Explain how the Fundamental Theorem of Calculus connects area and antiderivatives.

The area under a function's curve is equal to the value of its antiderivative, calculated with the same bounds.

How do you determine where a function is increasing using its derivative?

A function is increasing where its first derivative is positive.

How do you determine where a function is decreasing using its derivative?

A function is decreasing where its first derivative is negative.

How do you find relative extrema using the first derivative?

Relative extrema occur where the first derivative is zero or undefined and changes sign.

How do you determine concavity using the second derivative?

A function is concave up where its second derivative is positive and concave down where it is negative.

How do you find inflection points using the second derivative?

Inflection points occur where the second derivative changes sign.

What does the area below the x-axis represent when calculating definite integrals?

Area below the x-axis is considered negative when calculating definite integrals.

Explain how the graph of $f'(x)$ relates to the increasing/decreasing behavior of $f(x)$ .

When $f'(x) > 0$ , $f(x)$ is increasing. When $f'(x) < 0$ , $f(x)$ is decreasing.

Explain how the graph of $f'(x)$ relates to the concavity of $f(x)$ .

The slope of $f'(x)$ indicates the concavity of $f(x)$ . Positive slope means concave up, negative slope means concave down.

Explain how to find absolute minimum/maximum on a closed interval.

Evaluate the function at critical points and endpoints; the smallest/largest value is the absolute minimum/maximum.

Given the graph of $f'(x)$ , what does the area between the curve and the x-axis represent?

The area represents the change in the original function $f(x)$ over the given interval.

Given the graph of $f'(x)$ , how do you identify intervals where $f(x)$ is increasing?

$f(x)$ is increasing where $f'(x)$ is above the x-axis (positive).

Given the graph of $f'(x)$ , how do you identify intervals where $f(x)$ is decreasing?

$f(x)$ is decreasing where $f'(x)$ is below the x-axis (negative).

Given the graph of $f'(x)$ , how do you identify relative extrema of $f(x)$ ?

Relative extrema occur where $f'(x)$ crosses the x-axis, changing sign.

Given the graph of $f'(x)$ , how do you identify inflection points of $f(x)$ ?

Inflection points occur where the slope of $f'(x)$ changes sign.

What does the x-intercept of the graph of $f'(x)$ represent?

It represents a critical point of the original function $f(x)$ .

How can you find the value of $f(b)$ if you have the graph of $f'(x)$ and the value of $f(a)$ ?

Use the formula $f(b) = f(a) + \int_a^b f'(x) dx$ . The integral is the area under the curve of $f'(x)$ from $a$ to $b$ .

Given the graph of $f'(x)$ , how do you determine the concavity of $f(x)$ ?

If $f'(x)$ is increasing, $f(x)$ is concave up. If $f'(x)$ is decreasing, $f(x)$ is concave down.

Given the graph of $f'(x)$ , what does a sharp corner or cusp indicate about $f(x)$ ?

It indicates a possible point where $f''(x)$ is undefined, but it's not necessarily an inflection point unless the concavity changes.

Given the graph of $f'(x)$ , how do you determine where $f(x)$ has a local maximum?

Look for points where $f'(x)$ changes from positive to negative.

Given $G(x) = \int_a^x f(t) dt$ , how do you find where $G(x)$ is increasing?

Find $G'(x) = f(x)$ . 2. Determine where $f(x) > 0$ .

Given $G(x) = \int_a^x f(t) dt$ , how do you find where $G(x)$ is concave up?

Find $G'(x) = f(x)$ . 2. Find $G''(x) = f'(x)$ . 3. Determine where $f'(x) > 0$ .

Given the graph of $f'(x)$ , how do you find the x-coordinates of inflection points of $f(x)$ ?

Identify points on the graph of $f'(x)$ where the slope changes sign (from positive to negative or vice versa).

Given $g(x) = f(x) - x$ and the graph of $f'(x)$ , how do you find where $g(x)$ is decreasing?

Find $g'(x) = f'(x) - 1$ . 2. Solve for $f'(x) < 1$ . 3. Identify the intervals where $f'(x)$ is less than 1.

Given $g(x) = f(x) - x$ and the graph of $f'(x)$ , how do you find the absolute minimum of $g(x)$ on $[a, b]$ ?

Find $g'(x) = f'(x) - 1$ . 2. Find critical points where $g'(x) = 0$ . 3. Evaluate $g(x)$ at critical points and endpoints $a$ and $b$ . 4. Choose the smallest value.

How to find the area of a region bounded by a curve and the x-axis?

Identify the interval [a, b]. 2. Integrate the function over the interval: $\int_a^b f(x) dx$ . 3. Take the absolute value if the region is below the x-axis.

How to determine if a function $f(x)$ has a relative maximum?

Find $f'(x)$ . 2. Find critical points where $f'(x) = 0$ or is undefined. 3. Check if $f'(x)$ changes from positive to negative at the critical point.

How to determine if a function $f(x)$ has a relative minimum?

Find $f'(x)$ . 2. Find critical points where $f'(x) = 0$ or is undefined. 3. Check if $f'(x)$ changes from negative to positive at the critical point.

How to find the value of $f(x)$ at a specific point given $f'(x)$ and an initial value?

Use the formula: $f(x) = f(a) + \int_a^x f'(t) dt$ , where $f(a)$ is the initial value.

How to determine intervals where a function is concave up or concave down?

Find $f''(x)$ . 2. Determine where $f''(x) > 0$ (concave up) and $f''(x) < 0$ (concave down).

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