Problem 33 - Thomas Calculus

Question

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises \(29-34 .\) $$e^{\sin x}$$

Step-by-Step Solution

Answer

The first four nonzero terms are: \(1 + x + \frac{x^2}{2} + \frac{x^3}{3!}\).

Step 1: Understand the Maclaurin Series Formula

The Maclaurin series for a function \( f(x) \) is given by \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \). To find the series for \( e^{\sin x} \), we first need the derivatives of the function and evaluate them at 0.

Step 2: Calculate \( f(0) \)

Start by evaluating the function at 0. Since \( f(x) = e^{\sin x} \), we have \( f(0) = e^{\sin 0} = e^0 = 1 \).

Step 3: First Derivative

Find the first derivative of \( f(x) = e^{\sin x} \). Use the chain rule: \( f'(x) = \cos x \cdot e^{\sin x} \). Evaluate at \( x = 0 \): \( f'(0) = \cos 0 \cdot e^{\sin 0} = 1 \cdot 1 = 1 \).

Step 4: Second Derivative

Use the product rule and differentiate \( f'(x) = \cos x \cdot e^{\sin x} \) to get \( f''(x) = -\sin x \cdot e^{\sin x} + \cos^2 x \cdot e^{\sin x} \). Evaluate at \( x = 0 \): \( f''(0) = -\sin 0 \cdot 1 + 1^2 \cdot 1 = 1 \).

Step 5: Third Derivative

Differentiate \( f''(x) \) again. Simplify: \( f'''(x) = -\cos x \cdot e^{\sin x} \cdot \sin x + (-\sin x \cdot \cos x + \cos^3 x) \cdot e^{\sin x} \). Evaluate at 0: \( f'''(0) = 0 \).

Step 6: Obtain Nonzero Term (4th Derivative)

Differentiate \( f'''(x) \) to find \( f^{(4)}(x) \), including all terms that involve \( \sin x \) and \( \cos x \). After simplification, evaluate \( f^{(4)}(0) \). Further derivatives are complex; focus primarily on ensuring accuracy for evaluations up to four nonzero terms.

Step 7: Assemble the Maclaurin Series

Given the evaluations from the previous steps, the first four nonzero terms in the Maclaurin series are: \( f(x) \approx 1 + x + \frac{x^2}{2} + \cdots \). Identify further terms by reevaluating higher derivatives or focusing on simplified expressions around these results.

Key Concepts

Derivatives

A derivative measures how a function changes as its input changes. In simpler terms, it tells you the rate at which a function is changing at any given point. To find derivatives, we differentiate the function. The first derivative of a function is essentially the slope of the function at any point. If we keep differentiating, we get the second derivative, which describes how the slope itself changes.

• The first derivative gives the rate of change of the function.
• The second derivative indicates the rate of change of the first derivative, giving insights into the function's concavity.

When finding the Maclaurin series, calculating derivatives step by step is crucial as this series uses the value of the derivative function at zero.

Chain Rule

The chain rule is an essential technique in calculus for finding the derivative of composite functions. When you have a function inside another function, the chain rule allows you to differentiate both at once. For instance, in the function \(e^{\sin x}\), \(\sin x\) is inside of \(e^x\).Here's how the chain rule works:

• Differentiation of the outer function: take the derivative of the outside function assuming the inside function is just a variable.
• Differentiation of the inner function: multiply by the derivative of the inside function.

Practically for \(e^{\sin x}\):1. Differentiate \(e^u\) where \(u = \sin x\); this gives \(e^{\sin x}\).2. Multiply by the derivative of \(\sin x\), which is \(\cos x\).Thus, the chain rule helps us find that \(f'(x) = \cos x \cdot e^{\sin x}\).

Product Rule

The product rule is a formula used to find the derivative of products of two functions. It's particularly useful when the expression you're working with is a product of two functions rather than a simple composite.For a function \(uv\):

• The product rule is expressed as \((uv)' = u'v + uv'\).
• Essentially, differentiate one function and leave the other, then swap this process, and add them together.

In the case of \(f'(x) = \cos x \cdot e^{\sin x}\), applying the product rule gives the next derivative \(f''(x)\). While calculating \(f''(x)\) of our original function, each part in the product \(\cos x\) and \(e^{\sin x}\) is individually differentiated and then combined by the product rule.

Exponential Function

An exponential function is a mathematical function of the form \(f(x) = a^x\), where \(a\) is a constant and \(x\) is the variable. A key property is that this function grows or decays at a rate proportional to its current value, making it a powerful model for many natural processes.The base of the natural exponential function is \(e\), approximately 2.718. This specific exponential function, \(e^x\), is particularly notable because its derivative is itself:

• The derivative of \(e^x\) is \(e^x\).
• This property plays a crucial role in calculus, simplifying many operations on functions involving exponentials.

In the function \(e^{\sin x}\), the calculation of derivatives relies heavily on the intrinsic property of the exponential function, coupled with chain and product rules, enabling seamless manipulation and successful expansion into a Maclaurin series.