Rates of Change Rectilinear Motion Lesson 3.4 Rate of Change Consider the linear function y = m x + b rate at which y is changing with respect to x is.

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Presentation transcript:

Rates of Change Rectilinear Motion Lesson 3.4

Rate of Change Consider the linear function y = m x + b rate at which y is changing with respect to x is the slope, m The slope, the rate of change is constant

Rate of Change Consider a quadratic function For this function the rate of change is … not a constant changing different for different values of x

Average Rate of Change For any function, f(x), the average rate of change is

Instantaneous Rate of Change The instantaneous rate of change is the derivative Given a function f(x) and a point x 0 the instantaneous rate of change = f ‘(x 0 ) evaluated at the point x 0

Rectilinear Motion The object is moving in a straight line Position is a function of time s(t) Rate of change of position is s‘(t) = v(t) Rate of change of position is the velocity

Velocity Velocity is also a function speed is the absolute value of velocity The rate of change of velocity is acceleration v’(t) = a(t) Consider s(t) = 3t 2 + 2t – 5 What is velocity? What is acceleration?

Velocity and Acceleration For s(t) = 3t 2 + 2t – 5 Velocity = v(t) = s’(t) = 6t + 2 ft/sec Acceleration = v’(t) = 6 ft/sec 2 Demonstrate in data matrix Column 1 has values 1 – 10 Column 2 has s(c1) Column 3 has d(s(x),x) | x=c1 Column 4 has d(s(x),x,2) | x = c1

Velocity and Acceleration Results: Why is there only one value showing for column 4? Now plot the ordered pairs

Velocity and Acceleration Setting up plots Why the “dimension mismatch” error message? Position Velocity Acceleration

Falling Objects When an object falls we know Where s 0 is the initial height v 0 is the initial velocity g is the acceleration due to gravity 32 or 9.8

Falling Objects Given a cannon shooting straight up v 0 = 320 ft/sec assume initial height = 5 What is its velocity after 3 seconds? Which direction is it heading at that time … up or down? How long until it hits the ground? What is its velocity at that time?

Relative Rate of Change Relative rate of change at a point is instantaneous rate of change quantity at that point Example: given aerobic rating What is the relative rate of change at x = 20?

Assignment Lesson 3.4 Page 125 Exercises 1 – 51 odd