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Define average rate of change.

Slope of the secant line between two points on a curve.

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Define average rate of change.

Slope of the secant line between two points on a curve.

Define instantaneous rate of change.

Slope of the tangent line at a single point; found using the derivative.

What is a secant line?

A line that intersects a curve at two or more points.

What is a tangent line?

A line that touches a curve at a single point, having the same slope as the curve at that point.

What is the derivative?

A measure of the instantaneous rate of change of a function.

Define limit in the context of derivatives.

The value that a function approaches as the input approaches some value.

What does $f'(c)$ represent?

The instantaneous rate of change of $f(x)$ at $x = c$ .

What is the relationship between the tangent line and the derivative?

The derivative gives the slope of the tangent line at a specific point.

What is the average velocity?

The average rate of change of position with respect to time over an interval.

What is instantaneous velocity?

The instantaneous rate of change of position with respect to time at a specific moment.

How to find the average rate of change of $f(x) = x^3$ on $[0, 2]$ ?

Calculate $f(2)$ and $f(0)$ . 2. Apply the formula: $\frac{f(2) - f(0)}{2 - 0}$ . 3. Simplify to get the answer.

How to find the instantaneous rate of change of $f(x) = 3x^2$ at $x = 1$ using the limit definition?

Set up the limit: $\lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}$ . 2. Substitute $f(x) = 3x^2$ . 3. Simplify and evaluate the limit.

Steps to find the equation of the tangent line to $f(x)$ at $x=a$ ?

Find $f(a)$ . 2. Find $f'(x)$ . 3. Find $f'(a)$ . 4. Use point-slope form: $y - f(a) = f'(a)(x - a)$ .

How to determine where a function has a horizontal tangent line?

Find the derivative $f'(x)$ . 2. Set $f'(x) = 0$ . 3. Solve for $x$ .

How to find the average velocity of a particle given its position function $s(t)$ over an interval $[a, b]$ ?

Calculate $s(b)$ and $s(a)$ . 2. Apply the formula: $\frac{s(b) - s(a)}{b - a}$ . 3. Simplify to get the answer.

How to determine if a function is increasing or decreasing at a point?

Find the derivative $f'(x)$ . 2. Evaluate $f'(x)$ at the point. 3. If $f'(x) > 0$ , increasing; if $f'(x) < 0$ , decreasing.

How to approximate the instantaneous rate of change using average rate of change?

Choose a small interval around the point. 2. Calculate the average rate of change over that interval. 3. This is an approximation of the instantaneous rate of change.

How to solve for the limit definition of a derivative?

Substitute the function into the limit definition. 2. Simplify the numerator. 3. Cancel out the h term in the denominator. 4. Evaluate the limit.

How to find the instantaneous velocity at t=1 if $s(t) = t^2 + 3t$ ?

Find the derivative $s'(t) = 2t + 3$ . 2. Substitute t=1 into $s'(t)$ . 3. $s'(1) = 2(1) + 3 = 5$ .

How to find the average rate of change of $f(x) = x^2 + 1$ from x = 0 to x = 2?

Find $f(2) = 2^2 + 1 = 5$ . 2. Find $f(0) = 0^2 + 1 = 1$ . 3. Apply formula: $\frac{5-1}{2-0} = 2$ .

Formula for average rate of change of $f(x)$ over $[a, b]$ ?

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

Formula for instantaneous rate of change of $f(x)$ at $x = c$ ?

$f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$

What is the power rule for derivatives?

If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$

What is the constant multiple rule for derivatives?

If $f(x) = cf(x)$ , then $f'(x) = cf'(x)$

What is the sum rule for derivatives?

If $h(x) = f(x) + g(x)$ , then $h'(x) = f'(x) + g'(x)$

Formula for the derivative of $x^2$ ?

If $f(x) = x^2$ , then $f'(x) = 2x$

Formula for the derivative of $\sqrt{x}$ ?

If $f(x) = \sqrt{x}$ , then $f'(x) = \frac{1}{2\sqrt{x}}$

Formula for the derivative of a constant $c$ ?

If $f(x) = c$ , then $f'(x) = 0$

What is the point-slope form of a line?

$y - y_1 = m(x - x_1)$

What is the slope-intercept form of a line?

