1. 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?
a) 1770
b) 1729
c) 0
d) 1
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
= 1770
2. Find the Taylor series expansion of the function cosh(x) centered at x = 0.
a) \(1-\frac{x^2}{2!}+\frac{x^4}{4!}+….\infty\)
b) \(\frac{x}{1!}+\frac{x^3}{3!}+\frac{x^5}{5!}….\infty\)
c) \(1+\frac{x^2}{2!}+\frac{x^4}{4!}+….\infty\)
d) \(1+\frac{x}{1!}+\frac{x^2}{2!}+….\infty\)
Explanation: We know the general expression for the expansion of the Taylor series
\(\tau[f(x)]=f(a)+\frac{x.f^{(1)}(a)}{1!}+\frac{x^2.f^{(2)}(a)}{2!}+…..\infty\)
Given a=0 we substitute in the equation to get
\(\tau[f(x)]=f(0)+f^{(1)}(0)\times \frac{x}{1!}+f^{(2)}(0)\times \frac{x^2}{2!}+….\infty\)
Now the nth derivatives can be calculated as
\(f^{(n)}(x)=(\frac{e^x+e^{-x}}{2})^{(n)}\)
\(=\frac{e^x+(-1)^ne^{-x}}{2}\)
Substituting x=0 yields the final expansion
\(f^{(n)}(x)=\frac{1+(-1)^n}{2}\)
We get
\(\tau[f(x)]=1+(0)\times \frac{x}{1!}+(1)\times \frac{x^2}{2!}+(0)\times \frac{x^3}{3!}+…..\infty\)
\(\tau[f(x)]=1+\frac{x^2}{2!}+\frac{x^4}{4!}+…\infty\)
3. To find the value of sin(9) the Taylor Series expansion should be expanded with center as ___________
a) 9
b) 8
c) 7
d) Some delta (small) interval around 9
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.
4. f(1) (n) = g(n) (0) holds good for some functions f(x) and g(x). Now let the coordinate axes containing graph g(x) be rotated by 30 degrees clockwise, then the corresponding Taylor series for the transformed g(x) is?
a) g(0)+\(\frac{e^x – 1}{\sqrt{3}}+\frac{\sum_{n=1}^\infty f^{(1)}(n)x^n}{n!}\)
b) \(g(0) + \frac{g^{(1)}.x}{1!} + \frac{g^{(2)}(1).x^2}{2!}+…\infty\)
c) No unique answer exist
d) Such function is not continuous
Explanation: We have f(1) (n) = g(n) (0)
As the coordinate axes containing f(x) is rotated the tan(30) term gets added to the derivative of f(1)new(n)=g(n)(0)-tan(30)
We have
g(n)(0)=f(1)(n)+tan(30)
The Taylor expansion centered at 0 for g(x) is given by
g(x)=\(g(0)+\frac{g^{(1)}(0).x}{1!}+\frac{g^{(2)}(0).x^2}{2!}+…..\infty \)
Now substituting g(n)(0)=f(1)(n)+tan(30) we have
g(x)=\(g(0)+\frac{(f^{(1)}(1)+tan(30)).x}{1!}+\frac{(f^{(1)}(2)+tan(30)).x^2}{2!}+….\infty\)
g(x)=\(g(0)+(\frac{f^{(1)}(1).x}{1!}+\frac{f^{(1)}(2).x}{2!}+…\infty)+tan(30)\times(\frac{x}{1!}+\frac{x^2}{2!}+….\infty)\)
=\(g(0)+\frac{\sum_{n=1}^{\infty}f^{(1)}(n).x^n}{n!}+\frac{1}{\sqrt{3}}\times(e^x-1)\)
5. 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?
a) 1729
b) -1
c) 1
d) 0
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)
= 0.
6. Let Mclaurin series of some f(x) be given recursively, where an denotes the coefficient of xn in the expansion. Also given an = an-1 / n and a0 = 1, which of the
following functions could be f(x)?
a) ex
b) e2x
c) c + ex
d) No closed form exists
Explanation: Observing the recurrence relation we have
an=\(\frac{a_{n-1}}{n}=\frac{a_{n-2}}{n(n-1)}\)
an=\(\frac{a_0}{n(n-1)(n-2)…3\times 2 \times 1}\)
Thus one could deduce that
an=\(\frac{1}{n!}\)
Putting this into the Mclaurin expansion we have
f(x)=a0+a1x+a2x2+a3x3….∞
f(x)=\(1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+….\infty\)
Which is the well known expansion of ex.
7. 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(1)(0) = 1.
a) True
b) False
Explanation: Writing out the Mclaurin series we have
f(x)=f(0)+\(\frac{f^{(1)}(0).x}{1!}+\frac{f^{(2)}(0).x^2}{2!}+…\infty\)
Now sustituting f(n+1)(0) = \(\sqrt{f^{(n)}(0)}\)we have
f(x) = f(0) + \(\frac{f^{(1)}(0).x}{1!}+\frac{\sqrt{f^{(1)}(0)}.x^2}{2!}+\frac{\sqrt{\sqrt{f^{(1)}(0)}.x^3}}{3!}…\infty\)
Now f(1)(0)=1 and f(0)=a
Substituting this we have
f(x)=a+\(\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+…\infty\)
f(x)=a-1+ex
This is a well defined function.
8. Expansion of function f(x) is?
a) f(0) + x⁄1! f‘ (0) + x2⁄2! f” (0)…….+xn⁄n! fn (0)
b) 1 + x⁄1! f‘ (0) + x2⁄2! f” (0)…….+xn⁄n! fn (0)
c) f(0) – x⁄1! f‘ (0) + x2⁄2! f” (0)…….+(-1)^n xn⁄n! fn (0)
d) f(1) + x⁄1! f‘ (1) + x2⁄2! f” (1)…….+xn⁄n! fn (1)
Explanation: By Maclaurin’s series, f(0) + x⁄1! f‘ (0) + x2⁄2! f” (0)…….+xn⁄n! fn (0)
9. The necessary condition for the maclaurin expansion to be true for function f(x) is __________
a) f(x) should be continuous
b) f(x) should be differentiable
c) f(x) should exists at every point
d) f(x) should be continuous and differentiable
Explanation: By Maclaurin’s series, f(0) + x⁄1! f‘ (0) + x2⁄2! f” (0)…….+xn⁄n! fn (0)
Where, f(x) should be continuous and differentiable upto nth derivative.
10. The expansion of f(a+h) is ______
a) \(f(a)+\frac{h}{1!} f'(a)+\frac{h^2}{2!} f”(a)…….+\frac{h^n}{n!} f^n (a)\)
b) \(f(a)+\frac{h}{1!} f'(a)+\frac{h^2}{2!} f”(a)…….\)
c) \(hf(a)+\frac{h^2}{1!} f'(a)+\frac{h^3}{2!} f”(a)…….+\frac{h^n}{n!} f^n (a)\)
d) \(hf(a)+\frac{h^2}{1!} f'(a)+\frac{h^3}{2!} f”(a)………\)
Explanation: By taylor expansion,
f(a+h) = f(a) + h⁄1! f'(a) + h2⁄2! f” (a)….