Consider the following two Taylor Series (about x = 0): x3 x7 -1 tan x=x- + + 3 5 7 x3 3x5 5x7 -1 sin x=x+ + + 6 40 112 (a) Find the Taylor Series expansion for f(x) and including the xⓇ term. = -1 (tan¹x) (sin x) about x = 0, up to -1 (b) Use terms up to and including the x¹³ term in the expansion for tan‍ x, together with an appropriate value for x, to find an approximation for π. Write your approximation correct to 4 decimal places.

Find the first four terms of the Taylor series for the function 2 tan(x) about x = П 4

Find the Taylor expansion of the function f(x) = 4x - 2x-x about c = 0. (Express numbers in exact form. 'Use symbolic notation and fractions where needed.) f(x) =

CAOUST 9 Write the Taylor Series for f(x) = sin(x) at x = 1/2/2 as p n não Cn (x-7) Find the first 5 coeffecients. 2

Let f(x)=e(x-1)^2-1/(x-1)2  for x ≠ 1 and f(1)=1 a. Write the first four nonzero terms and the general term of the Taylor series for e(x-1)^2 about x=1 b. Use the series found in (a) to write the first four nonzero terms and the general term for the Taylor series for f about x=1 c. Determine the interval of convergence for the series given in (b). d. Use the series for f about x=1 to determine if the graph of f has any points of inflection.

What is the coefficient of (x-1)^5 in the Taylor series for xlnx about x=1?

Evaluate the Taylor Series for f(x) = x³ - 10x² +6 at x = 3.