2.7 Derivatives and Rates of Change1

Limits and Derivatives

3.1 Derivatives and Rates of Change

2.7 Derivatives and Rates of Change2

If a curve C has equation y = f (x) and we want to find the tangent line to C at the point P (a, f (a)), then we consider a nearby point Q (x, f (x)), where x a, and compute the slope of the secant line PQ:

Then we let Q approach P along the curve C by letting x approach a.

Slope and Tangents

2.7 Derivatives and Rates of Change3

If mPQ approaches a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. )

Slope and Tangents

2.7 Derivatives and Rates of Change4

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

provided the limit exists.

Definition 1

( ) ( )limx a

f x f am

x a

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Another expression for the tangent line that is easier to use

Definition (Equation 2)

0

( ) ( )limh

f a h f am

h

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Find an equation for the tangent line to the curve y = 2/x at the point (2,1) on this curve.

Example

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We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point.

The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line.

Let’s zoom in on the point (1, 1) on the parabola y = x2

Tangent

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The average velocity over this time interval is

which is the same as the slope of the secant line PQ.

Definition – Average Velocity

displacement ( ) ( )( )

time

f a h f aAV v a

h

Note: This is just the difference quotient whenx = a!

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Now suppose we compute the average velocities over shorter and shorter time intervals [a, a + h].

In other words, we let h approach 0. As in the example of the falling ball, we define the velocity (or instantaneous velocity) v (a) at time t = a to be the limit of these average velocities:

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

Definition – Instantaneous Velocity

0

( ) ( )( ) lim

h

f a h f aIV v a

h

Note: This is just the slope of the tangent line!!

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Verify that she hasn’t hit the ground at 6 seconds. Find the instantaneous velocity at 6 seconds.

Example: Free Fall

Sue decided to jump out of a perfectly good plane that is 2.5 miles (13200 feet) above ground. Neglecting air resistance and assuming initial velocity is zero, the distance fallen is denoted by the function

2( ) 13200 16s t t

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

if this limit exists.

Definition

0

( ) ( )'( ) lim

h

f a h f af a

h

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

Note

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Find the derivative with respect to a of the function below

Example

3( )f x x x

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Instantaneous rate of change

Definition

2 1

2 1

02 1

( ) ( )lim limx x x

f x f xyIRC

x x x

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Find the IRC of the function below with respect to x when x1=-4

Example

2( ) 1f x x

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a) Find the slope of the tangent line through the point (1, 3) of the parabola .

i. using Definition 1.

ii. using Equation 2.

b) Find an equation of the tangent line in part (a).

c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point (1,3) until the parabola and the tangent line are indistinguishable.

Book Example – Page 119 #3

24y x x

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Shown are the graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie.

a) Describe and compare how the runners run the race.

b) At what time is the distance between the runners the greatest?

c) At what time do they have the same velocity?

Book Example – Page 120 #12

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If , find and use it to find an equation of the tangent line to the curve

at the point (2,2).

Illustrate the first part by graphing the curve and the tangent line on the same screen.

Book Example – Page 121 #23

2

5( )

1

xF x

x

'(2)F

2

5

1

xy

x

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If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by . Find the velocity when t=2.

Book Example – Page 120 #13

2( ) 40 16f t t t

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Each limit represents the derivative of some function f at some number a. State such an f and a in the following case:

Book Example – Page 121 #32

4

0

16 2limh

h

h

2.7 Derivatives and Rates of Change21

Page 120 #5-50 multiples of 5

Assignment