How do I sketch $f^{\prime}$ and $f^{\prime\prime}$ ?

Curve Sketching: Derivatives

3-111.

CURVE ANALYSIS

On the Lesson 3.3.3 Resource Page (or a large sheet of graph paper), set up three sets of axes—one above the other so that the $y$ -axes are vertically aligned. On each set of axes, use a scale of $–5$ to $5$ for the $x$ -axis. On the top set of axes, sketch the graph of a function $y = f(x)$ such that:
- $f$ is a continuous, smooth function on $(–\infty, \infty)$
- $f$ has zeros at $x = -4, -\frac { 3 } { 2 }, 2$ , and $5$
- $f$ has local maxima at $(−3, 4)$ and $(2, 0)$
- $f$ has local minima at $(0, −2)$ and $(4, −3)$
- $f$ has points of inflection at $(−2, 3), (1, −1),$ and $(3, −1)$
- $f$ is increasing on $(−∞, −3)\cup(0, 2)\cup (4, \infty)$
- $f$ is decreasing on $\left(−3,0\right)\ \cup\ \left(2,4\right)$
Obtain a sheet of stickers or colored markers. Choose one color to represent local maxima another color to represent local minima, and a third color to represent points of inflection. On the $x$ -axis, use these colors to draw dots at the $x$ -values where the maxima, minima, and points of inflection occur.
Use a straightedge to draw tangent lines to your function at every integer value from $-5\le x\le5$ . Approximate the slopes of the tangent lines and record these slopes in a table of data: $x$ -value vs. slope.
On the middle set of axes, plot the points from your table of data. Connect these points to create a smooth, continuous curve. Be sure to consider what happens as $x$ approaches positive infinity and negative infinity. This graph represents the derivative, $f^\prime$ , of your function. Mark the local maxima and local minima values with the same colors you used before. Then choose a fourth color and label the zeros of $f^\prime$ . Compare these points to their corresponding $x$ -values on the graph of $y = f(x)$ . What appears to be significant about these points?
Sketch tangent lines on $f^\prime$ and approximate their slopes. Record these slopes in a table of data. Plot this data on the third set of axes and connect the points in a smooth, continuous curve. Again, be sure to consider what happens as $x$ approaches positive infinity and negative infinity. This graph represents $f^{\prime\prime}$ , the second derivative of $f$ . Mark all of the zeros with the same color you used on $f^\prime$ . What do you notice about the graphs of $y = f(x)$ and $y^\prime = f^\prime(x)$ where $f^{\prime\prime}$ is zero?

3-112.

As shown on the diagram below, the graph of $y = f^\prime(x)$ , the first derivative of $y = f(x)$ , has roots at $x = \text{A}$ and $x = \text{B}$ .

Upward parabola, labeled, f prime of x, turning in third quadrant, passing through the negative x axis at highlighted point labeled, A, & through the positive x axis at highlighted point labeled, B.

With your team, find two different ways to justify that $f(x)$ has a local maximum at $x = \text{A}$ . One method should involve $f ^\prime(x)$ and the other should involve $f ^{\prime \prime}(x)$ . Be prepared to share your justifications with your class.
How many local maxima, local minima and inflection points does $f$ have? Justify your answer.

The graph of $y = g^\prime(x)$ , the first derivative of $y = g(x)$ , has a root at $x = 2$ , as shown at right. Based on this graph, does $g(x)$ have local maximum, local minimum, or a point of inflection at $x = 2$ ? Justify your answer using $g^\prime (x)$ or $g^{\prime \prime} (x)$ .

Portion of first quadrant, with upward parabola labeled, g prime of x, with vertex at the point (2, comma 0),

Review and Preview problems below

3-113.

Use your observations from problem 3-98 to algebraically verify that $y = x^3 +\frac { 3 } { 2 }x^2 − 6x + 2$ is concave up when $x = 0$ . Homework Help ✎

3-114.

Theresa loves tangents! She drew several tangents to a function $g$ , and then erased $g$ ! Trace the graph with the tangents at right and sketch a possible function for $g$ . Is there more than one possibility? If so, sketch another possible function using a different color. Homework Help ✎

First Segment with highlighted center point at (1, comma 2), with one end point at about (1.3, comma 2.8), second segment with highlighted center point at about (3, comma 1.2), with one end point at about (2, comma 1.5).

3-115.

The graph at right shows the distance from a fixed point traveled by a toy car. Use the graph to sketch a velocity graph for the car. Homework Help ✎

First quadrant, x axis labeled time, seconds, y axis labeled distance, meters, continuous curve starting at the origin, opening up, changing concavity, turning down, changing concavity, turning up, changing concavity, then leveling off higher than first turning point, continuing to the right.

3-116.

For each function below, write and evaluate a Riemann sum to calculate the area under the curve for $–2 ≤ x ≤ 1$ using $24$ left endpoint rectangles. Homework Help ✎

$f(x) = 2^x$

$f ( x ) = \sqrt { x + 2 }$

3-117.

Sketch a function $f$ for which the following statements are true about its slope function $f^\prime$ . Homework Help ✎

For $x < −3, f^\prime (x) > 0$ and the slope is increasing.
For $−3 < x < −1, f ^\prime (x) > 0$ and the slope is decreasing.
At $x = −1, f^\prime (x) = 0$ .
For $x > −1, f^\prime (x) < 0$ and the derivative is decreasing.

3-118.

Multiple Choice: If $f^\prime (x) = 6x^2 − 4$ , then which of the following could be $f$ ? Homework Help ✎

$f(x) = 2x^3 − 4x + 1$

$f(x) = 2x^3 − 4x − 4$

$f(x) = 2x^3 − 4x + 4$

$f(x) = 2x(x^2 − 2)$

all of these

3-119.

Compare the limit statements below. What do you notice? Homework Help ✎

$\lim\limits _ { h \rightarrow 0 } \large\frac { ( x + h + 1 ) ( x + h + 2 ) - ( x + 1 ) ( x + 2 ) } { h }$

$\lim\limits _ { h \rightarrow 0 } \large\frac { ( x + h + 1 ) ( x + h + 2 ) - ( x - h + 1 ) ( x - h + 2 ) } { 2 h }$

What do they have in common? How are they different?
Evaluate these limits using any method you choose.

3-120.

Compute the limits below. What do you notice? Homework Help ✎

$\lim\limits _ { h \rightarrow 0 } \large\frac { ( 5 + h ) ( 6 + h ) - 30 } { h }$

$\lim\limits _ { x \rightarrow 4 } \large\frac { ( x + 1 ) ( x + 2 ) - 30 } { x - 4 }$

3-121.

While testing the brakes of a new car, Badru recorded the following speeds traveled (in miles per hour) over time (in seconds). Approximately how far did Badru’s car travel before stopping? Homework Help ✎

$\boldsymbol t$ (seconds)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$\boldsymbol {v(t)}$ (mph)	$50$	$48$	$46$	$43$	$40$	$37$	$34$	$29$	$23$	$14$	$0$

3-122.

Determine all values of $x$ where $f^\prime (x) = 2$ given $f(x) = \frac{1}{3}x^3 − \frac{5}{2}x^2 − 4x + 13$ . 3-122 HW eTool Homework Help ✎

3-123.

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 ✎

$\lim\limits _ { x \rightarrow \infty } \large\frac { x ^ { 3 } - x ^ { - 3 } } { 5 x ^ { 3 } + x ^ { - 3 } }$

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

$\lim\limits _ { x \rightarrow 6 } \large\frac { 2 x ^ { 2 } - 12 x } { x ^ { 2 } + x - 42 }$

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

Calculus Third Edition