Taylor Polynomials

Utilizing taylor polynomials is an effective way to approximate functions at a certain point. The higher degree you make your taylor polynomial, the more accurate your answer will be compared to the actual answer you could compute on a calculator.

Steps:
1. Compute the derivatives of the given function, meaning solve up to the nth derivative when solving an nth-degree taylor polynomial (for second degree polynomial, solve up to second derivative)
2. Plug in the given x value into each derivative
3. Plug the values you computed into the given form below

Form:\[T_n\left(x\right) = f(x) + f^1(x)(x-a) + (f^2(x)/2)(x-a)^2 +... (f^n(x)/n!)(x-a)^n\]


Question 1: \[f(x) = x^{1/2},T_2\left(4\right)\]

Question 2: \[f(x) = sin(x),T_2\left(0\right)\]

Question 3: \[f(x) = cos(x),T_2\left(\pi/4\right)\]