What is the formula for the average rate of change of a function f(x) over the interval [a, b]?

$\frac{f(b) - f(a)}{b - a}$

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

What is the formula for the average rate of change of a function f(x) over the interval [a, b]?

$\frac{f(b) - f(a)}{b - a}$

What is the condition stated by the Mean Value Theorem?

$f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c \in (a, b)$

How do you find critical points?

Solve $f'(x) = 0$ or find where $f'(x)$ is undefined.

How do you determine concavity using the second derivative?

$f''(x) > 0$ implies concave up; $f''(x) < 0$ implies concave down.

How do you find inflection points?

Solve $f''(x) = 0$ or find where $f''(x)$ is undefined, and verify concavity changes.

What does f'(x) > 0 imply?

f(x) is increasing.

What does f'(x) < 0 imply?

f(x) is decreasing.

What does f''(x) = 0 imply?

Possible inflection point.

What does the second derivative test tell us about local extrema?

If $f'(c) = 0$ and $f''(c) > 0$ , then $f(c)$ is a local minimum. If $f'(c) = 0$ and $f''(c) < 0$ , then $f(c)$ is a local maximum.

What is the general approach to solving optimization problems?

Define the objective function. 2. Identify constraints. 3. Find critical points. 4. Test for extrema.

Define the Mean Value Theorem (MVT).

If a function is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Define the Extreme Value Theorem (EVT).

If a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a maximum and a minimum value on that interval.

What is a critical point of a function f(x)?

A value c in the domain of f(x) such that either f'(c) = 0 or f'(c) does not exist.

Define global (absolute) extrema.

The highest and lowest points of a function over its entire domain.

Define local (relative) extrema.

The highest and lowest points of a function over a specific subinterval of its domain.

Define concavity.

The curvature of a function at a given point; indicates whether the function is 'bending up' or 'bending down'.

Define an inflection point.

A point on a curve where the concavity changes.

Define optimization problems.

Mathematical problems that involve finding the best solution (minimum or maximum) among a set of possible solutions.

Define the first derivative test.

A method used to determine whether a function is increasing or decreasing on a specific interval by analyzing the sign of its first derivative.

Define the Candidates Test.

A method used to determine the absolute extrema of a continuous function on a closed interval by evaluating the function at critical points and endpoints.

How do you find intervals where f(x) is increasing or decreasing?

Find f'(x). 2. Determine critical points where f'(x) = 0 or is undefined. 3. Test intervals between critical points using f'(x) to determine if it's positive (increasing) or negative (decreasing).

How do you find local extrema using the first derivative test?

Find critical points. 2. Test the sign of f'(x) to the left and right of each critical point. 3. If f'(x) changes sign, a local extremum exists.

How do you determine the concavity of a function?

Find f''(x). 2. Determine where f''(x) = 0 or is undefined. 3. Test intervals using f''(x) to determine if it's positive (concave up) or negative (concave down).

How do you find inflection points?

Find f''(x). 2. Solve for f''(x) = 0 or where it is undefined. 3. Verify that the concavity changes at those points.

How do you find absolute extrema on a closed interval?

Find critical points within the interval. 2. Evaluate f(x) at critical points and endpoints. 3. Compare values to find the absolute maximum and minimum.

How do you solve an optimization problem?

Define the objective function. 2. Identify constraints. 3. Express the objective function in terms of one variable. 4. Find critical points. 5. Test for extrema.

How do you apply the Mean Value Theorem to a problem?

Verify the function is continuous and differentiable on the given interval. 2. Calculate the average rate of change: (f(b)-f(a))/(b-a). 3. Find c such that f'(c) equals the average rate of change.

How do you use the second derivative test to find local extrema?

Find critical points. 2. Calculate the second derivative, f''(x). 3. Evaluate f''(x) at each critical point. 4. If f''(x) > 0, local minimum. If f''(x) < 0, local maximum.

How do you determine if a critical point is a local max, min, or neither?

Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)).

How do you sketch a graph of f(x) given f'(x) and f''(x)?

Identify critical points from f'(x). 2. Determine intervals of increasing/decreasing from f'(x). 3. Identify inflection points from f''(x). 4. Determine concavity from f''(x). 5. Sketch the graph using this information.

