Calculus Derivative Tests Flashcards

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Questions and Answers

What does Rolle's Theorem state?

It guarantees the existence of extrema on an interval if f is continuous and differentiable on the interval, and f(a) = f(b) then f'(c) = 0.

How do you find increasing and decreasing of a function?

Find all the critical numbers. 2. Make a sign line with the critical number and establish test intervals. 3. Determine the sign for each test interval. 4. Use the sign to decide if the function is increasing, decreasing, or constant.

What is a critical number?

A critical number occurs when the derivative is zero or undefined.

Where do relative extrema occur?

Relative minima and maxima only occur at critical numbers. Signup and view all the answers

Where do absolute extremes occur?

Absolute extrema can only come from endpoints or critical numbers. Signup and view all the answers

What does the first derivative test state?

If f'(x) changes from negative to positive at c (c = critical number), then f(c) is a relative minimum. If f'(x) changes from positive to negative at c, then f(c) is a relative maximum. Signup and view all the answers

What is extrema?

The maximum and minimum values of the function. Signup and view all the answers

What is the Extrema Value Theorem?

If f is continuous on the closed interval [a, b], then f has both a maximum and a minimum on the interval. Signup and view all the answers

What is the Mean Value Theorem?

If f(x) is continuous and differentiable, the slope of the tangent line equals the slope of the secant line at least once in the interval (a, b). Signup and view all the answers

What is concavity?

Where f is curving upward or curving downward. Signup and view all the answers

What test do you associate concavity with?

The second derivative test. Signup and view all the answers

What test do you associate with increasing and decreasing?

The first derivative test. Signup and view all the answers

What is a point of inflection?

A point where the graph of f changes from concave upward to concave downward or vice versa. Signup and view all the answers

What is a hypercritical point?

Where the second derivative equals 0. Signup and view all the answers

What is the second derivative test?

The second derivative test helps find relative minima and maxima by plugging the critical numbers into the second derivative. Signup and view all the answers

Study Notes

Rolle's Theorem

Ensures the existence of extrema on a closed interval if the function is continuous and differentiable.
If the function values at the endpoints are equal (f(a) = f(b)), then there exists at least one point c in the interval such that f'(c) = 0.

Finding Increasing and Decreasing Functions

Identify critical numbers where the derivative is zero or undefined.
Create a sign line with critical numbers to define test intervals.
Check the sign of the derivative in each interval to determine if the function is increasing, decreasing, or constant.

Critical Numbers

Defined as points where the derivative of a function is either zero or undefined (f'(x) = 0 or f'(x) is undefined).

Relative Extrema

Occur exclusively at critical numbers, determining local minima and maxima of the function.

Absolute Extrema

Absolute extrema can be found at endpoints of the interval or at critical numbers.

First Derivative Test

Indicates a relative minimum at c if f'(x) changes from negative to positive at that critical number.
Indicates a relative maximum at c if f'(x) changes from positive to negative.

Extrema

Refers to the maximum and minimum values that a function can attain.

Extreme Value Theorem

If a function is continuous on a closed interval [a, b], it must have both a maximum and a minimum within that interval.

Mean Value Theorem

For a continuous and differentiable function, there exists at least one c in the interval (a, b) where the slope of the tangent line equals the average slope of the function (f'(c) = [f(b) - f(a)] / (b - a)).

Concavity

Describes the direction in which a function curves: upward or downward.

Second Derivative Test

This test is used to analyze concavity and helps in identifying relative minima and maxima by evaluating the second derivative.

Relationship of Tests

First derivative test determines intervals of increase or decrease.
Second derivative test determines concavity and identifies inflection points.

Point of Inflection

A point on the graph where the concavity changes, occurring only when the second derivative equals zero (f''(x) = 0).

Hypercritical Point

Occurs when the second derivative of a function is zero, which may indicate potential inflection points.

Second Derivative Test for Extrema

Evaluate the second derivative at critical numbers:
- If the result is greater than zero, a relative minimum is present.
- If less than zero, a relative maximum is indicated.
- If equal to zero, the test fails, necessitating the use of the first derivative test for further analysis.

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Description

This quiz focuses on the concepts of the first and second derivative tests, as well as Rolle's Theorem. You'll learn to identify extrema and the behavior of functions using critical numbers and sign tests. Ideal for those studying calculus concepts related to derivatives.