Calculus Notes 7.5 Inverse Trigonometric Functions & 7.7 Indeterminate Forms and L’Hospital’s Rule

Start up:1. (AP question) Compute

• (Text Question) In Example 2, why wasn’t the Quotient Rule used when taking the derivatives?

21

x

x

edx

e

x

x

let u e

du e dx

2

1

1du

u arctan u C

We are not taking the derivative of . We are using l’Hospital’s Rule to evaluate

2

xe

x

'

lim lim'x x

f x f x

g x g x

arctan xe C

Calculus Notes 7.5 Inverse Trigonometric

1sin sin2 2

x y y x and y

1

1

sin sin2 2

sin sin 1 1

x x for x

x x for x

1

2

1sin 1 1

1

dx x

dx x

1cos cos 0x y y x and y

1

1

cos cos 0

cos cos 1 1

x x for x

x x for x

1

2

1cos 1 1

1

dx x

dx x

1tan tan2 2

x y y x and y

1 1limtan lim tan2 2x x

x x

12

1tan

1

dx

dx x

1csc 1 csc

30, ,2 2

y x x y x

and y

1sec 1 sec

30, ,2 2

y x x y x

and y

1cot cot

0,

y x x y x

and y

Calculus Notes 7.5 Inverse Trigonometric

1

2

1sin

1

dx

dx x

Table of Derivatives of Inverse Trigonometric Functions

1

2

1cos

1

dx

dx x

12

1tan

1

dx

dx x

12

1cot

1

dx

dx x

1

2

1sec

1

dx

dx x x

1

2

1csc

1

dx

dx x x

1

2

1sin

1dx x C

x

1

2

1tan

1dx x C

x

12 2

1 1tan

xdx C

x a a a

Calculus Notes 7.7 Indeterminate Forms and L’Hospital’s Rule lim

x a

f x

g x0

0Indeterminate form of type

'

lim lim'x a x a

f x f x

g x g x

lim 0 lim 0x a x a

f x and g x

' 0g x L’Hospital’s Rule: Suppose f and g are differentiable and near a

(except possibly at a). Suppose that

or that

(In other words, we have an indeterminate form of type or .) Then

if the limit on the right side exists (or is or ).

0

0

lim limx a x a

f x and g x

'

'

f c f b f a

g c g b g a

' 0g x Cauchy’s Mean Value Theorem: Suppose that the functions f and g are continuous on [a,b] and differentiable on (a,b), and for all x in (a,b). Then there is a number c in (a,b) such that

Calculus Notes 7.5 Inverse Trigonometric Functions

Example 1: Find the exact value.

1tan 3A

1arcsin

2B

Example 2: Simplify 1sin tan x2 1

x

x

√3

2

30 ̊

160 ̊

90 ̊

√245

1

4590 ̊

1

3

4

1

θ

x? ̊

90 ̊

2 1x

Example 3: Prove using implicit differentiation: 1

2

1cos

1

dx

dx x

coslet y x

sin 1dy

ydx

1

sin

dy

dx y

2 2 2sin cos 1 sin 1 cosy y so y y

2

1

1 cos

dy

dx y

Differentiate it implicitly with respect to x.

2

1

1

dy

dx x

1

2

1cos

1

dx

dx x

Calculus Notes 7.5 Inverse Trigonometric Functions

Example 4: Find the derivative of 21 tanH x x arc x

2 2

1' 2 arctan 1

1H x x x x

x

Example 5: Find the limit limarctan x

xe

lim x

xe

' 2 arctan 1H x x x

2

limarctan2x

x

Calculus Notes 7.7 Indeterminate Forms and L’Hospital’s Rule

Example 6: Find the limit. Us l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

8

41

9lim

5x

x

x

9

51

1lim

1x

xA

x

lim

x

x

eB

x lim

1

x

x

e

0

' '0

l Hospital s Rule

9

5

' 'l Hospital s Rule

PS 7.5 pg.483 #1, 4, 12, 15, 18, 23, 25, 26, 32, 43, 47, 49, 53, 59 (14)PS 7.7 pg.501 #1, 5, 8, 15, 20, 38, 41, 63, 65, 78, 79, 85, [92] (12-[13])Review Chapter 7 Worksheet #1-20