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Lecture 20: Graphical Interpretation of the Derivative

Goal: Recover information about a function \( f \) from the graph of \( f' \).

1 Implementing the I/D and 1st Derivative Test

Problem 1.

Use the graph of \( f' \) shown in Figure 1 to determine the following about \( f \):

Graph of f prime as a straight line through the origin with slope 1, negative left of 0 and positive right of 0.
Figure 1
A coordinate grid with the x-axis labeled from -4 to 4 and the y-axis from -4 to 4. A yellow straight line labeled f prime passes through the origin with slope 1, rising from the lower left to the upper right. A hand-drawn blue squiggly arrow along the negative x-axis is labeled f prime less than 0, indicating the line is below the x-axis for x less than 0. A red squiggly arrow along the positive x-axis is labeled f prime greater than 0, indicating the line is above the x-axis for x greater than 0. A small open circle marks the crossing point at the origin.
  1. Intervals of increase/decrease

    We need to determine the critical numbers of \(f\). I.e., when the derivative is \(0\) or undefined. When we are looking at the graph of \(f'\), critical numbers occur when the graph of \(f'\) touch the \(x\)-axis. Therefore, for this example, there is only one critical number (namely \(0\)).

    After that, we need to make a sign chart for \(f'\). I.e., determine when \(f'\) is positive and negative. Looking at the graph of \(f'\) we have:

    1. \(f'>0\) when the graph of \(f'\) is above the \(x\)-axis;
    2. \(f'< 0 \) when the graph of \(f'\) is below the \(x\)-axis.
    Sign line for f prime with critical number 0, negative arrow to the left and positive arrow to the right.
    Figure 2
    A hand-drawn horizontal number line with arrows at both ends. A single tick point is marked near the center and labeled 0 below the line. To the left of 0 a blue downward-slanting arrow indicates the function is decreasing; to the right of 0 a red upward-slanting arrow indicates the function is increasing. A small curved bracket is drawn above the point at 0.

    Using the sign chart in Figure 2, \(f\) is increasing on \((0,\infty)\) while \(f\) is decreasing on \((-\infty, 0)\).

  2. Positions of relative max/mins:

    3. By the 1st Derivative Test, \(f\) only has a relative minimum at \(x = 0\) (no relative maximums).

Crit. Nums \( \leftarrow f' = 0 \leftarrow \) when \( f \) touches the x-axis

Problem 2.

Use the graph of \( f' \) in Figure 3 to determine the following about \( f \):

V-shaped graph of f prime with vertex below the x-axis near x equals 1, crossing at 0 and 2.
Figure 3
A coordinate grid with x-axis labeled from -4 to 4 and y-axis labeled from -1 to 4. A yellow V-shaped graph labeled f prime descends steeply from the upper left, crosses the x-axis at x equals 0, reaches a minimum around y equals -1 near x equals 1, then rises crossing the x-axis at x equals 2 and continuing upward to the right. Hand-drawn squiggly arrows along the x-axis are labeled: f prime greater than 0 in blue on the far left region, f prime less than 0 in red between 0 and 2, and f prime greater than 0 on the right region.
  1. Intervals of increase/decrease

    The critical numbers of \(f\) are \(0\) and \(2\) (See Problem 1 for explanation why). Constructing the sign chart in a similar manner to Problem 1 we get Figure 4:

    Sign line for f prime with critical numbers 0 and 2, positive, negative, then positive.
    Figure 4
    A hand-drawn horizontal number line with arrows at both ends. Two marked points are labeled 0 and 2 below the line. To the left of 0 a blue upward arrow indicates increasing; between 0 and 2 a red downward arrow indicates decreasing; to the right of 2 a blue upward arrow indicates increasing. A small arch is drawn above the point 0 and a small cup shape is drawn above the point 2.

    Therefore, \(f\) is increasing on \( (-\infty, 0) \cup (2, \infty) \) while decreasing on \((0, 2)\).

