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Test: Taylor Series - Civil Engineering (CE) MCQ


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10 Questions MCQ Test Engineering Mathematics - Test: Taylor Series

Test: Taylor Series for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: Taylor Series questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Taylor Series MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Taylor Series below.
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Test: Taylor Series - Question 1

Let f (x) = . Then f(100)(54) is given by

Detailed Solution for Test: Taylor Series - Question 1

Concept:

Taylor’s series method:

The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and all of its derivatives, are known at a single point.

Taylor's series expansion for f (x + h) is
f(x+h) = f(x) + hf′(x) + h2/2!f″(x) + h3/3! + f‴(x)+…∞
f(x) = f(a) + (x−α)f′(x) +(x) +........∞

Calculation:

Given:

f(100)(54) = ?
Using Taylor series expansion for Sin x at a = 54

Now the function transforms into:

After Observing carefully the first term in the above infinite series, the (x - 54) term is always in the denominator, which will become zero when we put x = 54.

Every derivative will also have the same term till infinite.

So, every term will have zero in its denominator after putting x = 54.

⇒ f(100)(54) is Undefined.

Test: Taylor Series - Question 2

Which of the following is not true?

Detailed Solution for Test: Taylor Series - Question 2

Concept:

Taylor series expansion

Option 1:
The standard expansion of log(1 + z) is given as 

Hence, Option 1 is true
Option 2:
Given complex function is 
→ Let’s Resolve f(z) into partial fractions

For expanding about z = 2, let z – 2 = t ⇒ z = 2 + t


Option 3:
Cauchy’s Integral Formula:

If f(z) is an analytic function within a closed curve and if a is any point within C, then

Residue Theorem:

If f(z) is analytic in a closed curve C except at a finite number of singular points within C, then

⁡f(z)dz = 2πi × [sum of residues at the singualr points with in C]

Formula to find residue:

1. If f(z) has a simple pole at z = a, then

2. If f(z) has a pole of order n at z = a, then

Given complex integral is 
where Cis the circle |z-1| = 2;

Now for the given complex function, the pole is -4 with order 2;

The pole - 4 lies outside the given circle C;

Therefore, no residue inside the circle, hence integration will be zero.

Option 3 is also correct

Option 4:

The given complex function is f(z) = 
In this function, the singularities are z = 0, +i, -i;

Therefore, the given function has 3 singularities...

Option 4 is incorrect

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Test: Taylor Series - Question 3

The Maclaurin's series expansion of esin x is

Detailed Solution for Test: Taylor Series - Question 3


f''(0) = 1


fiv(0) = -2 -1 -1 + 1 = -3
Substitue in Maclaurin Series

Test: Taylor Series - Question 4

The series expansion of sin⁡x/x near origin is

Detailed Solution for Test: Taylor Series - Question 4

Concept:

Taylor series:

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number ‘a’ is the power series.

Expression of Taylor series is:

Calculation:

Given:

We have to find the series expansion of sin⁡x/x near origin, or a = 0.

Let f(x) = sin x

f(0) = sin (0) = 0,

f'(0) = cos (0) = 1,

f''(0) = -sin(0) = 0,

f'''(0) = -1 .... so on

Putting all the values in Taylor series expansion, we get:

Series expansion of sin x  will be:
sinx = x − x3/3!+…
Therefore the series expansion of sin⁡ x/x near origin will be:

Test: Taylor Series - Question 5

If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces to

Detailed Solution for Test: Taylor Series - Question 5

Taylor Series:

If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-


Laurent Series:
If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C1 and C2 having centre at 'a' & respective radii r1 and r2 (r1 > r2).


The negative part of Laurent's series i.e is called the singular part, and if that vanishes the terms that remain will be , which is nothing but Taylor series.

Test: Taylor Series - Question 6

In the neighbourhood of z = 1, the function f(z) has a power series expansion of the form

f(z) = 1 + (1 − z) + (1 − z)2 + ....... ∞

Then f(z) is

Detailed Solution for Test: Taylor Series - Question 6

Concept:

Taylor series:

Calculation:
f(z) = 1 + (1 – z) + (1 – z)2
The above series is in the form of G.P.
a = 1, r = (1 – z)

Test: Taylor Series - Question 7

What is the expansion of y = sin-1 x?

Detailed Solution for Test: Taylor Series - Question 7

Concept:

Binomial expansion of (1 - x)-n is given by
(1 - x)-n = 1 + nx + 
Formula used:

Calculation:
Given,
y = sin-1 x
....(1)
Using the above binomial expansion formula

Integrating both sides with respect to x, 

Hence, option c is the correct answer.

Test: Taylor Series - Question 8

The value of y at x = 0.1 to five places of decimals, by Taylor's series method, given that dy/dx = x2y−1, y(0) = 1, is

Detailed Solution for Test: Taylor Series - Question 8



Hence by Taylor series:

1 - 0.1 + 0 + 0.00033 + ......
= 0.90031 ≈ 0.90033

Test: Taylor Series - Question 9

The Taylor series expansion of 3 sin x + 2 cos x is

Detailed Solution for Test: Taylor Series - Question 9

Concept:
Taylor series expansion for sin x and cos x are respectively:

Calculation:

Similarly,

Adding equation (1) and equation (2), we get:
3sin⁡x + cos⁡x = 2 + 3x −

Test: Taylor Series - Question 10

The Taylor series expansion of is

Detailed Solution for Test: Taylor Series - Question 10


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