3.4 Rates of Change Tues Sept 29 Do Now Find the derivative of each: 1) 2)

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Presentation transcript:

3.4 Rates of Change Tues 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 #