3.4 Rates of ChangeTues Sept 29

Do Now

Find the derivative of each:

1)

2)

Rates of Change

• Review: Average Rate of Change

Instantaneous Rate of Change

• Because slope is a rate of change, we can use derivatives to compute the instantaneous rate of change using different variables

• Note: dA/dR means you are differentiating A in terms of R

Ex

• Let A be the area of a circle with radius r.

• Compute dA/dr at r = 2 and r = 5

F’(x) and 1 unit change

• For small values of h, slope of two points represents the derivative

• F’(x) can be used to approximate the change in f(x) caused by a one-unit change

• Note: you can always just plug in the 2 values into f(x)

Ex• For speeds x between 30 and 75 mph, the

stopping distance of an automobile after the brakes are applied is approximately F(x) = 1.1x +0.05x^2 ft

• For x = 60mph, estimate the change in stopping distance if the speed is increased by 1mph

Marginal Cost in Economics

• To study the relation between cost and production, the marginal cost is the cost of producing one additional unit

• Ex 4 p.152

Linear Motion

• Linear motion - motion along a straight line

• S(t) denotes the distance from the origin at time t

• Velocity v(t) can be computed by ds/dt

Linear Motion

• Derivatives can also show us the relationship between position and velocity (and acceleration)

• If s(t) = position function then

• Velocity v(t) = ds/dt

Velocity

• The sign of velocity indicates going forward or backwards

• We can look at position graphs to determine the velocity of the function

• Ex 5 p.153

Ex5

• A truck enters the off-ramp of a highway at t = 0. Its position after t seconds is s(t) = 25t – 0.3t^3 m for [0,5]

• (A) How fast is the truck going at the moment it enters the off-ramp?

• (B) Is the truck speeding up or slowing down?

Velocity and Gravity

• The height s(t) of an object tossed vertically in the air is

• Its velocity is given by

#22

• The height (in meters) of a helicopter at time t (in min) is s(t) = 600t – 3t^3 for [0,12]

• (b) Find the velocity at t = 8, 10

• (c ) Find the maximum height

Closure

• Journal Entry: What other applications of derivatives did we learn about today? How r position and velocity related?

• HW: p.156 #6 7 11 16 21 27 39 43