Chapter 4.5, Problem 100E

a.

To determine

Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window

Answer to Problem 100E

  $\boxed{\text{good approximation}}$

Explanation of Solution

Given information:

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials

  $\text{sinx }\approx \text{ x - }\frac{x^3}{3\text{!}}+\frac{x^5}{5\text{!}}$

And,

  $\begin{array}{l}\cos x\approx 1-\frac{x^2}{\text{2!}}+\frac{x^4}{\text{4!}}\\ \end{array}$

where $\text{x}$ is in radians.

Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare?

Calculation:

The sine function $\sin x$ and its polynomial function $\text{ x - }\frac{x^3}{3\text{!}}+\frac{x^5}{5\text{!}}$ are graphed as follows,

From the graph above, we see that

Hence, the polynomial function is $\boxed{\text{good approximation}}$ of sine function when $x$ is close to $0$ .

b.

To determine

Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window.

Answer to Problem 100E

  $\boxed{\text{good approximation}}$

Explanation of Solution

Given information:

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials

  $\text{sinx }\approx \text{ x - }\frac{x^3}{3\text{!}}+\frac{x^5}{5\text{!}}$

And,

  $\begin{array}{l}\cos x\approx 1-\frac{x^2}{\text{2!}}+\frac{x^4}{\text{4!}}\\ \end{array}$

Where $\text{x}$ is in radians.

Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare?

Calculation:

The cosine function $\cos x$ and its polynomial function $1-\frac{x^2}{\text{2!}}+\frac{x^4}{\text{4!}}$ are graphed as follows,

From the graph above, we see that

Hence, the polynomial function is $\boxed{\text{good approximation}}$ of cosine function when $x$ is close to $0$ .

c.

To determine

How does the accuracy of the approximations change when an additional term is added?

Answer to Problem 100E

Accuracy is increased as the additional term is added to the approximation.

Explanation of Solution

Given information:

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials

  $\text{sinx }\approx \text{ x - }\frac{x^3}{3\text{!}}+\frac{x^5}{5\text{!}}$

And,

  $\begin{array}{l}\cos x\approx 1-\frac{x^2}{\text{2!}}+\frac{x^4}{\text{4!}}\\ \end{array}$

Where $\text{x}$ is in radians.

Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How does the accuracy of the approximations change when an additional term is added?

Calculation

Studying the pattern of polynomial function from part(a), it can be written including the next term as follows,

  $\sin x\approx x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}$

Above polynomial function is graphed along with the sine function as follows,

From the graph above, we see that

  $\boxed{\text{Accuracy is increased}}$ as the additional term is added to the approximation.

Studying the pattern of polynomial function from part (b), it can be written including the next term as follows,

  $\cos x\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}$

Above polynomial function is graphed along with the sine function as follows,

From the graph above, we see that

Hence, Accuracy is increased as the additional term is added to the approximation.