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AP Calculus AB: 8.4.2 Cusp Points and the Derivative

Mathematics8 CardsCreated 8 months ago

This content explains how functions with fractional exponents can have cusp points—sharp turns where the derivative is undefined. It guides through analyzing critical points, concavity, and inflection points using first and second derivatives to identify cusp behavior on graphs.

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Cusp Points and the Derivative

Functions with fractional exponents could potentially have cusp points. A cusp point is a point where the curve abruptly
changes direction.
• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3. Find inflection points using the second derivative.
4. Determine where the function is concave up or concave down.

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Key Terms

Term
Definition

Cusp Points and the Derivative

Functions with fractional exponents could potentially have cusp points. A cusp point is a point where the curve abruptly
changes direction.
•...

note

  • Expect to see strange behavior in the graphs of functions with fractional exponents.

  • First, find critical points by setting ...

Which of the following graphs describes a function with these characteristics?
f ′(x) < 0 when x < 1.
f ′(1) is undefined. f ′(x) > 0 when x > 1.
f ″(x) < 0 when x ≠ 1.

This graph has negative slope when x < 1, positive slope when x > 1, an undefined slope at x = 1, and is concave down everywhere but x = 1. I...

A cusp point (or a point where the curve changes direction abruptly instead of smoothly) can occur when:

The first derivative is undefined

True or false?

The graph of y = 3x^ 5/3 − x ^2/3 has a cusp at x = 0.

true

Which of the following statements about the curve y = 1/3 (x 2−1)^2/3 is true?

The curve has a cusp at x = 1

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AP Calculus AB: 8.4.2 Cusp Points and the Derivative
TermDefinition

Cusp Points and the Derivative

Functions with fractional exponents could potentially have cusp points. A cusp point is a point where the curve abruptly
changes direction.
• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3. Find inflection points using the second derivative.
4. Determine where the function is concave up or concave down.

note

  • Expect to see strange behavior in the graphs of functions with fractional exponents.

  • First, find critical points by setting the derivative equal to zero and solving for x. The derivative is undefined at x = 0 even though the function exists, so it is a critical point, too.

  • Second, make a sign chart for the derivative, including arrows for the behavior of the function, and labels for the critical points.

  • Third, find possible inflection points by setting the second derivative equal to zero and solving for x.

  • In this case, the second derivative is never equal to zero. However, it is undefined for x = 0, which is an inflection point candidate.

  • Fourth, make a sign chart for the second derivative. Since the concavity does not change at x = 0, it is not an inflection point.

  • Finally, assimilate the information from the sign charts to sketch the graph of the function. From left to right the graph first decreases, then increases, then decreases, and finally increases.

  • Since the derivative is undefined at x = 0, the line tangent to the curve is vertical. This forms a sharp turn, called a cusp point, at (0, 0).

  • On either side of the cusp point the function is concave up.
    There is no inflection point.

Which of the following graphs describes a function with these characteristics?
f ′(x) < 0 when x < 1.
f ′(1) is undefined. f ′(x) > 0 when x > 1.
f ″(x) < 0 when x ≠ 1.

This graph has negative slope when x < 1, positive slope when x > 1, an undefined slope at x = 1, and is concave down everywhere but x = 1. It fits the description of f (x).

A cusp point (or a point where the curve changes direction abruptly instead of smoothly) can occur when:

The first derivative is undefined

True or false?

The graph of y = 3x^ 5/3 − x ^2/3 has a cusp at x = 0.

true

Which of the following statements about the curve y = 1/3 (x 2−1)^2/3 is true?

The curve has a cusp at x = 1

Which of the following is the graph of y=∣x^3−3x−2∣?

The graph of y=∣x3−3x−2∣y = |x^3 - 3x - 2|y=∣x3−3x−2∣ is the graph of y=x3−3x−2y = x^3 - 3x - 2y=x3−3x−2 reflected above the x-axis wherever x3−3x−2x^3 - 3x - 2x3−3x−2 is negative.

Which of the following is the graph of the equation f(x)=x^5/3−4x^2/3?

The graph has a cusp at x=0x=0x=0 and local extrema near x=1x=1x=1 and x=4x=4x=4. It is neither symmetric nor smooth at zero.