What are the differences between a critical point and a point of inflection?

Critical Point: $f'(x) = 0$ or undefined, potential for max/min. | Point of Inflection: $f''(x)$ changes sign, change in concavity.

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What are the differences between a critical point and a point of inflection?

Critical Point: $f'(x) = 0$ or undefined, potential for max/min. | Point of Inflection: $f''(x)$ changes sign, change in concavity.

What are the differences between relative and absolute extrema?

Relative Extrema: Local max/min within an interval. | Absolute Extrema: Overall max/min over the entire domain.

What are the differences between increasing and concave up?

Increasing: $f'(x) > 0$ , function is rising. | Concave Up: $f''(x) > 0$ , function curves upwards.

What are the differences between decreasing and concave down?

Decreasing: $f'(x) < 0$ , function is falling. | Concave Down: $f''(x) < 0$ , function curves downwards.

What are the differences between the graph of a function and its derivative?

Function: Represents the value of the function at each point. | Derivative: Represents the rate of change of the function at each point.

What is the difference between a definite and an indefinite integral?

Definite Integral: Computes the area under a curve between two limits, resulting in a numerical value. | Indefinite Integral: Finds the antiderivative of a function, resulting in a family of functions.

What is the difference between $f'(x)$ and $\int f(x) dx$ ?

$f'(x)$ : The derivative of $f(x)$ , representing the instantaneous rate of change. | $\int f(x) dx$ : The antiderivative of $f(x)$ , representing the accumulation of $f(x)$ .

What is the difference between using the first derivative test and the second derivative test to find relative extrema?

First Derivative Test: Examines the sign change of $f'(x)$ around a critical point. | Second Derivative Test: Uses the sign of $f''(x)$ at a critical point to determine concavity and thus whether it is a max or min.

What is the difference between a local extremum and an endpoint extremum?

Local Extremum: A maximum or minimum within the interior of an interval. | Endpoint Extremum: A maximum or minimum that occurs at the boundary of an interval.

What is the difference between average rate of change and instantaneous rate of change?

Average Rate of Change: The slope of the secant line between two points. | Instantaneous Rate of Change: The slope of the tangent line at a single point, given by the derivative.

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).

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

What are the differences between a critical point and a point of inflection?

Critical Point: $f'(x) = 0$ or undefined, potential for max/min. | Point of Inflection: $f''(x)$ changes sign, change in concavity.

What are the differences between relative and absolute extrema?

Relative Extrema: Local max/min within an interval. | Absolute Extrema: Overall max/min over the entire domain.

What are the differences between increasing and concave up?

Increasing: $f'(x) > 0$ , function is rising. | Concave Up: $f''(x) > 0$ , function curves upwards.

What are the differences between decreasing and concave down?

Decreasing: $f'(x) < 0$ , function is falling. | Concave Down: $f''(x) < 0$ , function curves downwards.

What are the differences between the graph of a function and its derivative?

Function: Represents the value of the function at each point. | Derivative: Represents the rate of change of the function at each point.

What is the difference between a definite and an indefinite integral?

What is the difference between $f'(x)$ and $\int f(x) dx$ ?

$f'(x)$ : The derivative of $f(x)$ , representing the instantaneous rate of change. | $\int f(x) dx$ : The antiderivative of $f(x)$ , representing the accumulation of $f(x)$ .

What is the difference between using the first derivative test and the second derivative test to find relative extrema?

What is the difference between a local extremum and an endpoint extremum?

Local Extremum: A maximum or minimum within the interior of an interval. | Endpoint Extremum: A maximum or minimum that occurs at the boundary of an interval.

What is the difference between average rate of change and instantaneous rate of change?

Average Rate of Change: The slope of the secant line between two points. | Instantaneous Rate of Change: The slope of the tangent line at a single point, given by the derivative.

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).

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?

$lw$