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Section 11.7 Graphing with Derivatives (AD7)

Subsection 11.7.1 Activities

Remark 11.87.

In Section 11.5 and Section 11.6 we learned how the first and second derivatives give us information about the graph of a function. Specifically, we can determine the intervals where a function is increasing, decreasing, concave up, or concave down as well as any local extrema or inflection points. Now we will put that information together to sketch the graph of a function.

Activity 11.88.

Which of the following features best describe the curve graphed below?
Figure 128.
  1. Increasing and concave up
  2. Increasing and concave down
  3. Decreasing and concave up
  4. Decreasing and concave down

Activity 11.89.

(a)
Which of the following features best describe the curve graphed below?
Figure 129.
  1. \(f'>0\) and \(f''>0\)
  2. \(f'>0\) and \(f''<0 \)
  3. \(f'<0\) and \(f''>0\)
  4. \(f'<0\) and \(f''<0 \)
(b)
For each of the other three answer choices, sketch a curve that matches that description.

Activity 11.90.

For each prompt that follows, sketch a possible graph of a function on the interval \(-3 \lt x \lt 3\) that satisfies the stated properties.
(a)
A function \(f(x)\) that is increasing on \(-3 \lt x \lt 3\text{,}\) concave up on \(-3 \lt x \lt 0\text{,}\) and concave down on \(0 \lt x \lt 3\text{.}\)
(b)
A function \(g(x)\) that is increasing on \(-3 \lt x \lt 3\text{,}\) concave down on \(-3 \lt x \lt 0\text{,}\) and concave up on \(0 \lt x \lt 3\text{.}\)
(c)
A function \(h(x)\) that is decreasing on \(-3 \lt x \lt 3\text{,}\) concave up on \(-3 \lt x \lt -1\text{,}\) neither concave up nor concave down on \(-1 \lt x \lt 1\text{,}\) and concave down on \(1 \lt x \lt 3\text{.}\)
(d)
A function \(p(x)\) that is decreasing and concave down on \(-3 \lt x \lt 0\) and is increasing and concave down on \(0 \lt x \lt 3\text{.}\)

Observation 11.91.

To draw an accurate sketch, we must keep in mind additional characteristics of a function, such as the domain and the horizontal and vertical asymptotes (when they exist). The next problem Activity 11.92 includes those aspects in addition to increasing, decreasing, and concavity features.

Activity 11.92.

The following chart describes the values of \(f(x)\) and its first and second derivatives at or between a few given values of \(x\text{,}\) where \(\nexists\) denotes that \(f(x)\) does not exist at that value of \(x\text{.}\)
\begin{equation*} \begin{array}{c|cccccccccccccccccccc} x & & -8 & & -6 & & -3 & & 0 & & 2 & & 5 & & 8 & & 11 & & 13 \\\hline f(x) & & 3 & & 5 & & \nexists & & -5 & & \nexists & & 4 & & \nexists & & -5 & & -3 \\ f'(x) & + & & + & & - & & - & & - & & - & & + & & + & & + & & + \\ f''(x) & + & & - & & - & & + & & - & & + & & + & & - & & - & & - \\ \end{array} \end{equation*}
Assume that \(f(x)\) has vertical asymptotes at each \(x\)-value where \(f(x)\) does not exist, that \(\displaystyle \lim_{x\to-\infty}f(x)= 1\text{,}\) and that \(\displaystyle \lim_{x\to\infty}f(x)= -1\text{.}\)
(a)
List all the asymptotes of \(f(x)\) and mark them on the graph.
(b)
Does \(f(x)\) have any local maxima or local minima? If so, at what point(s)?
(c)
Does \(f(x)\) have any inflection points? If so, at what point(s)?
(d)
Use the information provided to sketch a reasonable graph of \(f(x)\text{.}\) Watch changes in behavior due to changes in the sign of each derivative.

Remark 11.93. A guide to curve sketching.

  1. Identify the domain of the function.
  2. Identify any vertical or horizontal asymptotes, if they exist.
  3. Find \(f'(x)\text{.}\) Then use it to determine the intervals where the function is increasing and the intervals where the function is decreasing. State any local extrema.
  4. Find \(f''(x)\text{.}\) Then use it to determine the intervals where the function is concave up and the intervals where the function is concave down. State any inflection points.
  5. Put everything together and draw sketch.

Activity 11.94.

(a)
\(f(x)=x^4-4x^3+10\)
(b)
\(f(x)=\frac{x^2-4}{x^2-9}\)
(c)
\(f(x)=x+2\cos x\) on the interval \([0,2\pi]\)
(d)
\(f(x)=\frac{x^2+x-2}{x+3}\)
(e)
\(f(x)=\frac{x}{\sqrt{x^2+2}}\)
(f)
\(f(x)=x^6+\frac{12}{5}x^5-12x^4+10\)

Subsection 11.7.2 Videos

No video is available for this learning outcome.

Subsection 11.7.3 Exercises

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