Instantaneous Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative: provided the limit exists.

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3.4 Velocity and Other Rates of Change

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Presentation transcript:

Instantaneous Rate of Change The (instantaneous) rate of change of f with respect to x at a is the derivative: provided the limit exists.

(INSTANTANEOUS) RATE OF CHANGE Rate of change with respect to radius, time, area, feet Ex. 1: Find the rate of change in the area of a circle with respect to its radius A) What is the rate of change at r = 2, 6? B) What is the rate of change as r increases without bound?

RECTILINEAR MOTION s = f (t) describes the motion on a horizontal line, such as the x-axis, where s is the position on the line, and t is time Displacement from t to t + is

Average velocity = displacement travel time Instantaneous velocity, or simply put, velocity:

VELOCITY : indicates a direction as well as speed Velocity > 0 : object moves right or forward object moves up Velocity < 0 : object moves left or backward object moves down

SPEED : Refer to figure 3.25 on page 129 of your textbook.

ACCELERATION:derivative of velocity with respect to time

Free-falling bodies: the distance a body released from rest falling freely is proportional to the square of the amount of time fallen!

Vertical Motion Problem A dynamite blast projects a rock upward at a launch velocity of 10 ft/sec. Height :

A) How high does the rock fly? B) When is the velocity 5 ft/sec? C) What is acceleration at time t? D) When does the rock strike the ground?

A particle moves along a line so that its position at any time is given by the function: where time is in seconds and x(t) is in meters. Describe the motion of the particle for. Find the average velocity on [ 1, 3 ]. When does the particle change direction? Describe the velocity and speed of the particle. Find the acceleration when the velocity is zero.

On page 136 # 11 reads: the accompanying figure shows the velocity v = ds/dt = f (t) in m/sec of a body moving along a coordinate line. v (m/sec) t (sec) When does the body reverse direction? When, about, is the body moving at a constant speed? Graph the body’s speed from 0 to 10. Graph the acceleration, where defined.