MAT 1234Calculus I

Section 2.4

Derivatives of Tri. Functions

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HW and Quiz

WebAssign HW 2.4 Quiz: 2.3, 2.4

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Skills• Formulas for the derivatives of tri. functions

• Find limits by change of variables

Concepts• Find limits by simple geometric insights

• an application of the squeeze theorem

Formulas

xxdx

d

xxxdx

d

xxxdx

d

xxdx

d

xxdx

d

xxdx

d

22 csccot

tansecsec

cotcsccsc

sectan

sincos

cossin

Formulas

sin cosd

x xdx

Why?

Let ( ) sin

( ) ( )

f x x

f x h f x

h

0 0

( ) ( )lim limh h

f x h f x

h

Formulas

0

cos( ) 1lim 0h

h

h

0

sin( )lim 1h

h

h

We are going to look at the first limit later.

Example 1

t

ttf

sin1

cos)(

sin cos

cos sin

dx x

dxd

x xdx

( )f t

Example 2

tan)( h

2tan secd

x xdx

( )

( )

h

h

Important Limit

1sin

lim0

Use to find the formulas for the derivatives of the tri. functions

Use to find other limits Use often in physics for approximations

e.g. mechanical system, optics

Example Simple Pendulum

When the angle is small, the motion can be modeled by

l

02

2

l

g

dt

d

2

2sin 0

d g

dt l

0

sinlim 1

sin

Important Limit

1sin

lim0

Evidence: Graphs Proofs (a) Geometric proof (Section 2.4)

(b) L’ hospital Rule (Section 6.8)

(c) Taylor Series (Section 11.10)

Why?

Important Limit 0

sinlim 1

Evidence: Graphs

Important Limit

Evidence: Graphs

0

sinlim 1

Important Limit

Proofs (a) Geometric proof (Section 2.4)

(b) L’ hospital Rule (Section 6.8)

(c) Taylor Series (Section 11.10)

0

sinlim 1

Example 3

x

xx

2

0

sinlim

0

sinlim 1

2

0

sinlimx

x

x

Example 4

xx

xx sin

sinlim

0

0

sinlim 1

0

sinlim

sinx

x

x x

Generalized Formula

0

sinlim 1 , where 0x

kxk

kx

0

sinlim 1

Why?

Example 5

0

sin 7lim

3x

x

x

0

sinlim 1x

kx

kx

0

sin 7lim

3x

x

x

Remark

It is incorrect to use the limit laws and write

0 0

sin 7 1 sin 7lim lim

3 3x x

x x

x x

since we do not know the existence of

0

sin 7limx

x

x

Example 6

x

xx sin

7sinlim

0

0

sinlim 1x

kx

kx

0

sin 7lim

sinx

x

x

Purposes (Skip if …)

Look at the interesting power of geometry.

Look at an application of the squeeze theorem.

Geometric Proof (Idea)

1sin

lim0

1sin

cos

Simplified Proof: 1

sincos

1

1?

angleradiusArc

Simplified Proof:

1

1sin

1sin

cos

1sin

sin

angleradiusArc

Simplified Proof:

1

?

1sin

cos

Simplified Proof:

1

tan

1sin

cos

sin cos

cos

sin

tan

Important Limit1

sinlim

0

1sin

lim

theorem,squeeze By the

11lim and 1coslim

1sin

cos

,small and 0 If

0

00

Important Limit1

sinlim

0

1

sinlim

)(

)sin(lim

sinlim

)(

)sin(sinsin ,0 If

0

00

0,0 As

-θ