Question

In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=e^{2 x}, \quad n=4 $$

Short answer

The Maclaurin polynomial of degree 4 for the function \(f(x) = e^{2x}\) is \(f(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{4}{3}x^4\).

Step-by-step solution

Write the general formula for a Maclaurin series

The Maclaurin series has a general formula: \(f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \frac{f''''(0)x^4}{4!}\). Thus, the objective is to calculate all these values for the given function \(f(x) = e^{2x}\).

Calculate the values

Starting with \(f(0)\), simply replace x per 0 in the equation \(f(x) = e^{2x}\), obtaining \(f(0) = e^{2*0} = 1\). Following, you do the same for the derivatives. They result in: \(f'(x) = 2e^{2x}\), \(f''(x) = 4e^{2x}\), \(f'''(x) = 8e^{2x}\) and \(f''''(x) = 16e^{2x}\). Now replace x per 0 in these equations, obtaining \(f'(0) = 2\), \(f''(0) = 4\), \(f'''(0) = 8\), and \(f''''(0) = 16\).

Substitute in the general equation of the Maclaurin series

By substituting the found values into the Maclaurin series, the polynomial of degree 4 is obtained: \(f(x) = 1 + 2x + \frac{4x^2}{2!} + \frac{8x^3}{3!} + \frac{16x^4}{4!}\). This simplifies to: \(f(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{4}{3}x^4\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. At its core, calculus is about understanding how things evolve over time or how they are related to one another spatially through functions. The two primary divisions of calculus are differential calculus and integral calculus.

Differential calculus is centered around the notion of a derivative, which measures the rate at which a quantity changes. Integral calculus, on the other hand, focuses on the concept of an integral, which measures the accumulation of quantities. In the exercise presented, the process involves finding the Maclaurin polynomial for the exponential function, which requires the use of differential calculus to find various derivatives of the function that will later be used in the series expansion.

Exponential function

The exponential function, commonly represented as (e^x), is a mathematical function where the number e (~2.71828) is raised to the power of a variable x. This function is widely used because of its unique properties, one of which is that its rate of growth is directly proportional to its value.

Exponential functions are ubiquitous in real-world applications such as population growth, compound interest, and natural phenomena like radioactive decay. In our exercise, our particular function of interest is (e^{2x}), which is a type of exponential function where the growth rate is doubled compared to a standard (e^x) function.

Series expansion

Series expansion is a mathematical technique used to approximate functions using the sum of terms from a sequence, typically polynomials. One of the most important series expansions is the Taylor series, which can approximate any function that is infinitely differentiable at a point.

A special case of the Taylor series is the Maclaurin series, which is centered around 0 (zero). The Maclaurin series is crucial for our exercise as it is used to represent the exponential function (e^{2x}) as a polynomial of degree n. A polynomial approximation can be very useful for calculations as it simplifies computations, especially when dealing with complicated functions.

Derivatives

Derivatives represent the cornerstone of differential calculus. A derivative of a function at any given point describes the rate at which the function value changes with respect to the change in the independent variable. Simply put, it measures the sensitivity to change of the function value (output) with respect to change in its argument (input).

In the context of our problem, we need to find the derivatives of the function (e^{2x}) up to the fourth derivative, as we are dealing with a Maclaurin polynomial of degree 4. Each derivative evaluated at x = 0 gives us a coefficient for the respective term of the polynomial, allowing us to construct the polynomial approximation of the function.