  2. Positions of relative max/mins

    3. By the First Derivative Test, there is a relative maximum at \(x=0\) and a relative minimum at \(x = 2\).

2 Implementing the Concavity and 2nd Derivative Test

Problem 3.

Use the graph of \( f' \) shown in Figure 5 to determine the following about \( f \):

Upward parabola labeled f prime with vertex near (2,5), annotated f double prime less than 0 on left and greater than 0 on right.
Figure 5
A coordinate grid with the x-axis labeled from \( -2 \) to \( 6 \) and the y-axis labeled at 5, 10, 15, and 20. A yellow upward-opening parabola labeled \( f' \) has its vertex at about \( (2, 5) \) and rises to about 20 at both \( x = -2 \) and \( x = 6 \). On the left branch, short cyan tangent segments are drawn at several points and the annotation \( f'' < 0 \) is written in cyan. On the right branch, short red tangent segments are drawn and the annotation \( f'' > 0 \) is written in red. At the vertex a short horizontal white tangent segment is drawn, indicating zero slope.
  1. When \( f \) is concave up VS. concave down

    2. We need to determine when \(f'' = 0\). Interpreting \(f''\) as \([f']'\), it is \(0\) when a line tangent \(f'\) has a slope of \(0\). One way to finding these is to find when \(f'\) has a relative maximum or minimum (since possible locations for Inflection Points are critical numbers of \(f'\)). Thus, the only possible location for an inflection point is \(x = 2\).

    We now need to determine when \(f''\) is positive or negative. To do so,

    1. \(f''(c) > 0\) at \(c\) when the slope of the tangent line at \(c\) is positive (i.e., the line "points upward");
    2. \(f''(c) < 0\) at \(c\) when the slope of the tangent line at \(c\) is negative (i.e., the line "points downward").
    Sign line with mark at 2, labeled CD to the left and CU to the right.
    Figure 6
    A horizontal number line with arrows at both ends. A single tick mark is placed near the middle and labeled 2. To the left of the tick the line is labeled CD, indicating concave down; to the right of the tick the line is labeled CU, indicating concave up.

    Using this, we can make a sign chart of \(f''\) as in Figure 6 to conclude \(f\) is CD on \((-\infty, 2)\) and CU on \((2, \infty)\).

  2. Position of any inflection points

    Since concavity changes at \(2\), there is an IP at \( x = 2 \).

    Figure 7 depicts a crude drawing of \(f\) near \(x = 2\), emphasizing that concavity needs to change for a point to be an inflection point.

    Small freehand sketch of a curve bending from concave down to concave up.
    Figure 7
    A small hand-drawn curve sketched freehand. It begins rising, arcs over as a concave-down hump, then dips and sweeps upward as a concave-up bowl, illustrating the shape of a graph passing through an inflection point.

Problem 4.

Use the graph of \( f' \) in Figure 8 to determine the following about \( f \):

Cubic-like graph of f prime with a local max near x equals negative 3 and local min near x equals 2.
Figure 8
A coordinate grid with the x-axis labeled from \( -6 \) to \( 4 \) and the y-axis labeled at \( -10 \), 10, and 20. A yellow curve labeled \( f' \) rises steeply from the lower left, reaches a local maximum of about 15 near \( x = -3 \), then falls crossing the x-axis near \( x = -5 \) and again near \( x = 0 \), reaching a local minimum of about \( -8 \) near \( x = 2 \), then rises steeply crossing the x-axis near \( x = 3.5 \). Green tangent segments are drawn on the outer rising portions where the slope increases, and red tangent segments are drawn on the middle falling portion. Short horizontal white tangent segments mark the local maximum and local minimum, and dotted vertical lines are drawn from those turning points.
  1. When \( f \) is concave up VS. concave down

    2. The only possible locations for Inflection Points are \(-3\) and \(2\) (see Problem 3 for an explanation on how these are found).

