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Instantaneous Rate of Change

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- The instantaneous rate of change of f with respect to x is

For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

Example 1:

For a circle:

Instantaneous rate of change of the area with

respect to the radius.

- What is the rate of change of the area of a circle with respect to the radius when r = 5cm? 10cm?
- If r is in inches what is an appropriate unit for dA/dr?

- Find the rate of change of the volume of a cube with respect to side length s.
- Find the rate when s = 4 inches.

from Economics:

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

Note that this is not a great approximation – Don’t let that bother you.

The actual cost is:

Example 13:

Suppose it costs:

to produce x stoves.

If you are currently producing 10 stoves, the 11th stove will cost approximately:

marginal cost

actual cost

Note that this is not a great approximation – Don’t let that bother you.

Marginal cost is a linear approximation of a curved function. For large values it gives a good approximation of the cost of producing the next item.

B

distance

(miles)

A

time (hours)

(The velocity at one moment in time.)

Consider a graph of displacement (distance traveled) vs. time.

Average velocity can be found by taking:

The speedometer in your car does not measure average velocity, but instantaneous velocity.

Acceleration is the derivative of velocity.

Speed is the absolute value of velocity.

If s(t) is the position function

Velocity is the first derivative of position.

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

- In general the vertical position of an object is
where g is –16ft/s or –4.9m/s

v0 is the initial velocity

s0 is the initial height

- Dynamite blasts a rock straight up with a velocity of 160 ft/s
- How high does the rock go?
- What is the velocity when it is 256 ft high on the way up? On the way down?
- What is the acceleration of the rock at any time t during the blast?
- When does the rock hit the ground?
- When is the velocity zero? What is the acceleration when v = 0?

- A silver dollar is dropped from a building 1362 feet tall
- determine the position and velocity functions
- find the instantaneous velocity at t = 1s and t = 2s
- Find the time it takes for the coin to reach the ground
- Find the velocity at impact

- A stone is dropped from a building and hits the ground below 6.8 s after it is dropped. What is the height of the building?

- A ball is thrown from a building 220 ft high at a velocity of -22ft/s. What is velocity after 3 s? After falling 108ft?

- The velocity of an object in m/s is
Find the velocity and acceleration of the object when t = 3

What can be said about the speed of the object when the velocity and acceleration have opposite signs?

- A car is traveling at a rate of 66 feet per second (45mph) when the brakes are applied. The position function for the car is s(t) = -8.25t2 + 66t. Complete the table and find the avg vel for each time interval.

- The distance in miles that a person drives to work is shown below. Sketch the velocity.

distance

2

2

time

- 1-14 omit 7
- 16,23,24,25,26,30,31