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Back to FlashcardsMathematics / AP Calculus AB: 8.4.3 Domain-Restricted Functions and the Derivative

AP Calculus AB: 8.4.3 Domain-Restricted Functions and the Derivative

Mathematics6 CardsCreated 8 months ago

This content explains how fractional exponents with even denominators restrict the domain of functions and how to analyze their behavior using first and second derivatives. It covers finding critical points, increasing/decreasing intervals, concavity, and endpoints within a restricted domain to accurately sketch the graph.

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Domain-Restricted Functions and the Derivative

If the variable of a function is raised to a fractional exponent that has an even denominator, the function may not be defined for all real numbers.
• 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.
• On an interval, the sign of the first derivative indicates whether the function is increasing or decreasing. The sign of the second derivative indicates whether the function is concave up or concave down

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Key Terms

Term
Definition

Domain-Restricted Functions and the Derivative

If the variable of a function is raised to a fractional exponent that has an even denominator, the function may not be defined for all real numbers...

note

  • If a function involves a fractional exponent that has an even denominator even when reduced, then the domain of the function may be restric...

Which of the following is the graph of y=x−√x−1 +2?

The graph of y=x−x−1+2y = x - \sqrt{x-1} + 2y=x−x−1​+2 is defined for x≥1x \geq 1x≥1 and looks like a curve starting at (1,2)(1, 2)(1,2) increasing...

Which of the following graphs is the best sketch of f(x)=√8−x?

The best sketch of f(x)=8−xf(x) = \sqrt{8 - x}f(x)=8−x​ is a decreasing curve starting at (0,8)(0, \sqrt{8})(0,8​) and ending at (8,0)(8, 0)(8,0), ...

All of the following are true except:

Domains must be restricted so that any cusp points are excluded.

Which of the following curves is the best sketch of f(x)=√x^2−1?

The best sketch of f(x)=x2−1f(x) = \sqrt{x^2 - 1}f(x)=x2−1​ is two branches starting from x=±1x = \pm 1x=±1, increasing symmetrically outward, with...

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AP Calculus AB: 8.4.3 Domain-Restricted Functions and the Derivative
TermDefinition

Domain-Restricted Functions and the Derivative

If the variable of a function is raised to a fractional exponent that has an even denominator, the function may not be defined for all real numbers.
• 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.
• On an interval, the sign of the first derivative indicates whether the function is increasing or decreasing. The sign of the second derivative indicates whether the function is concave up or concave down

note

  • If a function involves a fractional exponent that has an even denominator even when reduced, then the domain of the function may be restricted. This particular function is not defined for x-values greater than 6. Keep this in mind as you go through the steps for sketching its graph.

  • First, take the derivative using the product rule. Set it equal to zero to find the critical points. Then make a sign chart for first derivative. It is not defined for x-values greater than or equal to 6.

  • Next, take the second derivative, which requires the quotient rule. Set it equal to zero and solve for x. The only solution is x = 8, but that is not in the domain, so there are no inflection point candidates.

  • Now make a sign chart for the second derivative.

  • Once you have analyzed the function, sketch its graph.

  • The function is increasing to the left of x = 4 and decreasing from x = 4 to x = 6. It has a maximum at x = 4. There is an endpoint minimum at x = 6 because the function is decreasing at the end of its domain.

  • The graph is concave down everywhere, so there is no
    inflection point.

Which of the following is the graph of y=x−√x−1 +2?

The graph of y=x−x−1+2y = x - \sqrt{x-1} + 2y=x−x−1​+2 is defined for x≥1x \geq 1x≥1 and looks like a curve starting at (1,2)(1, 2)(1,2) increasing gradually, resembling a line shifted and slightly curved downward near x=1x=1x=1.

Which of the following graphs is the best sketch of f(x)=√8−x?

The best sketch of f(x)=8−xf(x) = \sqrt{8 - x}f(x)=8−x​ is a decreasing curve starting at (0,8)(0, \sqrt{8})(0,8​) and ending at (8,0)(8, 0)(8,0), defined for x≤8x \leq 8x≤8.

All of the following are true except:

Domains must be restricted so that any cusp points are excluded.

Which of the following curves is the best sketch of f(x)=√x^2−1?

The best sketch of f(x)=x2−1f(x) = \sqrt{x^2 - 1}f(x)=x2−1​ is two branches starting from x=±1x = \pm 1x=±1, increasing symmetrically outward, with a gap between −1-1−1 and 111 where the function is undefined.