    Sign line with marks at negative 3 and 2, labeled CU, CD, CU from left to right.
    Figure 9
    A horizontal number line with arrows at both ends and two tick marks labeled \( -3 \) and 2. The region to the left of \( -3 \) is labeled CU in green, the region between \( -3 \) and 2 is labeled CD in red, and the region to the right of 2 is labeled CU in green.

    Using the sign chart in Figure 9, \(f\) is CU on \( (-\infty, -3) \cup (2, \infty)\) and CD on \( (-3, 2) \).

  2. Position of any inflection points

    Since concavity changes at both points, there are Inflection Points at \( x = -3\) and \(x = 2 \).

  3. Implement the 2nd Derivative Test

    The critical numbers are \( x = -5, 0, 3.5 \). We need to determine the sign of \(f''\) at these points. Thus, we draw tangent lines at these points are determine if the slope is positive or negative. By the Second Derivative Test,

    \( f''(-5) > 0 \) \( \Rightarrow \) \(f\) has a relative minimum at \( x = 5 \)

    \( f''(0) < 0 \) \( \Rightarrow \)\(f\) has a relative maximum at \(x = 0\)

    \( f''(3.5) > 0 \) \( \Rightarrow \) \(f\) has a relative minimum at \( x = 5 \)

    The 1st Derivative Test will most likely to easier to implement for these kinds of problems. However, I did want to demonstrate how the 2nd Derivative Test is done in this context.

3 Putting Everything Together

We now want to combine everything, so if we have the graph of \(f'\) we can discuss the basic shape of \(f\).

NOTE:

For the remainder of the problems, the graphs, sign charts, and final answers will be given. For an in-depth explanation, see Problems 1 and 3.

Problem 5

Use the graph of \( f' \) in Figure 10 to determine the following about \( f \):

Graph of f prime, a hump peaking near 4 at x equals 12, crossing zero near x equals 9 and 18, annotated with sign labels.
Figure 10
A coordinate grid with the horizontal axis labeled 10, 12, 14, 16, 18, 20 and the vertical axis labeled 4, 2, and negative 2. A yellow curve representing f prime starts low on the left, rises steeply crossing the horizontal axis near x equals 9, reaches a maximum value of about 4 at x equals 12 (marked with a white dot and a dashed vertical line), then decreases to touch the horizontal axis near x equals 18 at a white dot, after which it rises again. Handwritten colored annotations label the regions: red text f prime less than 0 on the far left where the curve is below the axis, blue text f prime greater than 0 where the curve is above the axis, green text f double prime greater than 0 on the rising left portion, red text f double prime less than 0 on the falling portion after x equals 12, and green text f double prime greater than 0 near x equals 18. The curve itself is labeled f prime. Squiggly red and blue sign lines are drawn along the horizontal axis.
  1. Intervals of Increase/Decrease:

    Critical numbers are \( x = 9, 18 \). Using the sign chart:

    Sign line with marks at 9 and 18; decreasing arrow left of 9, increasing arrows to the right.
    Figure 12
    A hand-drawn number line with arrows on both ends and tick marks labeled 9 and 18. To the left of 9 a red downward arrow and hatch marks indicate the function is decreasing. Between 9 and 18 and to the right of 18, blue upward arrows indicate the function is increasing.

    \(f\) is decreasing on \((-\infty, 9)\) and increasing on \((9, \infty)\).

  2. When \( f \) is concave up VS concave down:

    Possible locations of Inflection Points are \(x = 12,18\). From the sign chart:

    Sign line with marks at 12 and 18 labeled CU, CD, CU from left to right.
    Figure 13
    A hand-drawn number line with arrows on both ends and tick marks labeled 12 and 18. The interval to the left of 12 is labeled CU in green for concave up, the interval between 12 and 18 is labeled CD in red for concave down, and the interval to the right of 18 is labeled CU in green for concave up.

    We see \(f\) is CU on \((-\infty, 12) \cup (18,\infty)\) and CD on \((12,18)\).

  3. State the locations of any relative max/mins as well as inflection points:

    4. By the 1st Derivative Test, \(f\) does not have any relative maxima, but has a relative minimum at \(x = 9\). There are also Inflection Points at \(x=12,18\).

