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Calculus for Team-Based Inquiry Learning: 2023 Early Edition

TBIL Institute Fellows

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Section 3.7 Graphing with derivatives (AD7)

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Subsection 3.7.1 Activities

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Remark 3.7.1.

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In Section 3.5 and Section 3.6 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.
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Activity 3.7.2.

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Which of the following features best describe the curve graphed below?
Figure 74.
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    Increasing and concave up
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    Increasing and concave down
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    Decreasing and concave up
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    Decreasing and concave down
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Activity 3.7.3.

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(a)
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Which of the following features best describe the curve graphed below?
Figure 75.
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    f′>0 and f″>0
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    f′>0 and f″<0
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    f′<0 and f″>0
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    f′<0 and f″<0
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(b)
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For each of the other three answer choices, sketch a curve that matches that description.
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Activity 3.7.4.

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For each prompt that follows, sketch a possible graph of a function on the interval −3<x<3 that satisfies the stated properties.
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(a)
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A function f(x) that is increasing on −3<x<3, concave up on −3<x<0, and concave down on 0<x<3.
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(b)
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A function g(x) that is increasing on −3<x<3, concave down on −3<x<0, and concave up on 0<x<3.
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(c)
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A function h(x) thatis decreasing on −3<x<3, concave up on −3<x<−1, neither concave up nor concave down on −1<x<1, and concave down on 1<x<3.
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(d)
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A function p(x) that is decreasing and concave down on −3<x<0 and is increasing and concave down on 0<x<3.
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Observation 3.7.5.

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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 3.7.6 includes those aspects in addition to increasing, decreasing, and concavity features.
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Activity 3.7.6.

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The following chart describes the values of f(x) and its first and second derivatives at or between a few given values of x, where ∄ denotes that f(x) does not exist at that value of x.
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x−8−6−302581113f(x)35∄−5∄4∄−5−3f′(x)++−−−−++++f″(x)+−−+−++−−−
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Assume that f(x) has vertical asymptotes at each x-value where f(x) does not exist, that limx→−∞f(x)=1, and that limx→∞f(x)=−1.
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(a)
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List all the asymptotes of f(x) and mark them on the graph.
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(b)
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Does f(x) have any local maxima or local minima? If so, at what point(s)?
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(c)
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Does f(x) have any inflection points? If so, at what point(s)?
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(d)
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Use the information provided to sketch a reasonable graph of f(x). Watch changes in behavior due to changes in the sign of each derivative.
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Remark 3.7.7. A guide to curve sketching.

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    Identify the domain of the function.
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    Identify any vertical or horizontal asymptotes, if they exist.
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    Find f′(x). Then use it to determine the intervals where the function is increasing and the intervals where the function is decreasing. State any local extrema.
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    Find f″(x). 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.
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    Put everything together and draw sketch.
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Activity 3.7.8.

Sketch the graph of each of the following functions using the guide to curve sketching found in Remark 3.7.7
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(a)
f(x)=x4−4x3+10
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(b)
f(x)=x2−4x2−9
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(c)
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f(x)=x+2cos⁡x on the interval [0,2π]
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(d)
f(x)=x2+x−2x+3
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(f)
f(x)=x6+125x5−12x4+10
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Subsection 3.7.2 Videos

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No video is available for this learning outcome.
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