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

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

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

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

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

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

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

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

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

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

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

1. 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.

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

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

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

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

1. 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.

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

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

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

The average rate of change shows the average amount that a function changes over a given interval.

The instantaneous rate of change shows the exact amount that a function is changing at a specific point.

The slope of the secant line connecting two points on the graph of the function.

The slope of the tangent line at a specific point on the graph of the function.

The instantaneous rate of change is the limit of the average rate of change as the interval approaches zero.

The derivative of a function at a point is equal to the instantaneous rate of change at that point.

It provides a rigorous way to define the derivative and understand its meaning.

The sign of the derivative indicates whether the function is increasing (positive) or decreasing (negative).

It can be used to calculate average speeds, growth rates, or changes in quantities over time.

It can be used to determine velocity at a specific time, reaction rates in chemistry, or the rate of change of a stock price.