Taylor Series and Cosine Functions

The video tutorial explains how to find the first four non-zero terms of the Taylor series for cosine of x about x equals 0. It then uses this series, along with the series for sine of x squared, to derive the Taylor series for a function f(x) = sine(x^2) + cosine(x). The process involves calculating derivatives, identifying non-zero terms, and simplifying the series to achieve the final result.

What is the task in part b of the problem?

• Find the derivative of cosine of x.

• Find the integral of sine of x.

• Find the Taylor series for sine of x.

• Find the first four non-zero terms of the Taylor series for cosine of x about x equals 0.

What is the value of the first derivative of cosine of x at x equals 0?

• -1

• 0

• 2

• 1

Which term is not included in the first four non-zero terms of the Taylor series for cosine of x?

• 1

• -x^2/2!

• x^4/4!

• x^3/3!

What is the purpose of approximating cosine of x with the first four non-zero terms?

• To find the integral of cosine of x.

• To approximate cosine of x using a polynomial.

• To simplify the calculation of cosine of x.

• To find the exact value of cosine of x.

What is the function f(x) defined as in the second part of the problem?

• cosine of x squared

• sine of x squared plus cosine of x

• sine of x

• cosine of x

What is the lowest degree term in the Taylor series for f(x) = sine of x squared + cosine of x?

• x

• x^3

• x^2

• 1

How is the x^6 term in the Taylor series for f(x) simplified?

• By dividing x^6/3! by x^6/6!

• By adding x^6/3! and x^6/6!

• By subtracting x^6/3! from x^6/6!

• By multiplying x^6/3! by x^6/6!

What is the coefficient of the x^4 term in the Taylor series for f(x)?

• 1/6!

• 1/2!

• 1/3!

• 1/4!

Which of the following is a non-zero term in the Taylor series for f(x)?

• x^7

• x^2

• x^3

• x^5

What is the final expression for the first four non-zero terms of f(x)?

• 1 + x^2/2 + x^4/4! + 121x^6/6!

• 1 - x^2/2 + x^4/4! - 121x^6/6!

• 1 + x^2/2 - x^4/4! + 121x^6/6!

• 1 + x^2/2 + x^4/4! - 121x^6/6!