Calculus - Steelton-Highspire High School



AP Calculus AB 3.4 Velocity and Other Rates of Change

Objectives: Instantaneous Rates of Change; Motion along a line; Sensitivity to Change; Derivatives in Economics

Procedure:

1. Instantaneous Rates of Change:

a. Definition 1: Instantaneous Rate of Change:

The (instantaneous) rate of change of f with respect to x at a is the derivative

b. Example 1: Enlarging circles:

a. Find the rate of change of the area A of a circle with respect to its radius r.

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

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

2. Motion along a Line:

Suppose that an object is moving along a coordinate line (say an s-axis) so that we know its position s on that line as a function of time t:

The __________________ of the object over the time interval from [pic]

and the ____________________________ of the object over that time interval is

a. Definition 2: Instantaneous Velocity:

The instantaneous velocity is the derivative of the position function [pic] with respect to time. At time t the velocity is

b. Definition 3: Speed:

Speed is the absolute value of velocity.

c. Example 2: Reading a velocity graph:

A student walks around in front of a motion detector that records her velocity at 1-second intervals for 36 seconds. She stores the data in her graphing calculator and uses it to generate the time-velocity graph shown below. Describe her motion as a function of time by reading the velocity graph. When is her speed a maximum?

d. Definition 4: Acceleration:

Acceleration is the derivative of velocity with respect to time. If a body’s velocity at time t is [pic], then the body’s acceleration at time t is

e. Free-fall Constants:

English units:

Metric units:

f. Example 3: Modeling vertical motion:

A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec (about 109 mph). It reaches a height of [pic] ft after t seconds.

a. How high does the rock go?

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

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

d. When does the rock hit the ground?

g. Example 4: Studying particle motion:

A particle moves along a line so that its position at any time t ≥ 0 is given by the function [pic] 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 4 seconds.

c. Find the instantaneous velocity of the particle when t = 4.

d. Find the acceleration of the particle at t = 4.

e. Describe the motion of the particle. At what values of t does the particle change directions?

3. HW: Day I: p. 135 (1, 9, 10)

Day II: p. 136 (14, 15, 18, 19)

Day III: p. 137 (21, 24, 25)

Classwork: p. 138 (32, 34, 35, 38, 47)

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