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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 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.