8.4.2 Cusp Points and the Derivative

Question 1

Cusp Points and the Derivative

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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.

Answer 3

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).

Question 4

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

Answer 4

The first derivative is undefined

Question 5

True or false?

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

Answer 5

true

Question 6

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

Answer 6

The curve has a cusp at x = 1

Question 7

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

Answer 7

-

Question 8

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

Answer 8

-