MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Change - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §2.1: Derivatives and Rates of Change

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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Introduction

Our main goal in this section is to define and determine the relationships between several quantities: Velocity (average and instantaneous) Slope (of secant lines and tangent lines) Rate of Change (average and instantaneous) Derivative of a Function We will see that velocity is actually a special case of a rate of change, so we will concentrate on the last three concepts.

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Velocity

Recall that as an introduction to limits, we considered average velocity on an interval and defined instantaneous velocity at a specific time as a limit

• f average velocities on smaller and smaller time intervals.

More precisely, suppose an object is moving along a straight line so that its position at time t is given by s(t). The average velocity on the interval t = a to t = a + h is given by vavg = ∆s ∆t = s(a + h) − s(a) (a + h) − a = s(a + h) − s(a) h . The instantaneous velocity at time t = a is then v(a) = lim

h→0 vavg = lim h→0

s(a + h) − s(a) h . The instantaneous velocity v(a) is a measure of how fast the position is changing with respect to time at the time t = a. In other words, it is the “rate of change” of position with respect to time.

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Rates of Change

More generally, we make the following definition:

Definition

Let y = f (x) be a function and let a be a number in the domain of f . The average rate of change of y with respect to x on the interval x = a to x = a + h is Ravg = ∆y ∆x = f (a + h) − f (a) h . The instantaneous rate of change of y with respect to x at the point x = a is defined to be Rinst = lim

h→0 Ravg = lim h→0

f (a + h) − f (a) h .

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Rates of Change

Notes: Average or instantaneous velocity is the average or instantaneous rate

• f change of position with respect to time.

The units of the rate of change of y with respect to x are y-units per x-unit. Notice that the slope of the line passing through the points P(a, f (a)) and Q(a + h, f (a + h)) on the graph of f is m = ∆y ∆x = f (a + h) − f (a) (a + h) − a = f (a + h) − f (a) h = Ravg. Thus average rate of change on an interval is the slope of the line passing through the points on the graph at the endpoints of the interval.

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Tangent Lines

Let y = f (x) be a function and let a be a number in the domain of f . A line passing through two points on the graph is called a secant line. For example, the following figure illustrates the secant line through the points P(a, f (a)) and Q(a + h, f (a + h)) on the graph of f : §2.1 Figure 6 Notice that the “rise” (∆y) is f (a + h) − f (a) and the “run” (∆x) is h, and so the slope of the secant line is f (a + h) − f (a) h , which is the average rate of change of y with respect to x on the interval [a, a + h].

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Tangent Lines

Now instantaneous rate of change of y with respect to x is found by taking the limit as h → 0 of the average rate of change on [a, a + h]. This is also the limit as h → 0 of the slope of the secant line through P(a, f (a)) and Q(a + h, f (a + h)). What is the graphical meaning of this limit? As h → 0, the point Q moves along the graph toward the point P, as shown in the following figure: §2.1 Figure 5 In the limit, we think of Q moving all the way to P and the secant line becomes a tangent line at the point P(a, f (a)) (line t in the figure).

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Tangent Lines

Definition

Let y = f (x) be a function and let a be a number in the domain of f . The tangent line to the graph of f at x = a is the line passing through the point (a, f (a)) with slope m = lim

h→0

f (a + h) − f (a) h . Notes: The tangent line exists only if lim

h→0 f (a+h)−f (a) h

exists. By the Point-Slope formula, the equation of the tangent line is y − f (a) = m(x − a). This says that the instantaneous rate of change of y with respect to x at x = a is equal to the slope of the tangent line to the graph

• f f at x = a.

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Tangent Lines

Slope of Secant Line Through (a, f (a)) and (a+h, f (a+h)) = f (a + h) − f (a) h = Average Rate

• f Change on

[a, a + h]

❄

h → 0

❄

h → 0

❄

h → 0 Slope of Tangent Line at x = a = lim

h→0

f (a + h) − f (a) h = Instantaneous Rate of Change at x = a

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The Derivative

The expression lim

h→0 f (a+h)−f (a) h

has come up in enough different situations (instantaneous velocity, instantaneous rate of change, slope of a tangent line) that we want to give it its own name.

Definition

Let y = f (x) be a function and let a be a number in the domain of f . The derivative of f evaluated at x = a is f ′(a) = lim

h→0

f (a + h) − f (a) h , if this limit exists. We can now add f ′(a) to our chart of related quantities:

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The Derivative

Slope of Secant Line Through (a, f (a)) and (a+h, f (a+h)) = f (a + h) − f (a) h = Average Rate

• f Change on

[a, a + h]

❄

h → 0

❄

h → 0

❄

h → 0 Slope of Tangent Line at x = a = f ′(a) = lim

h→0 f (a+h)−f (a) h

= Instantaneous Rate of Change at x = a

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