2.1 Derivatives and Rates of Change

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§2.1 Derivatives and Rates of Change

Tangent Lines

axes, curve C

Consider a smooth curve C.

A line tangent to C at a point P both intersects C at P and has the same slope as C at P. add line 

The Tangent Line Problem

Given point P on curve C, how do you find the tangent line?

Example. Consider

-,-, 

What is the equation of the line tangent to the curve at ? add , line

Point-slope form for a straight line passing through

What is the slope ?

What is the slope of the secant passing through and ? add 

What is the slope of the secant line passing through and ?

this ratio is called a difference quotient

As long as ,

The slope of the tangent line is the limit of the difference quotient as .

The equation of the tangent line is

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Example. Find an equation of the line tangent to the curve at .

Point slope form of the tangent line

where

simplify the difference quotient

multiply by 1 to rationalize the numerator

cancel factors of

true if

Thus

Equation of tangent line

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The Velocity Problem

Drive to Spokane airport (~85 miles)

Start at noon

Drive slowly through Colfax

Have lunch at Harvester

Arrive 2pm

The speedometer 5 miles north of Colfax reads 65 mph. This is the “instantaneous velocity.”

Mathematical definition of instantaneous velocity?

Galileo drops a ball off the leaning Tower of Pisa

sketch ground, tower, coordinate with origin at top

ball falls distance at time after release.

meters, seconds

-,-, curve

What is the average velocity between and ?

Average velocity

What is the average velocity between and a variable ?

Average velocity

as long as .

Table

Define instantaneous velocity at as the limit of average velocities over shorter and shorter time intervals around .

Denote instantaneous velocity .

Derivatives

Define the derivative of a function at a number , denoted

[1]

From the example above .

Alternatively, introduce

and insert in equation [1] to get

[2]

§2.2 The Derivative as a Function

Replace the symbol in [2] by . Regard as a variable.

Regard as a new function.

Example. Let . Find

Simplify the difference quotient

this step assumes

Then

Graph and compare and

-, -, 

add tangent segments at to graph of 

add dots at to graph of  ■

Example. Let . Find .

Simplify the difference quotient

rationalize numerator by multiplying by 1

multiply terms in numerator

divide through by (assumes

Then

Show transparency comparing and

?? Transparency: match and

Notations for Derivative

original function:

derivative function:

prime notation emphasizes idea of derivative as a new function

the prime means differentiate with respect to function argument

evaluate at no.

Leibniz notation emphasizes idea of derivative as the limit of a ratio



evaluate at no.

Operator notation for derivative

Sometimes we write

view and as operators: machines that convert the functions they operate on into other functions

Differentiability

differentiable at means exists

differentiable on an open interval means is differentiable at every point in

Example (a function not differentiable at a point)

Is differentiable at ?

if so

does this limit exist?

find the limit from the right

find the limit from the left

the right and left hand limit do not agree.

conclude does not exist

Geometrical Idea

axes, graph of |x|, no tangent line here (at origin)

To be differentiable at point, the graph must have a unique tangent line at that point. ■

Three ways that a function can fail to be differentiable

(a)at any discontinuity

, function with dcty at 

DNE

(b)at any corner or kink

, function with a kink at 

DNE

 function w/ a vertical tangent at 

, DNE

Relationship between differentiability and continuity

We have shown: if is not continuous then is not differentiable

Let be the statement

is continuous at a no.

Let be the statement

is differentiable at

We have shown

If (not ) then (not )

This is logically equivalent to

If then

If is differentiable at then it is continuous at

Higher Derivatives

Consider

First derivative of

Regarded as a function, may itself be differentiable.

Second derivative of

If is differentiable, form the third derivative

If is differentiable, form the fourth derivative

Notation for the derivative, with :

Application of higher derivatives

Let be the position of an object at time .

is the velocity of the object

is the acceleration of the object

First and second derivatives are the most important in applications

Example. Mechanics

momentum = mass velocity

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2.3 Basic Differentiation Rules

We first consider those rules that will enable us to differentiate polynomials.

Derivative of a Constant Function

-,-, line 

slope of tangent line?

Derivative of

-, -, 

slope of tangent line?

Derivative of

We have seen that

Derivative of

simplify the difference quotient

assumes

thus

The Power Rule.

Let be a positive integer

Example. . ■

Proof.

Preliminary fact:

______

in other words

notice there are terms on the right hand side

Let

Regard as a variable. Replace by .

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The Power Rule (general version)

Let be any real number.

Examples.

recall

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The Constant Multiple Rule

Let be a constant and a differentiable function

a constant passes through the limit symbol

Examples.

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The Sum Rule

If and are both differentiable

In words: “the derivative of a sum is the sum of the derivatives”

prime notation

shorthand

the sum rule applies to the sum of any number of functions

Example

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The Difference Rule

If and are both differentiable

In words: “the derivative of a difference is the difference of the derivatives”

prime notation

shorthand

Example.

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We can now differentiate any polynomial

Example. Let

then

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We can differentiate other functions too.

Example. Let

Find

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Example. Find an equation of the line tangent to the curve

at the point

point slope form for straight line

where

what is ?

answer

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?? Example. A ball is thrown straight up from the ground at 20 meters/second. Its height is given by

(a) Find the velocity at time .

(b) Find the velocity at sec.

(d) What is the average velocity between and ?

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Economics – Marginal cost

cost to produce widgets

average rate of change of cost

marginal cost

Example. Jeans manufacture.

Let cost of producing pairs of jeans.

where

capital costs (sewing machines)

cost of labor, materials, rent

Cost of producing 100 pairs of jeans

What is the cost of producing one additional pair of jeans?

______

Cost of producing the 101ST pair

Compare with the marginal cost at 100th pair

is often a very good approximation to the cost of producing one additional widget. ■

Derivatives of Sine and Cosine

Recall the limits

Recall the addition formula for cosine

Now use the limit definition of derivative

addition formula for cosine

difference law for limits

constant multiple law of limits

recalling the limits above

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The derivative of may be found using a similar argument (see our text).

In summary:

?? Differentiate the following

§2.4 The Product and Quotient Rules

Product Rule

If and are both differentiable

or alternately (as I personally prefer)

prime notation

shorthand

WARNING: The derivative of a product is not the product of derivatives

This is a common mistake!

Example. By the power rule

Now let and

then and

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Proof of the Product Rule

Suppose and are both differentiable functions.

Let

then

subtract and add the same term in the numerator

algebra

sum and product laws forlimits

continuity of and definition of derivative

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Extension to a Product of Three Functions

If , and are all differentiable

Example. Let

then

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Example. Differentiate .

Law of exponents:

Then

Product rule

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Quotient Rule

If and are differentiable at a point where then

shorthand

terms in numerator in same order as my product rule (but take difference)

Proof of quotient rule.

Let

Then

By product rule

Solve for

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Example. Differentiate .

where is shorthand for

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Example. Find the equations of the tangent lines to the curve

that are parallel to the line

Solution. Parallel means same slope. Slope of line?

slope is

where does have slope ?

solve for

form of equation for tangent line

Consider .

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?? Class practice product and quotient rules