Explanation Formula used: Taylor series expansion of a function: f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) ( x − a ) 2 2 ! + … . Calculation: Consider the function f ( x ) = ln ( cos x ) with a = π 4 The derivatives and its values are obtained as follows. Differentiate the function f ( x ) = ln ( cos x ) with respect to x as shown below. f ′ ( x ) = d d x ( ln ( cos x ) ) = 1 cos x ( − sin x ) = − tan x Again, differentiate the function f ′ ( x ) with respect to x as follows. f ″ ( x ) = d d x ( − tan x ) = − sec 2 x = − ( 1 + tan 2 x ) = − 1 − tan 2 x Now differentiate the function f ″ ( x ) with respect to x as shown below. f ″ ( x ) = d d x ( − 1 − tan 2 x ) = 0 − 2 tan x ( sec 2 x ) = − 2 tan x ( 1 + tan 2 x ) = − 2 tan x − 2 tan 3 x Substitute x = π 4 in f ( x ) and obtain f ( π 4 ) as shown below. f ( π 4 ) = ln ( cos ( π 4 ) ) = ln ( 1 2 ) = ln ( 2 ) − 1 2 = − 1 2 ln ( 2 ) Substitute x = π 4 in f ′ ( x ) and obtain f ′ ( π 4 ) as shown below. f ′ ( π 4 ) = − tan ( π 4 ) = − 1 Again substitute x = π 4 in f ″ ( x ) and obtain f ″ ( π 4 ) as shown below

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 30, Problem 32RE

To determine

The first three terms of the Taylor expansion of ln(cosx) with a=π4.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 30, Problem 31RE

Chapter 30, Problem 33RE

Students have asked these similar questions

Microsoft Excel snapshot for random sampling: Also note the formula used for the last column 02 x✓ fx =INDEX(5852:58551, RANK(C2, $C$2:$C$51)) A B 1 No. States 2 1 ALABAMA Rand No. 0.925957526 3 2 ALASKA 0.372999976 4 3 ARIZONA 0.941323044 5 4 ARKANSAS 0.071266381 Random Sample CALIFORNIA NORTH CAROLINA ARKANSAS WASHINGTON G7 Microsoft Excel snapshot for systematic sampling: xfx INDEX(SD52:50551, F7) A B E F G 1 No. States Rand No. Random Sample population 50 2 1 ALABAMA 0.5296685 NEW HAMPSHIRE sample 10 3 2 ALASKA 0.4493186 OKLAHOMA k 5 4 3 ARIZONA 0.707914 KANSAS 5 4 ARKANSAS 0.4831379 NORTH DAKOTA 6 5 CALIFORNIA 0.7277162 INDIANA Random Sample Sample Name 7 6 COLORADO 0.5865002 MISSISSIPPI 8 7:ONNECTICU 0.7640596 ILLINOIS 9 8 DELAWARE 0.5783029 MISSOURI 525 10 15 INDIANA MARYLAND COLORADO

The spread of an infectious disease is often modeled using the following autonomous differential equation: dI - - BI(N − I) − MI, dt where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of transmission, and μ is the rate at which people recover from infection. Close a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria. b) (5 points) For the equilbria in part a), determine whether each is stable or unstable. c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the dt function by hand.) Identify the equilibria as stable or unstable in the graph. d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.

Find the indefinite integral. Check Answer: 7x 4 + 1x dx

Chapter 30 Solutions

Basic Technical Mathematics

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The first three terms of the Taylor expansion of ln ( cos x ) with a = π 4 .

Videos

The first three terms of the Taylor expansion of ln(cosx) with a=π4.

Want to see the full answer?

Chapter 30 Solutions