Understanding the Maclaurin Series of (f(x) 4^x)

The Maclaurin series is a particular form of the Taylor series, centered around the point (x 0). For the function f(x) 4^x, the Maclaurin series expansion involves the function and its derivatives evaluated at (x 0).

Step 1: Calculate (f(0))

The initial step is to find the value of f(0).

f(0) 4^0 1

Step 2: Derivatives and Evaluations

To find the non-zero terms, we need to calculate the derivatives of f(x) 4^x at (x 0).

First Derivative

The first derivative of (f(x) 4^x) can be found using the general derivative formula for exponential functions:

(f'(x) 4^x ln 4)

Evaluating at x 0: (f'(0) 4^0 ln 4 ln 4)

Second Derivative

The second derivative involves taking the derivative of the first derivative:

(f''(x) (4^x ln 4) ln 4 4^x (ln 4)^2)

Evaluating at x 0: (f''(0) 4^0 (ln 4)^2 (ln 4)^2)

Third Derivative

Continuing this pattern, the third derivative is:

(f'''(x) (4^x (ln 4)^2) ln 4 4^x (ln 4)^3)

Evaluating at x 0: (f'''(0) 4^0 (ln 4)^3 (ln 4)^3)

General (n)-th Derivative

From the pattern, we can generalize the (n)-th derivative:

(f^{(n)}(x) 4^x (ln 4)^n)

Evaluating at x 0: (f^{(n)}(0) 4^0 (ln 4)^n (ln 4)^n)

Step 3: Writing the Maclaurin Series

Now that we have the values of the derivatives, we can write the Maclaurin series for (f(x) 4^x):

(f(x) sum_{n0}^{infty} frac{f^{(n)}(0)}{n!}x^n sum_{n0}^{infty} frac{(ln 4)^n}{n!}x^n)

This simplifies to:

(f(x) e^{ln 4 cdot x} 4^x)

Conclusion

The non-zero terms in the Maclaurin series expansion of (f(x) 4^x) are as follows:

Constant term: 1 Linear term: (ln 4 cdot x) Quadratic term: (frac{(ln 4)^2}{2}x^2) Cubic term: (frac{(ln 4)^3}{6}x^3) And so on for higher-order terms

By recognizing that 4^x e^{ln 4 cdot x}, it's easy to take the consecutive derivatives and see the pattern in the Maclaurin series. The reader is encouraged to verify these steps for the next few derivatives to solidify understanding.