Understanding the Maclaurin Series

The video tutorial explains how to find the Maclaurin series for the function f(x) = 3x^2 * sin(9x). It begins by introducing the concept of Maclaurin series and the challenges of finding higher-order derivatives. Instead, the tutorial uses the known power series for sin(x) to simplify the process. The series is built by substituting 9x for x and multiplying by 3x^2. The series is then simplified using algebraic manipulation. The tutorial calculates the coefficients C3 through C7 and concludes with a graphical comparison of the original function and the polynomial approximation.

What is the function for which we are finding the Maclaurin series?

• f(x) = 3x^2 * cos(9x)

• f(x) = 3x^2 * sin(9x)

• f(x) = 2x^3 * sin(9x)

• f(x) = 3x^3 * sin(9x)

What is the center of the Maclaurin series?

• x = 0

• x = -1

• x = 1

• x = 2

Why is finding higher-order derivatives of the function directly considered complicated?

• Because it requires numerical methods

• Because it involves solving differential equations

• Because it requires the use of the product and chain rule repeatedly

• Because it involves integration

What known series is used to help construct the Maclaurin series for sin(9x)?

• The power series for exp(x)

• The power series for tan(x)

• The power series for cos(x)

• The power series for sin(x)

What substitution is made to the power series of sin(x) to find the series for sin(9x)?

• Substitute x with 3x

• Substitute x with x/9

• Substitute x with 9x

• Substitute x with x/3

What is the polynomial factor that multiplies the power series for sin(9x)?

• 4x^2

• 5x^2

• 3x^2

• 2x^2

How is the expression simplified using common bases?

• By using the base of 2

• By using the base of 3

• By using the base of 5

• By factoring out x

What is the coefficient C3 in the series?

• 108

• 27

• 54

• 81

What is the coefficient C5 in the series?

• 729/2

• 729/3

• 729/4

• 729/5

How does the polynomial approximation compare to the original function near x = 0?

• It is unrelated

• It is a good representation

• It is an exact match

• It is a poor representation