2.3 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 change in position s(b) - s(a) change in time b - a average velocity = =

Change in Time If we define ∆t to be b - a, then b = a + ∆t, and s(b) - s(a) b - a average velocity == 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 ∆t→ 0 s( a + ∆t ) - s(a) ∆t∆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 (x 1, y 1 ) and (x 2, y 2 ) is (y 2 -y 1 ) / (x 2 - x 1 ) Slope of tangent is slope as (x 2 - x 1 ) approaches 0. x1x1 x2x2 x1x1 x2x2 x1x1 x2x2

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: lim ∆t→ 0 s( a + ∆t ) - s(a) ∆t∆t s’(a) = dy dt = t = a 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?

Differentiablity not continous: s(t) = -1 if t = 0 s(t) = t 2 otherwise continuous but not differentiable: s(t) = |t| differentiable: s(t) = t 2 t s(t)s(t)

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 d 2 y/dt 2.