Section 4.1

Rates of Change

The derivative isn't just a geometric slope—it's the engine of change in the physical world. From physics to economics, tells us how fast a system is evolving.

The Many Faces of f'(x)

Physics

If is position, then:

v(t) = s'(t)Velocity
a(t) = v'(t)Acceleration

Flow Rates

If is volume in a tank:

V'(t)Rate of Flow
Negative result means draining!

Worked Example: The Draining Tank

Torricelli's Law

A tank holds 5000 gallons. It drains in 40 mins. Volume remaining is . Find the rate of draining at min.

Torricelli's Tank

Time (min): 0.0

Volume vs. Time

Volume:5000 gal

Rate of Change:-250.0 gal/min

Interactive: Slide time to

to see the tangent slope.

1. Understand the Request

"Rate of draining" means we need the derivative evaluated at .

2. Differentiate

Using Chain Rule (or expand first):

3. Solve

Negative sign confirms volume is decreasing.

Level Up Examples

Example A: Particle Motion

A particle moves with position . Find when the velocity is zero, and the total distance traveled in the first 8 seconds.

1. Find Velocity v(t):

2. Set v(t) = 0:

Stops at and .

3. Calculate Distance:

Calculate displacement in each interval .

Total Distance = 32 + 32 + 32 = 96 meters.

Example B: Marginal Cost

Cost function: . Find the marginal cost at items.

1. Marginal Cost Definition:

Marginal Cost is just the derivative .

2. Evaluate at x = 20:

Interpretation:

The cost to produce the 21st item is approximately $12.