Rates of Change Rectilinear Motion

1
Rates of ChangeRectilinear Motion

• Lesson 3.4

2
Rate of Change

• Consider the linear function y m x b
• rate at which y is changing with respect to xis
  the slope, m
• The slope, the rate of change is constant

3
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

4
Average Rate of Change

• For any function, f(x), the average rate of
  change is

5
Instantaneous Rate of Change

• The instantaneous rate of change is the
  derivative
• Given a function f(x) and a point x0 the
  instantaneous rate of change f (x0)

6
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

7
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) 3t2 2t 5
• What is velocity?
• What is acceleration?

8
Velocity and Acceleration

• For s(t) 3t2 2t 5
• Velocity v(t) s(t) 6t 2 ft/sec
• Acceleration v(t) 6 ft/sec2
• Demonstrate in data matrix
• Column 1 has values 1 10
• Column 2 has s(c1)
• Column 3 has d(s(x),x) xc1
• Column 4 has d(s(x),x,2) x c1

9
Velocity and Acceleration

• Results
• Why is there only one value showing for column 4?
• Now plot the ordered pairs

10
Velocity and Acceleration

• Setting up plots

Why the dimension mismatch error message?
11
Falling Objects

• When an object falls we know
• Where
• s0 is the initial height
• v0 is the initial velocity
• g is the acceleration due to gravity 32 or 9.8

12
Falling Objects

• Given a cannon shooting straight up
• v0 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?

13
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?

14
Assignment

• Lesson 3.4
• Page 125
• Exercises 1 51 odd