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 $f(x) > 0$ .

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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).

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.

What is the Fundamental Theorem of Calculus (Part 1)?

$\frac{d}{dx} \int_a^x f(t) dt = f(x)$

What is the formula for area under a curve?

$\int_a^b f(x) dx$

If $F(x) = \int_a^x f(t) dt$ , what is $F'(x)$ ?

$F'(x) = f(x)$

If $F(x) = \int_a^x f(t) dt$ , what is $F''(x)$ ?

$F''(x) = f'(x)$

How to find $f(0)$ given $f(4)$ and $f'(x)$ ?

$f(0) = f(4) - \int_0^4 f'(x) dx$

How to find $f(5)$ given $f(4)$ and $f'(x)$ ?

$f(5) = f(4) + \int_4^5 f'(x) dx$

Formula for $g'(x)$ if $g(x) = f(x) - x$ ?

$g'(x) = f'(x) - 1$

Area of a semicircle?

$\frac{\pi}{2}r^2$

Area of a triangle?

$\frac{1}{2}bh$

Area of a rectangle?

$lw$

All Flashcards

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).