All Flashcards

What is the formula for the average rate of change of a function f(x) over the interval [a, b]?

$\frac{f(b) - f(a)}{b - a}$

What is the condition stated by the Mean Value Theorem?

$f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c \in (a, b)$

How do you find critical points?

Solve $f'(x) = 0$ or find where $f'(x)$ is undefined.

How do you determine concavity using the second derivative?

$f''(x) > 0$ implies concave up; $f''(x) < 0$ implies concave down.

How do you find inflection points?

Solve $f''(x) = 0$ or find where $f''(x)$ is undefined, and verify concavity changes.

What does f'(x) > 0 imply?

f(x) is increasing.

What does f'(x) < 0 imply?

f(x) is decreasing.

What does f''(x) = 0 imply?

Possible inflection point.

What does the second derivative test tell us about local extrema?

If $f'(c) = 0$ and $f''(c) > 0$ , then $f(c)$ is a local minimum. If $f'(c) = 0$ and $f''(c) < 0$ , then $f(c)$ is a local maximum.

What is the general approach to solving optimization problems?

Define the objective function. 2. Identify constraints. 3. Find critical points. 4. Test for extrema.

Define the Mean Value Theorem (MVT).

If a function is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Define the Extreme Value Theorem (EVT).

If a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a maximum and a minimum value on that interval.

What is a critical point of a function f(x)?

A value c in the domain of f(x) such that either f'(c) = 0 or f'(c) does not exist.

Define global (absolute) extrema.

The highest and lowest points of a function over its entire domain.

Define local (relative) extrema.

The highest and lowest points of a function over a specific subinterval of its domain.

Define concavity.

The curvature of a function at a given point; indicates whether the function is 'bending up' or 'bending down'.

Define an inflection point.

A point on a curve where the concavity changes.

Define optimization problems.

Mathematical problems that involve finding the best solution (minimum or maximum) among a set of possible solutions.

Define the first derivative test.

A method used to determine whether a function is increasing or decreasing on a specific interval by analyzing the sign of its first derivative.

Define the Candidates Test.

A method used to determine the absolute extrema of a continuous function on a closed interval by evaluating the function at critical points and endpoints.

How do you find intervals where f(x) is increasing or decreasing?

Find f'(x). 2. Determine critical points where f'(x) = 0 or is undefined. 3. Test intervals between critical points using f'(x) to determine if it's positive (increasing) or negative (decreasing).

How do you find local extrema using the first derivative test?

Find critical points. 2. Test the sign of f'(x) to the left and right of each critical point. 3. If f'(x) changes sign, a local extremum exists.

How do you determine the concavity of a function?

Find f''(x). 2. Determine where f''(x) = 0 or is undefined. 3. Test intervals using f''(x) to determine if it's positive (concave up) or negative (concave down).

How do you find inflection points?

Find f''(x). 2. Solve for f''(x) = 0 or where it is undefined. 3. Verify that the concavity changes at those points.

How do you find absolute extrema on a closed interval?

Find critical points within the interval. 2. Evaluate f(x) at critical points and endpoints. 3. Compare values to find the absolute maximum and minimum.

How do you solve an optimization problem?

Define the objective function. 2. Identify constraints. 3. Express the objective function in terms of one variable. 4. Find critical points. 5. Test for extrema.

How do you apply the Mean Value Theorem to a problem?

Verify the function is continuous and differentiable on the given interval. 2. Calculate the average rate of change: (f(b)-f(a))/(b-a). 3. Find c such that f'(c) equals the average rate of change.

How do you use the second derivative test to find local extrema?

Find critical points. 2. Calculate the second derivative, f''(x). 3. Evaluate f''(x) at each critical point. 4. If f''(x) > 0, local minimum. If f''(x) < 0, local maximum.

How do you determine if a critical point is a local max, min, or neither?

Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)).

How do you sketch a graph of f(x) given f'(x) and f''(x)?

Identify critical points from f'(x). 2. Determine intervals of increasing/decreasing from f'(x). 3. Identify inflection points from f''(x). 4. Determine concavity from f''(x). 5. Sketch the graph using this information.