Problem 11 - Thomas Calculus in SI Units

Question

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(-\pi \sin \pi x\) b. \(3 \sin x\) c. \(\sin \pi x-3 \sin 3 x\)

Step-by-Step Solution

Answer

a. \(\cos \pi x\); b. \(-3 \cos x\); c. \(-\frac{1}{\pi} \cos \pi x + \cos 3x + C\).

Step 1: Understand Antiderivatives

The antiderivative of a function is a function whose derivative is the original function. For the expression \(-\pi \sin \pi x\), it means finding a function \(F(x)\) such that \(F'(x) = -\pi \sin \pi x\).

Step 2: Solve Part (a)

To find the antiderivative of \(-\pi \sin \pi x\), we need a function that differentiates to give that expression. Knowing the derivative of \(\cos x\) is \(-\sin x\), we get \(F(x) = \cos \pi x + C\), where \(C\) is the constant of integration. Differentiate \(F(x) = \cos \pi x\) to confirm: \(F'(x) = -\pi \sin \pi x\).

Step 3: Solve Part (b)

For the function \(3 \sin x\), find the antiderivative by recalling that the derivative of \(-\cos x\) is \(\sin x\). Thus, \(F(x) = -3\cos x + C\). Check by differentiating: \(F'(x) = 3\sin x\).

Step 4: Solve Part (c)

To find the antiderivative of \(\sin \pi x - 3 \sin 3x\), treat each term separately. For \(\sin \pi x\), the antiderivative is \(-\frac{1}{\pi}\cos \pi x\), and for \(-3 \sin 3x\), the antiderivative is \(\cos 3x\) multiplied by the constant. Therefore, \(F(x) = -\frac{1}{\pi} \cos \pi x + \cos 3x + C\). Differentiate \(F(x)\) to verify: \(F'(x) = \sin \pi x - 3 \sin 3x\).

Key Concepts

Differentiation

Differentiation is a process in calculus used to find the derivative of a function. The derivative represents the rate at which a function is changing at any given point. Understanding differentiation is crucial because finding an antiderivative often involves reversing the differentiation process.

When you differentiate a function, you are essentially looking for how a tiny change in the input of the function (usually denoted as \(x\)) changes the output. For a simple function like \(y=mx+b\), the derivative is \(m\), the slope. However, for more complex functions, especially involving trigonometric terms like \(\sin \pi x\), differentiation uses specific rules.

• The power rule: Helps in differentiating polynomial terms.
• The product rule: Used when the function is a product of two terms.
• The chain rule: Useful when dealing with composite functions, such as trigonometric functions that involve scaling or shifting by constants.

Differentiation tells us how to find rates of change in physics and economics, among many other fields. To verify an antiderivative, one checks the result of differentiation against the original function.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. Common trigonometric functions include sine, cosine, and tangent, with sine (\(\sin\)) and cosine (\(\cos\)) being particularly relevant in calculus problems involving antiderivation.

These functions have specific properties and derivatives that are helpful for solving calculus problems:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).

Understanding these properties allows us to reverse-engineer the process of differentiation, which is essentially what finding an antiderivative involves.
For instance, in part (a) of the original exercise, recognizing that the derivative of \(\cos x\) is \(-\sin x\) helps find the antiderivative of \(-\pi \sin \pi x\). By understanding these rules, you can more easily identify or confirm the antiderivative of any given trigonometric expression.

Constants of Integration

In calculus, the constant of integration is an important concept tied to finding antiderivatives. It represents an arbitrary constant added to an antiderivative because when differentiating, any constant term vanishes, leaving the rate of change unaffected.

When you integrate a function, you find a family of functions that differ only by a constant. This arises because the process of integration is the reverse of differentiation, and since differentiating a constant yields zero, the original constant is unknown.

• For example, the antiderivative of \(3 \sin x\) is \(-3\cos x + C\).
• \(C\) represents this unknown constant.

Including this constant is crucial for accurately representing all possible functions that could differentiate into the given function. While this might seem minor, in practical applications like physics, this constant could represent initial conditions or other specific factors relative to the problem at hand.