Notes Packet 2A - Differentiation and Derivatives.pdf

Name: _______________________________________ Period: _________ CALCULUS NOTES

UNIT 2A – DIFFERENTIATION AND DERIVATIVES

Rates of change play a role whenever we study the relationship between two changing quantities. A familiar example is velocity, which is the rate of change of position with respect to time.

If an object is traveling in a straight line, the average velocity over a given time interval from t₁ to t₂

is defined as

Average velocity =

( ) ( )

2 1

2 1 change in position

change in time

s t s t s

t t t

− ∆

= =

∆ −

For example, if a car travels 200 miles in 4 hours, then its average velocity during this 4-hour period is

200 50

4 = miles per hour. At any given moment, the car may be going faster or slower than the

average.

We cannot define instantaneous velocity as a ratio because we would have to divide by the length of the time interval, which is zero. However we can estimate the instantaneous velocity by computing the average velocity over successively smaller intervals and then determine the limit of the average

velocity as the lengths of the time intervals get closer and closer to zero.

s t

( )

If we look at a graph of the position of the object, a secant line can be drawn through the points

(

t s t₁,

( )

₁

)

and ,

(

t₂ s t

( )

₂

)

. The average

velocity of the object would be the slope of the secant line through

( )

(

t s t1, 1

)

and ,

(

t2 s t

( )

2

)

. If we move the secant line so that t2 gets closer and closer to t₁, the secant line gets closer and closer to

becoming a tangentline to the graph at

(

t s t1,

( )

1

)

. The slope of t the tangent line is the instantaneousvelocity of the object when the

time is t₁. Graph of Position Function

Velocity is only one of many examples of a rate of change. Our same reasoning applies to any quantity y that depends on a variable x. The average rate of change of y with respect to x over the interval x₁ to x₂ is the ratio

Average rate of change =

( ) ( )

2 1 2 1 change in

change in

y x y x

y y

x x x x

− ∆

= =

∆ −

The instantaneous rate of change at x=x₁ is the limit of the average rates of change as x₂ gets closer and closer to x₁. We must consider values of x₂ on both the left side of x₁ and on the right side of x₁

as we take the limit.

An expression such as

( ) ( )

2 1 2 1

s t s t t t

−

− or

( )

( )

f x f c x c

−

The instantaneous rate of change of a function is called its derivative and is defined as follows.

Definition of the derivative:

The derivative of f x

( )

at x = c is the limit of the difference quotients (if it exists):

( )

(

)

( )

0 lim

h

f x h f x

f x

h

→

+ −

′ =

When the limit exists, we say that f is differentiable at x = c. An equivalent definition of the derivative is called the alternative form of the definition of the derivative.

Alternative form of the definition of the derivative:

( )

lim

( )

( )

x c

f x f c

f c

x c

→

−

′ =

−

Ex. A diver jumps from a diving board that is 32 feet above the water. The position of the diver from the water is given by s t

( )

= −16t2+16t+32, where t is measured in seconds.

(a) Graph this function on your graphing calculator, and sketch the result below.

(b) Find the average velocity (average rate of change) at which the diver is moving for the time interval

[

1, 1.5 .

]

Average velocity =

(c) Compute the average velocity of the diver on the following time intervals.

Time interval

[

_{0.9, 1}

]

[

_{0.99, 1}

]

[

_{0.999, 1}

]

[

_{1, 1.001}

]

[

_{1, 1.01}

]

[

_{1, 1.1}

]

Average velocity

(d) Use your answers to (b) to estimate the instantaneous rate of change (instantaneous velocity) at which the diver is moving at time t

=

1 second.

Ex. Given f x

( )

=x2−4 .x

(a) Sketch the function and the tangent line to the graph at the point where x = 3.

(b) Based on your sketch, do you expect f′

( )

3 to be positive or negative?

(c) Compute f′

( )

3 two different ways: 1) by using the definition of the derivative

2) by using the alternative form of the definition of the derivative

(d) Write the equation of the tangent line to the graph of y= f x

( )

at x = 3. Leave your equation in

Graphing a Derivative Function

a) Draw a tangent line to the graph of f at each point, and find the slope of the tangent line. b) Fill in the table, using the slope as the f′

( )

x value.

c) Then graph the points

(

x f, ′

( )

x

)

on the graph of f′

( )

x grid. Connect the points to see the

graph of y= f′

( )

x .

Graph of f x

( )

Graph of f′

( )

x

x y _f′

( )

_x

0 1 2 - 1 - 2

x _f′

( )

_x

The Derivative and Differentiability

A function f is said to be differentiable at x = c if the derivative of f exists at x = c. In order for the derivative to exist at x = c, the graph of f must have local linearity there. In other words, if you zoom in over and over at x = c on the graph of f, the graph will look like a line (but not a vertical line).