This gives us enough information to make a (very crude) sketch of \(f\).
Hand-drawn sketch of f with marks at 9, 12 and 18, dipping to a minimum then rising.
Figure 11
A freehand sketch of the original function f on a set of hand-drawn axes. Tick marks on the horizontal axis are labeled 9, 12, and 18. The curve starts high on the left, falls steeply to a minimum marked with a small circle just past 9, stays low and flattens, then rises gradually through a marked point near 18 where an inflection occurs, and finally shoots upward with an arrow indicating continued increase.

Problem 6

Use the graph of \( f' \) in Figure 14 to determine the following about \( f \):

Graph of f prime, a W-shaped curve touching zero near x equals negative 1 and 1 with a local max of 1 at x equals 0.
Figure 14
A coordinate grid with the horizontal axis labeled negative 1.5, negative 1, negative 0.5, 0.5, 1, 1.5 and the vertical axis labeled 1 through 5. A yellow W-shaped curve representing f prime comes down steeply from the upper left, touches the horizontal axis near x equals negative 1, rises to a local maximum of 1 at x equals 0 (marked with a white dot), falls back to touch the horizontal axis at x equals 1 (marked with a white dot), and then rises steeply to the upper right. Handwritten colored annotations read f double prime less than 0 in red on the upper left branch, f double prime greater than 0 in green on the left rising portion, f prime greater than 0 in blue above the middle, and f double prime less than 0 in red to the right of the local maximum. A blue squiggly sign line runs along the horizontal axis indicating f prime is nonnegative throughout.
  1. Intervals of Increase/Decrease:

    The critical numbers are \(x = -1,1\). From the sign chart:

    Sign line with marks at negative 1 and 1, increasing arrows on all three intervals.
    Figure 16
    A hand-drawn number line with arrows on both ends and tick marks labeled negative 1 and 1. Blue upward arrows are drawn above each of the three intervals, to the left of negative 1, between negative 1 and 1, and to the right of 1, indicating the function is increasing on all intervals.

    We see \(f\) is always increasing. I.e., it is increasing on \( (-\infty, \infty) \)

  2. When \( f \) is concave up VS concave down

    The possible locations for the Inflection Points are \(x = -1, 0,1\). From the sign chart:

    Sign line with marks at negative 1, 0 and 1 labeled CD, CU, CD, CU from left to right.
    Figure 17
    A hand-drawn number line with arrows on both ends and marks at negative 1, 0 (drawn as an open circle), and 1. The interval to the left of negative 1 is labeled CD in red, the interval between negative 1 and 0 is labeled CU in green, the interval between 0 and 1 is labeled CD in red, and the interval to the right of 1 is labeled CU in green.

    One has \(f\) is CD on \((-\infty, -1) \cup (0,1)\) and CU on \((-1,0) \cup (1,\infty)\).

  3. State the locations of any relative max/mins as well as inflection points

    There are no relative extrema by the 1st Derivative Test. Moreover, all possible locations for Inflection Points are indeed inflection points. I.e., \(f\) has Inflection Points at \(x=-1,0,1\).

    4. We have enough information to make a (very crude) sketch of \(f\).

    Hand-drawn increasing sketch of f passing through the origin with marks at negative 2, negative 1, 0 and 1.
    Figure 15
    A freehand blue sketch of the original function f drawn on hand-drawn axes. Tick marks on the horizontal axis are labeled negative 2, negative 1, 0, and 1. The curve rises from the lower left, flattens briefly near x equals negative 1, passes through a marked point at the origin, flattens again near x equals 1 at another marked point, and continues rising to the upper right, showing the function is increasing everywhere with inflection points at the flat spots.

