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Chapter 8: Q. 9 (page 692)

If $f (x) = 4 x^{3} - 5 x^{2} + 6 x + 1 a n d P_{3} (x)$ is the third Taylor polynomial for f at 鈭�1, what is the third remainder $R_{3} (x)$ ? What is $R_{4} (x)$ ? (Hint: You can answer this question without finding any derivatives.)

Short Answer

Expert verified

The required values are $R_{3} (x) = 0 a n d R_{4} (x) = 0$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = 4 x^{3} - 5 x^{2} + 6 x + 1$

02

Step 2. Calculation

The formula to calculate the remainder is $R_{n} (x) = \frac{f^{n + 1} (c)}{(n + 1)!} {(x - x_{0})}^{n + 1}$

Substitute n as 3 to find the third remainder of the function.

$R_{3} (x) = \frac{f^{3 + 1} (c)}{(3 + 1)!} {(x - x_{0})}^{3 + 1} = \frac{f^{4} (c)}{4!} {(x - x_{0})}^{4}$

Since the given degree has highest degree 3 which implies that $f^{4} (c) = 0$

Hence, $R_{3} (x) = 0$

Similarly,

localid="1649315178808" $R_{4} (x) = \frac{f^{4 + 1} (c)}{(4 + 1)!} {(x - x_{0})}^{4 + 1} = \frac{f^{5} (c)}{5!} {(x - x_{0})}^{5} = \frac{0}{5!} {(x - x_{0})}^{5} = 0$

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Most popular questions from this chapter

If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?

Find the interval of convergence for power series: ${鈭�}_{k = 1}^{鈭�} {(- 1)}^{K} \frac{1}{k^{k}} {(x - 1)}^{k}$

In Exercises 23鈥�32 we ask you to give Lagrange鈥檚 form for the corresponding remainder, $R_{4} (x)$

$e^{x}$

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ .It may be shown that $J_{p} (x)$ is given by the following power series in x:

$J_{p} (x) = {鈭�}_{k = 0}^{鈭�} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

What is the interval of convergence for $J_{p} (x)$ where p is a non-negative integer

If a function f has a Taylor series at $x_{0}$ , what are the possibilities for the interval of convergence for that series?

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