$y = mx + b$

All Flashcards

Define average rate of change.

Slope of the secant line between two points on a curve.

Define instantaneous rate of change.

Slope of the tangent line at a single point; found using the derivative.

What is a secant line?

A line that intersects a curve at two or more points.

What is a tangent line?

A line that touches a curve at a single point, having the same slope as the curve at that point.

What is the derivative?

A measure of the instantaneous rate of change of a function.

Define limit in the context of derivatives.

The value that a function approaches as the input approaches some value.

What does $f'(c)$ represent?

The instantaneous rate of change of $f(x)$ at $x = c$ .

What is the relationship between the tangent line and the derivative?

The derivative gives the slope of the tangent line at a specific point.

What is the average velocity?

The average rate of change of position with respect to time over an interval.

What is instantaneous velocity?

The instantaneous rate of change of position with respect to time at a specific moment.

How to find the average rate of change of $f(x) = x^3$ on $[0, 2]$ ?

Calculate $f(2)$ and $f(0)$ . 2. Apply the formula: $\frac{f(2) - f(0)}{2 - 0}$ . 3. Simplify to get the answer.

How to find the instantaneous rate of change of $f(x) = 3x^2$ at $x = 1$ using the limit definition?

Set up the limit: $\lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}$ . 2. Substitute $f(x) = 3x^2$ . 3. Simplify and evaluate the limit.

Steps to find the equation of the tangent line to $f(x)$ at $x=a$ ?

Find $f(a)$ . 2. Find $f'(x)$ . 3. Find $f'(a)$ . 4. Use point-slope form: $y - f(a) = f'(a)(x - a)$ .

How to determine where a function has a horizontal tangent line?

Find the derivative $f'(x)$ . 2. Set $f'(x) = 0$ . 3. Solve for $x$ .

How to find the average velocity of a particle given its position function $s(t)$ over an interval $[a, b]$ ?

Calculate $s(b)$ and $s(a)$ . 2. Apply the formula: $\frac{s(b) - s(a)}{b - a}$ . 3. Simplify to get the answer.

How to determine if a function is increasing or decreasing at a point?

Find the derivative $f'(x)$ . 2. Evaluate $f'(x)$ at the point. 3. If $f'(x) > 0$ , increasing; if $f'(x) < 0$ , decreasing.

How to approximate the instantaneous rate of change using average rate of change?

Choose a small interval around the point. 2. Calculate the average rate of change over that interval. 3. This is an approximation of the instantaneous rate of change.

How to solve for the limit definition of a derivative?

Substitute the function into the limit definition. 2. Simplify the numerator. 3. Cancel out the h term in the denominator. 4. Evaluate the limit.

How to find the instantaneous velocity at t=1 if $s(t) = t^2 + 3t$ ?

Find the derivative $s'(t) = 2t + 3$ . 2. Substitute t=1 into $s'(t)$ . 3. $s'(1) = 2(1) + 3 = 5$ .

How to find the average rate of change of $f(x) = x^2 + 1$ from x = 0 to x = 2?

Find $f(2) = 2^2 + 1 = 5$ . 2. Find $f(0) = 0^2 + 1 = 1$ . 3. Apply formula: $\frac{5-1}{2-0} = 2$ .

Formula for average rate of change of $f(x)$ over $[a, b]$ ?

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

Formula for instantaneous rate of change of $f(x)$ at $x = c$ ?

$f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$

What is the power rule for derivatives?

If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$

What is the constant multiple rule for derivatives?

If $f(x) = cf(x)$ , then $f'(x) = cf'(x)$

What is the sum rule for derivatives?

If $h(x) = f(x) + g(x)$ , then $h'(x) = f'(x) + g'(x)$

Formula for the derivative of $x^2$ ?

If $f(x) = x^2$ , then $f'(x) = 2x$

Formula for the derivative of $\sqrt{x}$ ?

If $f(x) = \sqrt{x}$ , then $f'(x) = \frac{1}{2\sqrt{x}}$

Formula for the derivative of a constant $c$ ?

If $f(x) = c$ , then $f'(x) = 0$

What is the point-slope form of a line?

$y - y_1 = m(x - x_1)$

What is the slope-intercept form of a line?

$y = mx + b$