extrema at $x = a, x = c,$ and $x = e$ .

Note: If a function is increasing or decreasing over its entire domain, the function is monotonic.

When the slopes of the tangent lines increase on an interval, the graph is concave up because it curves up. However, when the slopes decrease on an interval, the graph is concave down because it curves down.

The graph of $y = f(x)$ , shown above, is concave up for $(−\infty, b)\cup(d, \infty)$ and concave down for $(b, d)$ .

A coordinate point where the concavity changes is called a point of inflection. At this point, the curve changes from concave up to concave down or vice versa.

3-98.

Review the graphs from problem 3-96. Which of the curves are concave up, which are concave down, and which are sometimes concave up and sometimes concave down?

3-99.

The graph at right is the slope function of $f$ . Examine the graph carefully as you complete the parts below.

At what $x$ -values is $f^\prime(x)=0$ ? What happens to $f$ at these $x$ -values?
Identify the part(s) of the domain on which $f$ is increasing. Explain which graphical clues you used to determine this.
At what $x$ -value(s) does $f$ have a local minimum (i.e. the lowest point on a local region of a curve)? How can you tell? Explain which graphical clues you used to determine this.
Approximate the interval(s) over which $f^\prime$ is increasing. What happens to $f$ at these $x$ -values?

3-100.

Describe the difference between stating, “ $f$ is increasing” and “ $f^\prime$ is increasing.” Which of the two statements indicates that $f^\prime$ is positive?

3-101.

Draw the following function, given its slope statement below.

The slope starts close to zero. When $x = −5$ , the slope increases quickly. Then at $x = 0$ the slope decreases quickly until $x = 5$ when the slope is close to zero again. Homework Help ✎

3-102.

Write the slope function for each of the following functions. Homework Help ✎

$f(x) = \frac { 2 } { 3 }(x − 5)^3 + x^2$

$f ( x ) = \sqrt [ 3 ] { x }$

$f ( x ) = \operatorname { sin } ( \frac { \pi } { 4 } )$

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

3-103.

The graph at right shows the distance a bicyclist travels from Oshkosh to a town $100$ miles away. Homework Help ✎

Describe the velocity of the bicyclist.
What is the bicyclist’s average velocity?
Approximate the bicyclist’s instantaneous velocity at $t = 5$ hours.

First quadrant, x axis labeled, time, hours, y axis labeled distance, miles, increasing curve starting at the origin, increasing more rapidly, changing from opening up to opening down at the point (5, comma 50), increasing more slowly, stopping at the point (10, comma 100).

3-104.

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

$h(x) = (2x − 5)^3$

$h(x)=\sin(3x−1)$

$h ( x ) = \sqrt [ 5 ] { \operatorname { tan } ( x ) }$

3-105.

If $f^\prime(x)=\cos(x)$ , write two different possible equations for $f$ . How many equations are possible? Homework Help ✎

3-106.

Examine the Riemann sum at right for the area under $f$ . Homework Help ✎

How many rectangles were used?
If the area being approximated is over the interval $a\le x\le b$ , what are the values of $a$ and $b$ ?

$\displaystyle\sum _ { i = 0 } ^ { 11 } \frac { 6 - 3 } { 12 } f ( 3 + \frac { 6 - 3 } { 12 } \cdot i )$

3-107.

Oliver is trying to determine the derivative of $f(x) = −4x^3$ at $x = 2$ . He uses substitution and gets $f(2) = −32$ . He then takes the derivative and gets $f^\prime(2) = 0$ . What went wrong? Why did his method not work? Homework Help ✎

3-108.

For each graph below:

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

3-109.

Curves can be labeled with descriptors such as “concave down” and “increasing.” On graph paper, graph each function and label its respective parts. Use different colors to represent concavity. Homework Help ✎

$f(x) = 2x^2 + x − 15$

$g(x) = x^3 − 12x − 1$

3-110.

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 ✎

$\lim\limits _ { x \rightarrow 0 } \large\frac { 1 - \sqrt { x } } { x - 1 }$
$\lim\limits _ { x \rightarrow 1 } \large\frac { 1 - \sqrt { x } } { x - 1 }$
$\lim\limits _ { h \rightarrow 0 } \large\frac { \sqrt { 25 + h } - 5 } { h }$
$\lim\limits _ { h \rightarrow - 2 } \large\frac { ( h + 2 ) - 2 } { 2 }$
Which of the problems above can be interpreted as the definition of the derivative at a point?

Calculus Third Edition

Curve Analysis