Chapter 8: Q. 9 (page 700)

If $f$ is a function such that $f (0) = 2$ and $f^{'} (x) = - 3 f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Expert verified

The Maclaurin series for the function $f (x)$ is.

$f (x) = 2 (1 - 3 x + \frac{9 x^{2}}{2!} - \frac{27 x^{3}}{3!} + \dots)$

Or, it can be written as

$f (x) = 2 \sum_{k = 0}^{\infty} \frac{(- 1)^{k} 3^{k} x^{k}}{k!}$

Step by step solution

given information

Let us consider the function such that that $f (0) = 1$ and $f^{'} (x) = - 3 f (x)$

calculation

The general formula to calculate the Maclaurin series for the function is:

$f (x) = f (0) + f^{'} (0) x + \frac{f^{''} (0)}{2!} x^{2} + \frac{f^{''} (0)}{3!} x^{3} + \dots$

As, localid="1650381164301" $f (0) = 2$

$f^{'} (0) = - 3 f (0) = - 3 \cdot 2 = - 6$

So,

$f^{''} (0) = - 3 f^{'} (0) = - 3 \cdot (- 6) = 18$

And

$f^{m} (0) = - 3 f^{''} (0) = - 3 \cdot 18 = - 54$

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If $f$ is a function such that $f (0) = 2$ and $f^{'} (x) = - 3 f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Step by step solution

given information

calculation

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If fis a function such that f(0)=2and f'(x)=-3f(x)for every value of x, find the Maclaurin series for f.

Short Answer

Step by step solution

given information

calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

If $f$ is a function such that $f (0) = 2$ and $f^{'} (x) = - 3 f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .