Taylor Polynomials and Derivatives

The video tutorial explains how to find a Taylor polynomial of degree 3 centered at C=4 for the function f(x) = sqrt(x+1). It covers the process of finding derivatives up to the third order, evaluating them at x=4, and constructing the Taylor polynomial. The tutorial also includes simplifying the polynomial expression and comparing its graph with the original function to demonstrate its accuracy around x=4.

What is the degree of the Taylor polynomial we are trying to find?

• 5

• 3

• 2

• 4

What is the center point C for the Taylor polynomial in this tutorial?

• 5

• 4

• 3

• 2

Which rule is typically applied when finding the derivative of a composite function?

• Power Rule

• Product Rule

• Quotient Rule

• Chain Rule

What is the first derivative of the function f(x) = sqrt(x + 1)?

• 2 * (x + 1)^(-1/2)

• 1/2 * (x + 1)^(1/2)

• 1/2 * (x + 1)^(-1/2)

• 2 * (x + 1)^(1/2)

How is the second derivative of the function expressed in terms of positive exponents?

• 1/4 * (x + 1)^(3/2)

• 1/4 * (x + 1)^(-3/2)

• 1/4 * (x + 1)^(-1/2)

• 1/4 * (x + 1)^(1/2)

What is the value of f(4) for the function f(x) = sqrt(x + 1)?

• 5^(1/2)

• 5^(5/2)

• 5^(3/2)

• 5^(1/4)

What is the coefficient of the (x - 4)^2 term in the Taylor polynomial?

• 1/8

• 1/16

• 1/4

• 1/2

How is the term (x - 4)^3 expressed in the Taylor polynomial?

• 1/16 * (x - 4)^3

• 1/8 * (x - 4)^3

• 1/4 * (x - 4)^3

• 1/2 * (x - 4)^3

What color is used to graph the original function in the comparison?

• Red

• Green

• Blue

• Yellow

What is the main purpose of using a Taylor polynomial in this context?

• To simplify the function

• To find the exact value of the function

• To change the function's domain

• To approximate the function around a specific point