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B. Derivatives and Their Applications · Curve Sketching: Connecting f, f′ and f″

The Second Derivative: The Rate of Change of the Rate of Change

B1.2 B1.1About 70 minutes
Learning goals
  • I can explain the second derivative as the rate of change of the rate of change, in words and from data.
  • I can connect the sign of f″ to concavity (concave up or concave down).
  • I can sketch the graphs of f′ and f″ from the graph of a smooth function f.
  • I can describe points of inflection as places where the concavity changes.

Words to know

second derivative
The derivative of f′, written f″(x), y″ or d²y/dx². It measures how fast the slope is changing.
concave up
Bending like ∪: slopes increase, and the graph lies above its tangent lines. Happens where f″ > 0.
concave down
Bending like ∩: slopes decrease, and the graph lies below its tangent lines. Happens where f″ < 0.
point of inflection
A point on the graph, where f is continuous, at which the concavity changes.
higher derivatives
f‴, f⁽⁴⁾ and so on: derivatives taken three or more times.

The rate of change of the rate of change

A news report might say: "Prices are still rising, but more slowly than last month." Prices are going up, so their rate of change is positive. But that rate is getting smaller. To describe this you need a second layer: the rate of change of the rate of change. In calculus this is the second derivative.

The second derivative is the derivative of the derivative:

f″(x) = (f′(x))′, also written d²y/dx² or y″.

You can keep going: f‴(x) is the third derivative and f⁽⁴⁾(x) the fourth. In Applications of Derivatives you will see that if s(t) is position, then s′(t) is velocity and s″(t) is acceleration.

Worked example

Example 1: A second derivative from data

During a snowstorm in Sudbury, the depth of fresh snow on a deck was recorded each hour: 0, 2, 5, 9, 12, 14 cm at t = 0, 1, 2, 3, 4, 5 h. Describe how the rate of snowfall changed.

  1. First differences (the average rate each hour): 2, 3, 4, 3, 2 cm/h. These estimate f′.
  2. Second differences: 3 − 2 = 1, 4 − 3 = 1, 3 − 4 = −1, 2 − 3 = −1 (cm/h per hour). These estimate f″.
  3. For the first 3 hours the rate was increasing (second differences positive): the storm was intensifying.
  4. After that the rate was decreasing (second differences negative), although snow kept piling up (first differences still positive).
  5. The switch, near t = 3 h, is where the snowfall was heaviest: an inflection point of the depth graph.

Depth always increased. The rate of snowfall increased until about t = 3 h, then decreased. Depth graph: concave up, then concave down.

Concavity

The sign of f″ tells you how the graph bends.

  • f is concave up on an interval if f′ is increasing there. The graph bends like a bowl (∪) and lies above its tangent lines. This happens when f″ > 0.
  • f is concave down on an interval if f′ is decreasing there. The graph bends like a dome (∩) and lies below its tangent lines. This happens when f″ < 0.
  • A point of inflection is a point on the graph where f is continuous and the concavity changes. At such a point f″ is 0 or undefined, and on the graph of f′ there is a local maximum or minimum.

Concavity is completely separate from increasing and decreasing: all four combinations are possible.

f′ > 0, f″ > 0xy11234rising fasterf′ > 0, f″ < 0xy11234rising slowerf′ < 0, f″ > 0xy11234falling slowerf′ < 0, f″ < 0xy11234falling faster
The four basic shapes. The sign of f′ says up or down; the sign of f″ says which way the curve bends: concave up (bowl side) when f″ > 0, concave down (dome side) when f″ < 0.
concave up: above its tangentsxy−2−112−11234concave down: below its tangentsxy−2−112−4−3−2−1
A concave-up curve lies above its tangent lines (dashed) and its slopes increase from left to right. A concave-down curve lies below its tangents and its slopes decrease.

Memory trick

Concave up holds water like a cup (∪), and a smile is concave up. f″ > 0 → smile; f″ < 0 → frown. Just remember that a smile can be on a rising or a falling part of the graph.

Sketching f′ and f″ from the graph of f

You already know how to sketch f′ from f. To sketch f″, do the same thing again, starting from the graph of f′:

  1. Sketch f′ from f (zeros at horizontal tangents; signs from rising and falling).
  2. Sketch f″ from f′: zeros of f″ where f′ has horizontal tangents (its turning points); f″ positive where f′ rises, negative where f′ falls.
  3. Check with concavity: f″ > 0 exactly where f looks like ∪, and f″ < 0 where f looks like ∩.
f(x) = x³ − 3x²xy−1123−5−4−3−2−112maxminIPf′(x) = 3x² − 6xxy−1123−5−4−3−2−112min of f′f″(x) = 6x − 6xy−1123−5−4−3−2−112zero
Read down the columns. On the shaded part (x < 1), f is concave down, f′ is decreasing and f″ is negative. At x = 1, f has a point of inflection, f′ has its lowest point and f″ crosses zero.
Worked example

Example 2: Sketching f′ and f″ for a cubic

The graph of f(x) = x³ − 3x² has a local maximum at (0, 0), a local minimum at (2, −4) and looks like ∩ to the left of x = 1 and ∪ to the right. Sketch f′ and f″ from this information, then check algebraically.