Problem 7

Use the graph of \( f' \) in Figure 18 to determine the following about \( f \):

Graph of f prime, an upward parabola crossing zero at x equals negative 2 and 1, with sign annotations.
Figure 18
A coordinate grid showing the yellow graph of \( f' \), an upward-opening parabola with vertex near \( (-0.5, -2) \). It crosses the x-axis at \( x = -2 \) and \( x = 1 \), where open circles are drawn. The axis is labeled with ticks at \( -3, -2, -1, 1, 2 \) horizontally and \( 2, 4, -2 \) vertically. The curve is labeled \( f' \) in cyan at the upper right. Red handwritten annotations read \( f'' < 0 \) on the upper-left branch and \( f' < 0 \) in the middle region below the axis; a cyan annotation \( f' > 0 \) appears at the lower right. Short red and green tangent segments are drawn along the curve. Cyan squiggles mark the intervals outside the roots along the x-axis, and a red squiggle marks the interval between them. Below the graph the handwritten label \( x = -\frac{1}{2} \) marks the vertex location.
  1. Intervals of Increase/Decrease

    The critical numbers are \( x = -2, 1 \). From the sign chart:

    Sign line with marks at negative 2 and 1; increasing left of negative 2, decreasing between, increasing right of 1.
    Figure 19
    A hand-drawn horizontal number line with arrowheads at both ends and tick marks labeled \( -2 \) and \( 1 \). To the left of \( -2 \) there is a cyan increasing arrow mark, between \( -2 \) and \( 1 \) a red decreasing mark, and to the right of \( 1 \) another cyan increasing arrow mark.

    We see \(f\) is increasing on \((-\infty, -2) \cup (1,\infty)\) and decreasing on \((-2,1)\).

  2. When \( f \) is concave up VS concave down:

    The only possible location for an Inflection Point is \(x = -\frac{1}{2}\). From the sign chart:

    Sign line marked at negative one half, labeled CD on the left and CU on the right.
    Figure 20
    A hand-drawn horizontal number line with arrowheads at both ends and a single tick mark labeled \( -\frac{1}{2} \). The region to the left of the tick is labeled CD in red for concave down, and the region to the right is labeled CU in green for concave up.
    \(f\) is CD on \((-\infty, -\frac{1}{2})\) and CU on \((-\frac{1}{2}, \infty)\).
  3. State the locations of any relative max/mins as well as inflection points

    By the 1st Derivative Test, \(f\) has a relative maximum at \(x = -2\) and a relative minimum at \(x =1\). Moreover, \(x = -\frac{1}{2}\) is indeed the location of an inflection point.

Problem 8

Use the graph of \( f' \) in Figure 21 to determine the following about \( f \):

Graph of f prime, an S-shaped cubic curve flattening at the origin, passing through negative 0.75 and 0.75 region.
Figure 21
A coordinate grid showing the yellow graph of \( f' \), an S-shaped cubic curve that rises from the lower left, flattens horizontally as it passes through the origin, and then rises steeply to the upper right. The curve is labeled \( f' \) in cyan at the upper right. The horizontal axis is labeled at \( -1, -0.5, 0.5, 1 \) and the vertical axis at \( -1, -0.5, 0.5, 1 \). Cyan circles mark points on the curve near \( (-0.75, -0.5) \), at the origin, and near \( (0.75, 0.5) \), with short cyan tangent strokes drawn through them.
  1. Intervals of Increase/Decrease:

    \(f\) only has one critical number, namely at \(x = 0\). From the sign chart:

    Sign line marked at 0 with increasing arrows to the right of 0 and decreasing arrows to the left of 0.
    Figure 22
    A hand-drawn red horizontal number line with arrowheads at both ends and a single tick mark labeled 0. An downward-slanting decreasing arrow is drawn to the left of 0 and another increasing arrow to the right of 0, indicating the function is switches from decreasing to increasing at 0.

    \(f\) is decreasing on \((-\infty, 0)\) and increasing on \((0, \infty)\).

  2. When \( f \) is concave up VS concave down:

    From the sign chart below, \(f\) is always CU.

