Chapter 8: Q.10 (page 659)

If $f$ is a function such that $f (0) = - 3$ and localid="1650438953513" role="math" $f' (x) = 2 f (x)$ every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Expert verified

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

$f (x) = - 3 (1 + 2 x + \frac{4 x^{2}}{2!} + \frac{8 x^{3}}{3!} + \dots)$

Or, it can be written as

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

Step by step solution

Given Information

Given equations:

$f (0) = - 3$

$f^{'} (x) = 2 f (x)$

Finding the Maclaurin series for f

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, $f (0) = - 3$ .

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

So,

$f^{''} (0) = 2 f^{'} (0) = 2 \cdot (- 6) = - 12$

And,

$f^{''} (0) = 2 f^{''} (0) = 2 \cdot - 12 = - 24$

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

$f (x) = - 3 - 6 x + \frac{- 12 x^{2}}{2!} + \frac{- 24 x^{3}}{3!} + \dots$

$= - 3 (1 + 2 x + \frac{4 x^{2}}{2!} + \frac{8 x^{3}}{3!} + \dots)$

Or, it can be written as

localid="1650438932580" $f (x) = - 3 \sum_{k = 0}^{\infty} \frac{2^{k} x^{k}}{k!}$

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If $f$ is a function such that $f (0) = - 3$ and localid="1650438953513" role="math" $f' (x) = 2 f (x)$ every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Step by step solution

Given Information

Finding the Maclaurin series for f

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If fis a function such that f(0)=−3and localid="1650438953513" role="math" f'(x)=2f(x)every value of x, find the Maclaurin series for f.

Short Answer

Step by step solution

Given Information

Finding the Maclaurin series for f

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Statistics

Applied Mathematics

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Pure Maths

Study anywhere. Anytime. Across all devices.

If $f$ is a function such that $f (0) = - 3$ and localid="1650438953513" role="math" $f' (x) = 2 f (x)$ every value of $x$ , find the Maclaurin series for $f$ .