3 3 derivatives of trig functions
This presentation is the property of its rightful owner.
Sponsored Links
1 / 12

§3.3 Derivatives of Trig Functions PowerPoint PPT Presentation


  • 67 Views
  • Uploaded on
  • Presentation posted in: General

§3.3 Derivatives of Trig Functions. The student will learn about:. Derivative formulas for trigonometric functions. Some Preliminary Work #1. Remember if a, b and c are positive 0 < a < b < c, then. If then. Reciprocals and inequalities. 2. Some Preliminary Work #2. Q.

Download Presentation

§3.3 Derivatives of Trig Functions

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


3 3 derivatives of trig functions

§3.3 Derivatives of Trig Functions

The student will learn about:

Derivative formulas for trigonometric functions.


Some preliminary work 1

Some Preliminary Work #1

Remember if a, b and c are positive

0 < a < b < c, then

If then

Reciprocals and inequalities.

2


Some preliminary work 2

Some Preliminary Work #2

Q

P

Consider the drawing to the right.

tan x

sin x

x

O

A

M (1, 0)

We need to establish the following important limit.

∆OMP is inside sector OMP which is inside ∆OMQ.

Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ.

3


Preliminary work 2

Preliminary Work #2

Q

P

tan x

sin x

x

O

A

M (1, 0)

Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ.

and

and since

So by the “squeeze” theorem

4


Some preliminary work 3

Some Preliminary Work #3

We will also need to establish that

= 0 ∙ 1 ∙ ½ = 0

5


Derivative of sin x

Derivative of sin x

Let f (x) = sin x and we will use the five step procedure to calculate the derivative.

2. f (x) = sin x

1. f (x + h) = sin x cos h + cos x sin h

3. f (x + h) – f (x) = sin x cos h + cos x sin h – sin x

= sin x cos h – sin x + cos x sin h

= sin x (cos h – 1) + cos x sin h

= (sin x) · (0) + (cos x) · (1) = cos x


Derivative formulas

Derivative Formulas for Sine and Cosine

Basic Form

General Form - Chain Rule

For u = u (x):

Derivative Formulas


Examples

Examples

a. y = sin 5x y’ =

5 cos 5x

b. y = cos x 2 y’ =

(2x)(- sin x2) = - 2x sin x 2

c. y = (cos x) 2 y’ =

(cos x)(- sin x) + (cos x) (-sin x

= 2 sin x cos x


Examples1

slope

Examples

Find the slope of the graph of f (x) = cos x at (π/4, 2/2), and sketch the tangent line to the graph at that point.

We will use our graphing calculator to do this problem.


Examples2

Examples

And the tangent goes through the point (π/4, 2/2), so

using the point-slope form of a line

Find the slope of the graph of f (x) = cos x at (π/4, 2/2), and sketch the tangent line to the graph at that point.

Or we can use algebra to solve this problem.

y’ = - sin x

y’ = - sin x so the m = y’ = - sin (π/4)

y’ = - sin x so the m = y’ = - sin (π/4) = -√2/2.

10


Summary

Derivative Formulas for Sine and Cosine

Basic Form

General Form

For u = u (x):

Summary


3 3 derivatives of trig functions

ASSIGNMENT

§3.3; Page 53; 1 to 21 odd.


  • Login