lec 1 admath review taylor

8/18/2019 Lec 1 Admath Review Taylor

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Advanced EngineeringMathematics

Engr. Rimson E. Junio

Instructor


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FEU Institute of TechnologyElectrical & Electronics Engineering (EEE)

Trigonometry

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EEMA211

Review:

Factoring and Algebraic Identities Trigonometric Identities Derivatives Integrals

Functions as Infinite Series: Taylor series Maclaurin series


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Trigonometry

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Factoring and Algebraic Identities

( ) 2 ( ) 2 ( )( )

( ) 3 3 ( ) 3 3 ( )( )


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Trigonometry

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P1.1

Factor:


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Trigonometry

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P1.2

Compute mentally:

a) 103 97 b) 96 c) 104


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Trigonometry

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EEMA211

P1.3

Is 1,000,001 a prime number? Why or why not?


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Trigonometry

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Trigonometric Identities

1 sin ± sin cos ± cos sin cos ± cos cos ∓ sin sin

tan ± tan ± tan 1 ∓ tan tan


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Trigonometry

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P1.4

Derive the double angle identity:

sin 2 2 sin cos


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Trigonometry

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P1.5

If sec tan 3, what is the value of sec tan ?


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Trigonometry

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Derivatives

P1.7

Solve the following derivative problems.

a) 10th derivative w.r.t. to x of b) 100th derivative w.r.t. to x of sin c) 100th derivative w.r.t. to x of sin


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Trigonometry

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EEMA211

Integrals

P1.8

Solve the following integrals.

a) sin() b)

+

∞

c) cos


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Trigonometry

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Seatwork

a) Factor 125 8 b) Derive the identity: cos 3θ 4cosθ 3 cos θ c) Find the 100th derivative with respect to x of

1 2 3 … ( 99)( 100) d) Solve the indefinite integral:

l n


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Trigonometry

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Taylor Series

expresses a function as an infinite seriesthat are calculated from the values of thefunction's derivatives at a particularpoint within its domain

()( )

!∞

=

′ 1! ( )

"()2! ( )

()3! ( ) ⋯


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Trigonometry

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1

1!

2!

3!

⋯

cos 1 2! 4! ⋯ sin

3!

5! ⋯

ln(1 )

2

3 ⋯

(1 )− 1 12! 1 2

3! ⋯

Maclaurin Series

obtained by setting a=0 in the Taylor series expansion ofa function.

Maclaurin series of some functions


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Trigonometry

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EEMA211

P1.9

Find the Taylor series of the following function:

a) sin at . Approximate the value of sin 30° using the series obtained byadding up to the 5th degree term.

T i t


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Trigonometry

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P1.10

Find the Maclaurin series of the following function:

a) sin Approximate the value of sin 30° using the series obtained byadding up to the 5th degree term.

T i t

EEMA211


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Trigonometry

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Other Applications of Maclaurin Series

sin 0

−

lim→ sin

T i t

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Trigonometry

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Seatwork 1

1) Find the Taylor series of

cos at a 6 and approximate the value of cos(72°) by adding up to the 5th degree term.

2) Find the Maclaurin series of ln(1 ) and approximate the valueof ln(1.2) by adding up to the third degree term.

T i t

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Trigonometry

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Homework 1

1) Determine the Maclaurin series of () andapproximate the value of (0.3) by adding up to the 3rd degree term of the series.

2) Find a shorter way to determine the Maclaurin series of thissomewhat complicated function:

sin() Hint: Find the Maclaurin series of sin and substitute to all the′. Multiply the result by .3) Find the Maclaurin series of ln +− . Approximate thevalue of ln(3) by adding up to the 7

th

degree term.

Hint: Use the property: ln() ln ln and determine theMaclaurin series separately for ln and ln .

T igonomet

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Trigonometry

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Homework 1

4) The energy of an electron at speed in special relativity theoryis −

where is the electron mass, and is the speed of light.The factor is called the rest mass energy (energy when 0).Find two terms of the series expansion of

−

, and multiply by

to get the energy at speed .Hint: Use the Maclaurin series of

(−).

lec 1 admath review taylor

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