Average Versus Instantaneous Change

Learn average rate of change, instantaneous rate.

Lesson 1 of 32 parts Practice Flashcards

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Average Rate of Change

Objective: Explain average rate of change, apply the stated method, and check its conditions.

The average rate from u to v, with u≠v, is [f(v)−f(u)]/(v−u). It is the secant slope through two graph points. Its units are output units per input unit; it need not equal a rate at either endpoint. A secant rate summarizes an interval, not necessarily an instant.

Formulas

Average rate = Δf/Δx

Worked examples

Try itFor f(x)=x², find average rate from x=7 to x=13.Show answer

Step 1: Use the secant slope [f(v)−f(u)]/(v−u). Step 2: Compute [(13)²−49]/6. Step 3: The average rate is 20.

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Average Rate of Change

Objective: Explain average rate of change, apply the stated method, and check its conditions.

Formulas

Average rate = Δf/Δx

Worked examples

Try itFor f(x)=x², find average rate from x=7 to x=13.Show answer

Step 1: Use the secant slope [f(v)−f(u)]/(v−u). Step 2: Compute [(13)²−49]/6. Step 3: The average rate is 20.

Instantaneous Rate

Objective: Explain instantaneous rate, apply the stated method, and check its conditions.

An instantaneous rate is the limit of average rates as the interval shrinks to zero, if the limit exists. For position s(t), this derivative s′(t) is velocity. Signed velocity includes direction; speed is its absolute value. Speed is |velocity|, not always velocity itself.

Formulas

$v(t)=s′(t)$

Worked examples

Try itFor s(t)=6t², find instantaneous velocity at t=7.Show answer

Step 1: Differentiate: v(t)=2(6)t. Step 2: Substitute t=7. Step 3: v(7)=84 position units per time unit.

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