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 −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

Step 1. Given Information

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

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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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 −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

Step by step solution

Step 1. Given Information

Step 2. Calculation

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If f(x)=4x3-5x2+6x+1andP3(x) is the third Taylor polynomial for f at −1, what is the third remainder R3(x)? What is R4(x)? (Hint: You can answer this question without finding any derivatives.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Calculus

Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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 −1, what is the third remainder $R_{3} (x)$ ? What is $R_{4} (x)$ ? (Hint: You can answer this question without finding any derivatives.)