Question 1 Write the first four terms of the Taylor expansion of the following function near the point x₀

i. f(x)=5x⁴ - 3x³ - x² + 7x + 14
ii. f(x) = 12-x
iii. f(x) = e^3x
iv. f(x) = xe^x

Question 2. Use the Taylor series to evaluate the following expressions:

i. √(54)

ii. e²

iii. ln 6

Question 3. Write a program to calculate the sine and cosine functions of different angles based on the Taylor expansion.

Question 4. Solve the following equations using both bisection and Newton's methods.

i. x³ - 5x + 14 = 0
ii. x³ - 4x² + 7x - 8 = 0
iii. xe^x - 12 = 0
iv. x⁴ - 3x³ + 5x² + x - 12

Question 5. Use the Excel routine to find the cube of the following numbers:

i. 456987, ii. 312701, iii. 8123455

After finding the cube root, move your initial guess closer to the final result and see the effect on the number of iterations needed.

Question 6. Write the first eight terms of the Taylor series of the following functions:

i. z = (x-y)/(x+y),

ii. z = e^2x-3y,

iii. z = ln(x+y)

Question 7. Suppose we have a model and have estimated E(y) by y^. Further assume that var(y^) = σ^². What would be a reasonable estimate of E(e^y) and its variance?