Derivatives

Rates of change

Connect changes in a quantity to the time or input interval over which they occur. Distinguish average, instantaneous, and relative rates.

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1. Average velocity

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Example graph

s(t)=t^2,\quad v_{avg}=\frac{b^2-1}{b-1}=b+1

With time t in seconds and position s(t)=t² meters, the average velocity from t=1 to t=b, b>1, is (b²−1)/(b−1)=b+1 m/s. It depends on the whole interval.

Δt: 1
Δs: 3
v avg: 3

The marked positions are at t=1 and t=b. The dashed secant slope is average velocity. The question uses t=3 regardless of the slider position.

2. Instantaneous velocity

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Example graph

s(t)=t^2,\quad v(t)=2t

For s(t)=t² meters and t in seconds, average velocities over shrinking intervals approach 2t. This instantaneous rate is the derivative s′(t), written v(t)=2t. A negative velocity describes direction, not negative speed.

s(a): 1
v(a): 2

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.

3. Relative growth

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Example graph

P(t)=e^{t/2},\quad\frac{P\prime(t)}{P(t)}=\frac12

For P(t)=e^(t/2), the absolute rate is P′(t)=P(t)/2, while P′/P=1/2 stays constant. A constant relative rate does not mean a constant absolute increase.

P(a): 1.648721
P′(a): 0.824361
P′/P: 0.5

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.