Rates of Change
Rectilinear Motion
Lesson 3.4
Rate of Change
y
x
y
m
x
• 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
y
x
Average Rate of Change
• For any function, f(x), the average rate of
change is
x
f ( x  x )  f ( x )
x
f ( x  x)  f ( x)
Instantaneous Rate of Change
• The instantaneous rate of change is the derivative
f ( x  x)  f ( x)
dy
lim
 f '( x0 ) 
x 0
x
dx
x  x0
evaluated at the point x0
• Given a function f(x) and a point x0 the
instantaneous rate of change = f ‘(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
Position
Velocity
Acceleration
Why the “dimension mismatch”
error message?
Falling Objects
• When an object falls we know
1
2
h(t )   g  t  v0  t  s0
2
• Where
• s0 is the initial height
• v0 is the initial velocity
• g is the acceleration due to gravity 32 or 9.8
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
f '( x0 )
f ( x0 )
• Example: given aerobic rating
ln x  2
A( x)  110
x
• What is the relative rate of change at x = 20?
Assignment
• Lesson 3.4
• Page 125
• Exercises 1 – 51 odd