(a) Calculate the average rate of change of temperature T from 6:11 AM to 9:05 AM.

(b) Use the figure to estimate the rate of change at t = 12:28 PM.

(a) Compute dA/dr at r = 2 and r = 5. (b) Why is dA/dr larger at r = 5?

The Effect of a One-Unit Change For small values of h, the difference quotient is close to the derivative itself:difference quotient derivative

Marginal Cost (the cost of producing or selling 1 more item) Marginal cost = C (x 0 + 1) − C (x 0 ) Cost of an Air Flight Company data suggests C (x) = x 3 − 0.38x x, where x is the number of passengers. (a) Estimate the marginal cost of an additional passenger if the flight already has 150 passengers.

Marginal Cost (the cost of producing or selling 1 more item) Marginal cost = C (x 0 + 1) − C (x 0 ) Cost of an Air Flight Company data suggests C (x) = x 3 − 0.38x x, where x is the number of passengers. (b) Compare your estimate with the actual cost of an additional passenger.

Actual Cost (c) Is it more expensive to add a passenger when x = 150 or when x = 200?

Linear Motion Speed is defined as |v (t)|. Speeding up or slowing down? The tangent lines get steeper in [0, 1], so the car was speeding up during the first hour. They get flatter in the interval [1, 2], so the car slowed down in the second hour. Standing still The graph is horizontal over [2, 3] (perhaps the driver stopped at a restaurant for an hour). Returning to the same spot The graph rises and falls in the interval [3, 4], indicating that the driver returned to the restaurant (perhaps she left her cell phone). Average velocity The graph rises more over [0, 2] than over [3, 4.5], so the average velocity was greater over the first two hours than during the last 1.5 hours.

Linear Motion 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 ≤ t ≤ 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?

Motion Under the Influence of Gravity (Galileo’s Quadratic Model) −g is acceleration due to gravity on the surface of the earth. Finding the Maximum Height A stone is shot with a slingshot vertically upward with an initial velocity of 50 m/s from an initial height of 10 m. (a) Find the velocity at t = 2 and at t = 7. Explain the change in sign. (b) What is the stone’s maximum height and when does it reach that height?

Finding Initial Conditions What initial velocity ν 0 is required for a bullet, fired vertically from ground level, to reach a maximum height of 2 km?