  1. f′: zeros at x = 0 and x = 2; positive, negative, positive. An upward parabola with vertex halfway, at x = 1.
  2. f′ falls until x = 1 and rises after. So f″ is negative for x < 1, zero at x = 1, positive for x > 1. f′ is a parabola, so f″ is a straight line.
  3. Concavity check: f is ∩ for x < 1 (f″ < 0) and ∪ for x > 1 (f″ > 0). ✓
  4. Algebra: f′(x) = 3x² − 6x = 3x(x − 2); f″(x) = 6x − 6, zero at x = 1. f(1) = 1 − 3 = −2. ✓

f′ is an upward parabola through (0, 0) and (2, 0) with vertex (1, −3); f″ is the line 6x − 6, crossing zero at x = 1. The inflection point of f is (1, −2).

Worked example

Example 3: Translating words into signs

A town's population P(t) is "increasing at a decreasing rate". A lake's water level L(t) is "falling, and falling faster each week". Give the signs of the first and second derivatives and the shape of each graph.

  1. Population increasing: P′(t) > 0. The rate is decreasing: P″(t) < 0. Rising and concave down.
  2. Water level falling: L′(t) < 0. Falling faster means the rate is becoming more negative, so L′ is decreasing: L″(t) < 0. Falling and concave down.

P′ > 0, P″ < 0 (rising, ∩); L′ < 0, L″ < 0 (falling, ∩). Careful: "falling faster" means L″ < 0, not L″ > 0.

Worked example

Example 4: Reading f″ from a graph of f′

The graph of f′ is a downward parabola with zeros at x = −1 and x = 5. What does it tell you about f″ and about the concavity of f?

  1. f′ rises until its vertex and falls after. The vertex of a parabola is halfway between its zeros: x = 2.
  2. So f″ > 0 for x < 2 and f″ < 0 for x > 2, with f″(2) = 0.
  3. f is concave up for x < 2, concave down for x > 2, with an inflection point at x = 2.

Concave up on (−∞, 2), concave down on (2, ∞), inflection at x = 2. (Also: f′ > 0 between −1 and 5, so f increases there, with a local min at −1 and a local max at 5.)

Just for fun

Funny but true: physicists call the third derivative of position jerk, and the next three are sometimes nicknamed snap, crackle and pop.

Hands-on activity

Fill a bottle, watch it bend

You need: an empty clear plastic bottle that narrows at the top (a 1 L or 2 L pop bottle), a measuring cup, a ruler or tape measure, water, a pencil and paper. Work over a sink.

  1. Pour in exactly 100 mL at a time. After each pour, measure the water height h in centimetres from the table to the water surface.
  2. Continue until the bottle is nearly full. Record volume V and height h in a table.
  3. Compute first differences of h (cm per 100 mL) and second differences.
  4. Plot h against V.

What to look for: where the bottle is wide, each 100 mL raises the water only a little; where it narrows near the top, the height climbs faster. So in the top part the first differences grow, the second differences are positive and the graph of h is concave up. If your bottle has a narrow waist and a wider part above it, the graph switches concavity: find that inflection point and compare it with where the bottle is narrowest.

Your turn

1

Find f″(x) for f(x) = 4x³ − 5x² + 2x − 9.

2

Find y″ for y = x⁵ − 3x⁴.

3

f″(2) = −3. Describe the graph of f near x = 2.

4

A car's speed is increasing at an increasing rate. Give the signs of v′ and v″, where v is speed.

5

The graph of f′ has a local minimum at x = 3. What feature does f have at x = 3?

6

For which values of x is f(x) = x³ concave up?

More practice: warm-up

1

Find f″(x) for f(x) = x².

2

Find f″(x) for f(x) = 7x + 3.

3

If f″ > 0 on an interval, is f concave up or concave down?

4

Is a graph shaped like ∩ concave up or concave down?

5

Write the second derivative of y in Leibniz notation.

More practice: on your own

1

Snow depths at t = 0, 1, 2, 3, 4 h are 0, 1, 3, 6, 10 cm. Use second differences to describe the concavity.

2

Sketch, in words, f′ and f″ for f(x) = −x² + 4x.

3

A company's revenue R(t) is falling, but falling more slowly each month. Give the signs of R′ and R″.

4

Find f‴(x) and f⁽⁴⁾(x) for f(x) = x⁴ − 2x³.

5

The graph of f′ is the line y = 2x − 6. Describe the concavity of f and where f has a local extremum.

Challenge

1

Show that every quadratic f(x) = ax² + bx + c (a ≠ 0) has the same concavity everywhere, and say how a decides it.

2

Find the value of x where f(x) = x³ − 6x² + 5 is decreasing most rapidly.

3

Find a and b so that f(x) = ax³ + bx² has an inflection point at (1, 4).

Check your understanding

Quiz

1.

What is f″(x) for f(x) = x⁴?

2.

A graph is increasing and concave down on an interval. Which description fits?

3.

f is concave up on an interval. Which statement must be true there?

4.

"Unemployment is rising, and rising faster each month." If U(t) is unemployment, which is correct?

5.

On the graph of f′, a point of inflection of f appears as:

6.

Second differences of a data set are all negative. The graph of the data is:

7.

For f(x) = x³ − 3x², where is f concave up?

For parents and teachers

Big idea (B1.2, B1.1): the second derivative is the rate of change of the rate of change. Its sign tells whether a graph bends up like a cup (f″ > 0) or down like a dome (f″ < 0). This is separate from whether the graph is going up or down.

How to help: listen for everyday phrases like "growing, but more slowly" or "dropping faster and faster" in the news, and ask your teen to give the signs of f′ and f″. The bottle activity is a good shared experiment.

Common difficulties:

  • Thinking concave up means increasing. A falling graph can be concave up (falling more and more slowly).
  • "Falling faster" means f″ < 0, because the slope is becoming more negative.
  • Forgetting to differentiate twice, or differentiating f instead of f′ the second time.