    Sign line marked at 0 labeled CU on both the left and right sides.
    Figure 23
    A hand-drawn red horizontal number line with arrowheads at both ends and a single tick mark labeled 0. The label CU for concave up appears above the line on the left side of 0 and again above the line on the right side of 0.
  3. State the locations of any relative max/mins as well as inflection points

    \(f\) has a relative minimum at \(x = 0\) by the 1st Derivative Test. However, \(f\) does not have an inflection point at \(x = 0\) since concavity does not change.

Problem 9.

Use the graph of \( f' \) in Figure 24 to determine the following about \( f \):

Graph of f prime, a sinusodial curve with two peaks near 2.8 and zeros at 0, one half, 1 and three halves.
Figure 24
A coordinate grid with the vertical axis labeled from \( -2 \) to 4 and the horizontal axis labeled at 0.5, 1 and 1.5. A yellow sinusodial curve labeled \( f' \) crosses the horizontal axis at \( x = 0 \), \( x = \frac{1}{2} \), \( x = 1 \), and \( x = \frac{3}{2} \); these crossings are circled in cyan. The curve rises to a peak near \( y \approx 2.8 \) just after \( x = 0.25 \) and again just after \( x = 1.25 \), dips to a minimum near \( y \approx -2.8 \) around \( x = 0.75 \), and falls again after the second peak. Green marks with short flat tangent segments sit at the peaks and the trough, and green and red tangent slope segments are drawn along the rising and falling portions of the curve, with red circles marking steep descending points.

NOTE:

For this problem and this problem only, assume the function does not extend past the viewing window \([-\frac{1}{4},\frac{7}{4}]\) and the endpoints are not included. This would make a difference when you are selecting test points to make your sign charts, but this should not change how to solve the problem for our purposes today

  1. Intervals of Increase/Decrease:

    \(f\) has 4 critical numbers, namely \(x = 0, \frac{1}{2}, 1\) and \(\frac{3}{2}\). From the sign chart:

    Sign line marked at 0, one half, 1 and three halves with alternating decreasing and increasing arrows.
    Figure 25
    A horizontal number line drawn in red with arrows at both ends and tick marks labeled 0, \( \frac{1}{2} \), 1 and \( \frac{3}{2} \). A downward arrow appears to the left of 0, an upward arrow between 0 and \( \frac{1}{2} \), a downward arrow between \( \frac{1}{2} \) and 1, an upward arrow between 1 and \( \frac{3}{2} \), and a downward arrow to the right of \( \frac{3}{2} \).

    \(f\) is decreasing on \((-\frac{1}{4}, 0) \cup (\frac{1}{2},1) \cup (\frac{3}{2}, \frac{7}{4})\) and increasing on \((0, \frac{1}{2}) \cup (1, \frac{3}{2})\).

  2. When \( f \) is concave up VS concave down
    Sign line marked at one fourth, three fourths and five fourths labeled CU, CD, CU, CD.
    Figure 26
    A horizontal number line drawn in red with arrows at both ends and tick marks labeled \( \frac{1}{4} \), \( \frac{3}{4} \) and \( \frac{5}{4} \). The interval to the left of \( \frac{1}{4} \) is labeled CU in magenta, the interval between \( \frac{1}{4} \) and \( \frac{3}{4} \) is labeled CD in cyan, the interval between \( \frac{3}{4} \) and \( \frac{5}{4} \) is labeled CU in magenta, and the interval to the right of \( \frac{5}{4} \) is labeled CD in cyan.

    \(f\) is CU on \((-\frac{1}{4}, \frac{1}{4})\cup(\frac{3}{4}, \frac{5}{4})\) and CD \((\frac{1}{4}, \frac{3}{4})\cup(\frac{5}{4}, \frac{7}{4})\).

  3. State the locations of any relative max/mins as well as inflection points

    Relative Minima at \( x = 0, 1 \)

    Relative Maxima at \( x = \frac{1}{2}, \frac{3}{2} \)

    IPs at \( x = \frac{1}{4}, \frac{3}{4}, \frac{5}{4} \)