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What does the Fundamental Theorem of Calculus (Part 1) state?

If $f$ is continuous on $[a, b]$ , then the function $F$ defined by $F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , and $F'(x) = f(x)$ .

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What does the Fundamental Theorem of Calculus (Part 1) state?

If $f$ is continuous on $[a, b]$ , then the function $F$ defined by $F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , and $F'(x) = f(x)$ .

What does the Fundamental Theorem of Calculus (Part 2) state?

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$ , then $\int_a^b f(x) dx = F(b) - F(a)$ .

How does the Fundamental Theorem of Calculus relate differentiation and integration?

It shows that differentiation and integration are inverse processes.

What is the Mean Value Theorem for Integrals?

If $f$ is continuous on $[a, b]$ , then there exists a number $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$ .

What is the Intermediate Value Theorem?

If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$ , then there exists at least one number $c$ in $(a, b)$ such that $f(c) = k$ .

How can the Fundamental Theorem of Calculus be used to find the area under a curve?

By finding the antiderivative of the function and evaluating it at the limits of integration.

What is the relationship between the derivative of an integral and the original function according to the Fundamental Theorem of Calculus?

The derivative of the integral of a function is the original function itself.

What is the Squeeze Theorem?

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$ ) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$ , then $\lim_{x \to a} g(x) = L$ .

What does the Extreme Value Theorem state?

If $f$ is continuous on a closed interval $[a, b]$ , then $f$ has both a maximum and a minimum value on that interval.

What is Rolle's Theorem?

If a function $f$ is continuous on the closed interval $[a, b]$ , differentiable on the open interval $(a, b)$ , and $f(a) = f(b)$ , then there exists at least one $c$ in the open interval $(a, b)$ such that $f'(c) = 0$ .

Define Accumulation Function.

A function that represents the accumulated area under a curve from a fixed point to a variable point.

What is an antiderivative?

A function whose derivative is the given function.

Define the Fundamental Theorem of Calculus.

The theorem that links the concept of the integral of a function with the concept of the derivative of a function.

What is a point of inflection?

A point on a curve where the concavity changes.

Define relative maximum.

A point where the function's value is greater than or equal to the values at all nearby points.

Define relative minimum.

A point where the function's value is less than or equal to the values at all nearby points.

Define Concave Up.

A curve that opens upwards.

Define Concave Down.

A curve that opens downwards.

What is the area under the curve?

The integral of a function between two points, representing the accumulation of the function's values.

Define critical point.

A point where the derivative of a function is either zero or undefined.

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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What does the Fundamental Theorem of Calculus (Part 1) state?

If $f$ is continuous on $[a, b]$ , then the function $F$ defined by $F(x) = \int_a^x f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , and $F'(x) = f(x)$ .

What does the Fundamental Theorem of Calculus (Part 2) state?

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$ , then $\int_a^b f(x) dx = F(b) - F(a)$ .

How does the Fundamental Theorem of Calculus relate differentiation and integration?

It shows that differentiation and integration are inverse processes.

What is the Mean Value Theorem for Integrals?

If $f$ is continuous on $[a, b]$ , then there exists a number $c$ in $[a, b]$ such that $\int_a^b f(x) dx = f(c)(b - a)$ .

What is the Intermediate Value Theorem?

If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$ , then there exists at least one number $c$ in $(a, b)$ such that $f(c) = k$ .

How can the Fundamental Theorem of Calculus be used to find the area under a curve?

By finding the antiderivative of the function and evaluating it at the limits of integration.

What is the relationship between the derivative of an integral and the original function according to the Fundamental Theorem of Calculus?

The derivative of the integral of a function is the original function itself.

What is the Squeeze Theorem?

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $a$ (except possibly at $a$ ) and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$ , then $\lim_{x \to a} g(x) = L$ .

What does the Extreme Value Theorem state?

If $f$ is continuous on a closed interval $[a, b]$ , then $f$ has both a maximum and a minimum value on that interval.

What is Rolle's Theorem?

Define Accumulation Function.

A function that represents the accumulated area under a curve from a fixed point to a variable point.

What is an antiderivative?

A function whose derivative is the given function.

Define the Fundamental Theorem of Calculus.

The theorem that links the concept of the integral of a function with the concept of the derivative of a function.

What is a point of inflection?

A point on a curve where the concavity changes.

Define relative maximum.

A point where the function's value is greater than or equal to the values at all nearby points.

Define relative minimum.

A point where the function's value is less than or equal to the values at all nearby points.

Define Concave Up.

A curve that opens upwards.

Define Concave Down.

A curve that opens downwards.

What is the area under the curve?

The integral of a function between two points, representing the accumulation of the function's values.

Define critical point.

A point where the derivative of a function is either zero or undefined.

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).