Rates of ChangeRectilinear 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

xy

ym

x

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

x

y

Average Rate of Change

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

change is( ) ( )f x x f x

x

x ( ) ( )f x x f x

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)

000

( ) ( )lim '( ) x xx

f x x f x dyf x

x dx

evaluated at the point x0evaluated at the point x0

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) = 3t2 + 2t – 5• What is velocity?• What is acceleration?

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) | 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?

Why the “dimension mismatch” error message?

Position

Velocity

Acceleration

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

20 0

1( )

2h t g t v t s

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?

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?

0

0

'( )

( )

f x

f x

ln 2( ) 110

xA x

x

Assignment

• Lesson 3.4

• Page 125

• Exercises 1 – 51 odd