Transcript 2.2 Rate of Change

2.2 Rate of Change
(Calc I Review)
Average Velocity
Suppose s(t) is the position of an object at
time t, where a ≤ t ≤ b. The average velocity,
or average rate of change of s with respect to
t, of the object from time a to time b is
average velocity =
change in position
change in time
s(b) - s(a)
=
b-a
Change in Time
If we define ∆t to be b - a, then b = a + ∆t, and
average velocity =
s(b) - s(a)
b-a
=
s(a + ∆ t) - s(a)
∆t
Limits
Suppose that as x approaches some number c,
the function f(x) approaches a number L. We
say that the limit of f(x) as x approaches c is L,
and we write
lim f(x) = L
x→c
Instantaneous Velocity
The instantaneous velocity, or the
instantaneous rate of change of s with respect
to t, at t = a is
lim s(a + ∆t) - s(a)
∆t→ 0
∆t
provided that the limit exists
Slope of Tangent Line
• We can think of decreasing ∆t as zooming in
closer on the graph of the function s.
• Q.: What happens to the appearance of the
function as we do this?
Slope of Tangent Line
• We can think of decreasing ∆t as zooming in
closer on the graph of the function s.
• Q.: What happens to the appearance of the
function as we do this?
• A.: The line appears to straighten out - i.e.,
we start seeing the linear slope as ∆t
approaches zero.
Slope of Tangent Line
• Formally, the slope of a (novertical) line through two
distinct points (x1, y1) and (x2, y2) is (y2-y1) / (x2- x1)
• Slope of tangent is slope as (x2- x1) approaches 0.
x1
x2
x1
x2
x1x2
Derivative
The derivative of y = s(t) with respect to t at
t = a is the instantaneous rate of change of s
with respect to t at a:
s’(a) =
lim s(a + ∆t) - s(a)
= ∆t→ 0
∆t
dt t = a
dy
provided that the limit exists. If the derivative of s
exists at a, we say the function is diffentiable at a.
What would it mean for a function not to be
differentiable?
s(t)
Differentiablity
t
not continous:
s(t) = -1 if t = 0
s(t) = t2 otherwise
continuous
differentiable:
but not
differentiable:
s(t) = |t|
s(t) = t2
Differential Equations
• A differential equation is an equation that
contains one or more derivatives.
• An initial condition is the value of the
dependent variable when the independent
variable is zero.
• A solution to a differential equation is a
function that satisfies the equation and initial
conditions.
Second Derivative
• Acceleration is the rate of change of velocity
with respect to time.
• The second derivative of a function y = s(t) is
the derivative of the derivative of y w.r.t. the
independent variable t. The notation for this
second derivative is s”(t) or d2y/dt2.