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3.4 Rates of change

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3.4 Rates of change

- Example 1:
- Find the rate of change of the Area of a circle with respect to its radius.

- If r is measured in inches and A is measured in square inches, what units would be appropriate for dA/dr?

in2/in

These are units, so we do not cancel them!!!

- Evaluate the rate of change of A at r = 5 and r = 10.

Displacement of an object is how far an object has moved over time.

- Average velocity is the slope of a displacement vs. time graph.

- Instantaneous Velocity is the velocity at a certain point.
- Instantaneous Velocity is the first derivative of the position function.
- Speed is the absolute value of velocity.

- Acceleration is the rate at which a particle’s velocity changes.
- Measures how quickly the body picks up or loses speed.
- 2nd derivative of the position function!!!

A particle moves along a line so that its position at any time

t ≥ 0 is given by the function s(t) = 2t2 – 5t + 3 where s is measured in meters and t is measured in seconds.

- a.) Find the displacement of the particle during the first 2 seconds.

- b.) Find the average velocity of the particle during the first 6 seconds.

A particle moves along a line so that its position at any time

t ≥ 0 is given by the function s(t) = 2t2 – 5t + 3 where s is measured in meters and t is measured in seconds.

- c.) Find the instantaneous velocity of the particle at 6 seconds.

- d.) Find the acceleration of the particle when t = 6.

A particle moves along a line so that its position at any time

t ≥ 0 is given by the function s(t) = 2t2 – 5t + 3 where s is measured in meters and t is measured in seconds.

- e.) When does the particle change directions?

distance

time

It is important to understand the relationship between a position graph, velocity and acceleration:

acc neg

vel pos &

decreasing

acc neg

vel neg &

decreasing

acc zero

vel neg &

constant

acc zero

vel pos &

constant

acc pos

vel neg &

increasing

velocity

zero

acc pos

vel pos &

increasing

acc zero,

velocity zero

Gravitational

Constants:

Free-fall equation:

s is the position at any time t during the fall

g is the acceleration due to Earth’s gravity (gravitational constant)

Example: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft after t seconds.

- a.) How high does the rock go?

- Find when position = 0 and divide by 2 (symmetric path)

- Since it takes 10 seconds for the rock to hit the ground, it takes it 5 seconds to reach it max height.

Example: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft after t seconds.

- a.) How high does the rock go?

- Find when velocity = 0 (this is when the rock changes direction)

Example: A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft after t seconds.

- b.) What is the velocity and speed of the rock when it is 256 ft above the ground on the way up?

- At what time is the rock 256 ft above the ground on the way up?

- b.) What is the velocity and speed of the rock when it is 256 ft above the ground on the way down?

- At what time is the rock 256 ft above the ground on the way down?

- c.) What is the acceleration of the rock at any time t at any time t during its flight?

from Economics:

Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

Marginal revenue is the first derivative of the revenue function, and represents an approximation of the revenue of selling one more unit.