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AP Calculus AB: 8.3.2 Using the Second Derivative to Examine Concavity

Mathematics6 CardsCreated 3 months ago

This flashcard set demonstrates how the second derivative helps determine where a function is concave up or down and how to find inflection points. By analyzing the sign of the second derivative and using sign charts, you can accurately identify changes in concavity, enhancing your ability to sketch and understand the shape of graphs.

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Using the Second Derivative to Examine Concavity

  • The second derivative can be used to determine where the graph of a function is concave up or concave down, and to find inflection points.

  • Knowing the critical points, local extreme values, increasing and decreasing regions, the concavity, and the inflection points of a function enables you to sketch an accurate graph of that function.

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Key Terms

Term
Definition

Using the Second Derivative to Examine Concavity

  • The second derivative can be used to determine where the graph of a function is concave up or concave down, and to find inflection points.<...

note

  • Knowing the concavity of a function can help you make a better sketch of its curve. Recall that the graph of a function is concave up if th...

Find the points of inflection for the curve determined by the equation y = x^ 3 − 12x ^2

(4, −128)

Find the points of inflection for the function  f (x) = x ^3 − 3x ^2 + 2x − 1.

(1, −1)

Which of the following could be the graph of f (x), given the graphs of f ′(x) and f ″(x) shown below?

The x values where the first derivative is positive (the graph of f ′ lies above the x‑axis) are the x values where f is increasing. Similarly, the...

Find all the points of inflection for the curve y = x^ 4 − 12x^ 2.

(−√2,−20), (√2,−20)

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AP Calculus AB: 8.3.2 Using the Second Derivative to Examine Concavity
TermDefinition

Using the Second Derivative to Examine Concavity

  • The second derivative can be used to determine where the graph of a function is concave up or concave down, and to find inflection points.

  • Knowing the critical points, local extreme values, increasing and decreasing regions, the concavity, and the inflection points of a function enables you to sketch an accurate graph of that function.

note

  • Knowing the concavity of a function can help you make a better sketch of its curve. Recall that the graph of a function is concave up if the derivative is increasing, and
    concave down if the derivative is decreasing.

  • To determine the behavior of the derivative, you will need its derivative, i.e. the second derivative of the function.

  • Set the second derivative equal to zero to determine possible inflection points, which are characterized by a change in concavity.

  • Then make a sign chart for the second derivative. If the second derivative is negative, then the derivative is decreasing and the function is concave down. Similarly, if the second derivative is positive, then the derivative is increasing and the function is concave up.

  • Since the concavity changes at the point where the second derivative equals zero, it is an inflection point, after all.

  • Once again, take the second derivative of the function in order to determine its curvature.

  • The second derivative is never equal to zero, but it is
    undefined at x = 0. An inflection point might exist there.

  • Make a sign chart for the second derivative. Since the function changes from concave up to concave down at x = 0, it is an inflection point. Notice that the tangent line at x = 0 has an undefined slope.

Find the points of inflection for the curve determined by the equation y = x^ 3 − 12x ^2

(4, −128)

Find the points of inflection for the function  f (x) = x ^3 − 3x ^2 + 2x − 1.

(1, −1)

Which of the following could be the graph of f (x), given the graphs of f ′(x) and f ″(x) shown below?

The x values where the first derivative is positive (the graph of f ′ lies above the x‑axis) are the x values where f is increasing. Similarly, the x values where the first derivative is negative (the graph of f ′ lies below the x‑axis) are the x values where f is decreasing.

The x values where the second derivative is positive (the graph of f ″ lies above the x‑axis) are the x values where f is concave up. Similarly, the x values where the second derivative is negative (the graph of f ″ lies below the x‑axis) are the x values where f is concave down.

Find all the points of inflection for the curve y = x^ 4 − 12x^ 2.

(−√2,−20), (√2,−20)