Taylor Polynomials and Derivatives

The video tutorial explains how to find the Taylor polynomial of degree 5 centered at zero for the function f(x) = cos(x). It covers the process of determining the first five derivatives, evaluating them at zero, and constructing the polynomial. The tutorial also discusses using the Taylor remainder theorem to estimate the error and find the range of x values for which the approximation is accurate within 0.0014 of the true function value. Finally, it provides a graphical comparison of the original function and the polynomial approximation.

What is the function for which we are finding the Taylor polynomial?

• f(x) = cos(x)

• f(x) = e^x

• f(x) = sin(x)

• f(x) = tan(x)

What is the first derivative of f(x) = cos(x)?

• cos(x)

• -sin(x)

• -cos(x)

• sin(x)

How many derivatives are evaluated to construct the degree 5 Taylor polynomial?

• 3

• 6

• 4

• 5

What is the coefficient of the x^2 term in the Taylor polynomial?

• 1/2

• -1/2

• 1/4

• -1/4

What is the simplified form of the degree 5 Taylor polynomial for f(x) = cos(x)?

• 1 - (1/2)x^2 + (1/24)x^4

• 1 + (1/2)x^2 - (1/24)x^4

• 1 - (1/2)x^2 - (1/24)x^4

• 1 + (1/2)x^2 + (1/24)x^4

What is the maximum value of the absolute value of the sixth derivative of f(x) = cos(x)?

• 0

• 1

• -1

• 2

What is the interval of x values for which the Taylor polynomial approximation is accurate within 0.0014?

• [-1.03, 1.23]

• [-1.13, 1.13]

• [-1.03, 1.03]

• [-1.23, 1.23]

What is the value of 6 factorial (6!) used in the remainder theorem calculation?

• 1440

• 5040

• 720

• 120

In the graphical representation, what color is used to represent the original function f(x) = cos(x)?

• Blue

• Red

• Green

• Yellow

What is the purpose of using the Taylor Remainder Theorem in this context?

• To find the exact value of the function

• To determine the degree of the polynomial

• To find the range of x values for accurate approximation

• To calculate the derivatives