A function f is not differentiable at x = c if:

1) f has a sharp turn at x = c. 2) f has a discontinuity at x = c.

Ex. f x

( )

= −x 2 Ex.

( )

(

)

2

1 2 f x x = −

f is not differentiable at x = 2 because there f is not differentiable at x = 2 because is a sharp turn at x = 2. there is a discontinuity at x = 2.

f is differentiable for all other values of x. f is differentiable for all other values of x. ______________________________________________________________________________ 3) f has a vertical tangent at x = c 4) f has a cusp at x = c.

Ex. _{f x}

( ) (

= _x−₁

)

13 _{Ex. f x}

( ) (

= _x−₁

)

23

f is not differentiable at x = 1 because f is not differentiable at x = 1 because there is a vertical tangent at x = 1. there is a cusp at x = 1.

f is differentiable for all other values of x. f is differentiable for all other values of x. ______________________________________________________________________________ To test for differentiability, you will use the use the alternative form of the definition of the derivative. You must know this formula.

Alternative form of the definition of the derivative:

( )

lim

( )

( )

x c

f x f c

f c x c → − ′ = −

Notice that this is the formula you learned previously for the slope of the tangent line.

1 2 3 4

1 2 3

1 2 3 4

1 2 3

−1 1 2 3

−1 1

−1 1 2 3

Ex. Use the alternative form of the derivative to find the derivative at x = c if it exists. (a) f x

( )

= 2x−6 c=3

______________________________________________________________________________

(b) _{f x}

( ) (

= −_{x 2}

)

23 _c=₂

______________________________________________________________________________

( )

2 2, 1

(c) 1

3 , 1

x x

f x c

x x

 + <

=_ =

1 2 3 4 5 6 7 8 9 10 1

2 3 4 5

x y

A

B

C D

E

F

More on Derivatives

Ex. List the x-values for which the function appears to be: (a) not continuous

(b) not differentiable

________________________________________________________________________________ Ex. At which labeled points is the slope of the graph:

(a) zero?

(b) positive?

(c) negative?

________________________________________________________________________________

Ex. The graph of derivative function y = f ‘ (x) is given on the right.

Identify the x-value(s) where f (x) is:

(a) Increasing

(b) Decreasing

(c) Is instantaneously constant (i.e. has a horizontal tangent)

(d) At a relative extrema and classify each as a maximum or a minimum

Ex. Let f be a function which satisfies f

(

2+ −h

)

f

( )

2 =5h−3h2+9h3 for all real numbers h.

Find f′

( )

2 .

________________________________________________________________________________

Ex. Let f be a function which satisfies the property f x

(

+y

)

= f x

( )

−4f y

( )

+9xy for all real

numbers x and y, and suppose that

( )

0 2

lim

h

f h h

→ = . Use the definition of the derivative

to find f′

( )

x .

__________________________________________________________________________________

Ex. Given

( )

( )

5

5 3. 5

lim

x

f x f

x

→

−

=

−

Which of the following MUST be true, MIGHT be true, or can NEVER be true?

(a) f′

( )

5 =3

(b) f′

( )

5 =0

(c) f

( )

5 =3

(d) f is continuous at x = 0.

(e) f is continuous at x = 5.

(f)

( )

( )

5 5

lim

DERIVATIVES USING DATA

Ex. Water is flowing into a tank over a 24-hour period. The amount of water in the tank is modeled by a differentiable function W for 0≤ ≤t 24, where t is measured in hours and W t

( )

is measured in gallons. Values of W t

( )

at selected values of time t are shown in the table below.

t (hours) 0 4 8 12 16 20 24

( )

W t (gallons) 150 184 221 257 294 327 357

(a) Use the data in the table to find W

( )

8 . Using appropriate units, explain the meaning of your answer.

(b) Use the data in the table to find W−1

(

257

)

. Using appropriate units, explain the meaning of your answer.

(c) Use the data in the table to find an approximation for W′

( )

15 . Show the computations that lead to your answer. Using appropriate units, explain the meaning of your answer.

(d) Use the data in the table to find the average rate of change of W t

( )

over the time period 4≤ ≤t 20 hours. Show the computations that lead to your answer.

(f) A model for the amount of water in the tank is given by

( )

1

(

3 30 2 1800 33, 750

)

225

A t = − +t t + t+

where A t

( )

is measured in gallons and t is measured in hours. Find A′

( )

15 .