This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Taylor Mclaurin Series – 1”.
2. What is the coefficient of x101729 in the series expansion of cos(sin(x))
Explanation: We know that the series expansion of cos(x) is
Observe that every term has odd powered series raised to an even term
Thus, we must have only even powered terms in the above series expansion. The coefficient of any odd powered term is zero.
3. Let τ(X) be the Taylor Series expansion of f(x) = x3 + x2 + 1019 centered at
a = 1019 , then what is the value of the expression 2(τ(1729))2 + τ(1729) * f(1729) – 3(f(1729))2 + 1770
Explanation: Observe first off that the given function is a polynomial and so any other representation (Taylor Series here) which is continuous and differentiable has to be the
same polynomial. This gives us
τ(x) = f(x)
We now evaluate the expression as follows
= 2(f(1729))2 + (f(1729))2 – 3(f(1729))2 + 1770
= 3(f(1729))2 – 3(f(1729))2 +1770
5. To find the value of sin(9) the Taylor Series expansion should be expanded with center as
d) None of these.
Explanation: The Taylor series gives accurate results around some point taken as center. As we need the value of 9 the center nearer to the point should be taken. Hence, 9 is the right answer.
6. sin(9) Holds good for some functions f(1) (n) = g(n) (0) and f(x) . Now let the coordinate axes containing graph g(x) be rotated by f(x) degrees clockwise, then the corresponding Taylor series for the transformed is
c) No unique answer exist
d) Such function is not continuous
7. Let τ(f(x)) denote the Taylor series for some function f(x). Then the value of τ(τ(τ(f(1729)))) – 2τ(τ(f(1729))) + τ(f(1729)) is
Explanation: We know that the Mclaurin Series for any given function always yields a polynomial (finite OR infinite).
Further the Mclaurin series of this polynomial (i.e.τ(τ(f(x))) ) is also a polynomial. Due to uniqueness of this polynomial, no matter how many nested Mclaurin series we might find, they are all equal. Thus, we have
τ(τ…….(f(x))….)) = f(x)
Substituting this into our required expression we have
= f(1729) – 2f(1729) + f(1729)
8. Let Mclaurin series of some f(x) be given recursively, where an denotes the coefficient of xnin the expansion. Also given an = an-1 / n and a0 = 1, which of the
following functions could be f(x)
c) c + ex
d) No closed form exists
9. A function f(x) which is continuous and differentiable over the real domain exists such that f(n) (x) = [ f(n + 1) (x) ]2 , f(0) = a and f(x)
Explanation: Writing out the Mclaurin series we have
This is a well defined function.
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