Derivatives of Trigonometric Functions - PowerPoint Presentation

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Derivatives of Trigonometric Functions

Chapter 3.5Slide2

Proving that

In section 2.1 you used a table of values approaching 0 from the left and right and concluded that

; but that was not a proof

Because you will need to know this limit (and a related one for cosine), we will begin this section by proving this through geometryWe will need the followingThe Squeeze TheoremThe formula for the area of a sector of a circle

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Proving that

The Squeeze Theorem says that if

are functions so that, for

-values in an open interval around some number

and

then

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Proving that

If a circle of radius 1 is subtended by an angle

in radians (creating a sector or “pie slice”), then the area of the sector is

NOTE: “subtend” means: to be opposite to and extend from one side to the other of; a hypotenuse subtends a right angle

(From Merriam-Webster Dictionary online)

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Proving that

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Proving that

We can simplify by multiplying through by 2; next divide through by

(we can do this without changing inequality signs; why?)

For positive numbers

and

, if

then

For example,

, but ; we use this to take the reciprocals 

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Proving that

This brings in

and also puts it between two functions within the first quadrant; we are now able to use the Squeeze Theorem

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Proving that

Since

and since

then by the Squeeze Theorem

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Proving that

Since sine is an odd function, then

We can conclude that

Hence,

QED

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Proving that

We can use the previous result to prove another special limit, specifically

We have

We want to be able to use the previous theorem. Note that the numerator is just

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Proving that

Now use the limit properties to get

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Derivative of the Sine and Cosine Functions

Having proved that

and that

, we can now use the derivative definition to determine the derivative of the sine function and the cosine function

We will also need to be reminded of the following trigonometric identity:

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Derivative of the Sine Function

THEOREM:

If , then

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Derivative of the Sine Function

PROOF

Using the definition of the derivative:

Use the previous identity to expand the numerator:

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Derivative of the Sine Function

PROOF

We can factor

Now use the limit properties

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Derivative of the Sine Function

PROOF

QED

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Derivative of the Cosine Function

THEOREM:

If , then

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Derivative of the Cosine Function

THEOREM:

PROOFWe will need the angle sum identity for cosine which is

Use the definition of the derivative:

Expand the numerator:

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Derivative of the Cosine Function

THEOREM:

PROOF

Factor out

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Derivative of the Cosine Function

THEOREM:

PROOF

Use the limit properties

QED

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Example 1: Using the Derivatives of Sine and Cosine Functions

Find the derivatives of

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Example 1: Using the Derivatives of Sine and Cosine Functions

You will need to use the Product Rule:

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Example 1: Using the Derivatives of Sine and Cosine Functions

Use the Quotient Rule:

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Example 2: The Motion of a Weight on a Spring

A weight hanging from a spring is stretched 5 units beyond its rest position (

) and released at time to bob up and down. Its position at any later time is

What are its velocity and acceleration at time

? Describe its motion. 24Slide25

Example 2: The Motion of a Weight on a Spring

Recall that velocity is the first derivative of position and acceleration is the second derivative of position (or the first derivative of velocity).

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Example 2: The Motion of a Weight on a Spring

Time

Time

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Jerk

A sudden change in acceleration is called a

jerkWhen a ride in a vehicle is jerky, it is caused by sudden changes in accelerationHence, jerk, sudden changes in acceleration, is the first derivative of acceleration, or the third derivative of position27Slide28

Example 3: A Couple of Jerks

What is the jerk caused by the acceleration due to gravity?

What is the jerk of the simple harmonic motion from Example 2?28Slide29

Example 3: A Couple of Jerks

What is the jerk caused by the acceleration due to gravity?

The acceleration due to gravity is constant (near the Earth’s surface):

feet per second per second. Since the derivative of a constant is zero, then

What is the jerk of the simple harmonic motion from Example 2?

Jerk is maximum when

, which occurs when the weight is at the rest position and acceleration changes sign

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Derivatives of the Other Basic Trigonometric Functions

We can use the derivatives of sine and cosine as well as the Product and/or Quotient Rules to determine the derivatives of the other basic trigonometric functions

Find

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Derivatives of the Other Basic Trigonometric Functions

We can use the derivatives of sine and cosine as well as the Product and/or Quotient Rules to determine the derivatives of the other basic trigonometric functions

Find

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Derivatives of the Other Basic Trigonometric Functions

Find

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Derivatives of the Other Basic Trigonometric Functions

THEOREM:

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Example 4: Finding Tangent and Normal Lines

Find equations for the lines that are tangent and normal to the graph of

at

.

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Example 4: Finding Tangent and Normal Lines

Find equations for the lines that are tangent and normal to the graph of

at

.

The slope of the tangent line at

is

and of the normal line is

. The equations are

and

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Example 5: A Trigonometric Second Derivative

Find

if .

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Example 5: A Trigonometric Second Derivative

Find

if .

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Derivative Theorems So Far

, where

is a constant

(Constant Multiple Rule)

(Sum/Difference Rule)

, for

(Power Rule for integers)

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Derivative Theorems So Far

(Product Rule)

(Quotient Rule)

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Derivative Theorems So Far

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Exercise 3.5

Online exercise

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