Chapter 3: Derivatives

3.1Derivatives and Rate of Change

Differential Calculus

• Study of how one quantity changes in relation to another quantity

• Central concept is the derivative– Uses the velocities and slopes of tangent lines

from chapter 2

Derivatives

• Special type of limit

• Like what we used to find the slope of a tangent line to a curve

• Or finding the instantaneous velocity of an object

• Interpreted as a rate of change

Slope of a Tangent Line

• If a curve has an equation of y = f(x), and we want to find the slope of a tangent line at some point P, then we would consider some nearby point Q and compute the slope of that “tangent line” as:

ax

afxfm

)()(

Slope of a Tangent Line (cont.)

• Then, we would pick a point Q that is even closer to P, and then closer, and closer

• The number that the slope would approach as we got closer to P (the limit) was the slope of the tangent line

Formal Definition

• The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope:

• As long as the limit exists!

ax

afxfm

ax

)()(lim

Example 1• Find an equation of the tangent line to the parabola

y = x2 at the point P(1,1).

Another definition for the slope of a tangent line…

h

afhafm

h

)()(lim

0

As x approaches a, h approaches 0 (because h = x – a)

So, the slope of the tangent line becomes the equation above!

Example 2• Find an equation of the tangent line to the

hyperbola y = 3/x at the point (3,1).

Average Velocity

• Ave velocity = displacementtime h

afhaf )()(

Instantaneous Velocity

• we take the average velocity over smaller and smaller intervals (as h approaches 0)

• Velocity v(a) is the limit of the average velocities:

h

afhafav

h

)()(lim)(

0

Instantaneous Velocity cont.

• This means that the velocity at time t = a is equal to the slope of the tangent line at P.

• Let’s reconsider the problem of the falling ball…

Example 3

• Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground.

• (a) what is the velocity of the ball after 5 seconds?• Remember the equation of motion s = f(t) = 4.9t2

Example 3 cont.

• Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground.

• (b) how fast is the ball traveling when it hits the ground?

Definition

• The derivative of a function f at a number a, denoted by f’(a), is:

• If that limit exists.

h

afhafaf

h

)()(lim)('

0

Equivalent way to write the derivative

ax

afxfaf

ax

)()(lim)('

Example 4

• Find the derivative of the function f(x) = x2 – 8x + 9 at the number a.

A note….

• The tangent line to y = f(x) at some point (a,f(a)), is the line through (a,f(a)) whose slope is equal to f’(a), the derivative of f at a.

Example 5• Find an equation of the tangent line to the parabola

y = x2 – 8x + 9 at the point (3,-6).

Rates of Change

• Suppose y is a quantity that depends on another quantity x

• This means y is a function of x, and we write y = f(x)• If x changes from x1 to x2, then the change in x

(called the increment of x) is:

• The corresponding change in y is:

12 xxx

)()( 12 xfxfy

Rates of Change (cont.)

• The difference quotient (slope) is:

• Called the average rate of change of y with respect to x

12

12 )()(

xx

xfxf

x

y

Instantaneous Rate of Change

12

12

0

)()(limlim

12 xx

xfxf

x

yxxx

Definition

• The derivative f’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a.

Homework

• P. 120

• 5, 7, 9 a&b, 13, 17, 25, 27, 29