Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Nd antiderivatives for the given functions. In other words, for each function \(f, \quad\) nd a function \(F\) such that \(F^{\prime}=f .\) (a) \(f(x)=\sin 2 x\) (b) \(f(x)=\cos (x / 3)\)

Step-by-Step Solution

Answer

The antiderivative of \(f(x)=\sin 2x\) is \(-\frac{1}{2}\cos 2x + C\), and the antiderivative of \(f(x)=\cos(x / 3)\) is \(3\sin\left(\frac{x}{3}\right) + C\). These are general solutions given the indefinite nature of the integrals; they may have different specific values of C when applied to initial value problems.

Step 1: Find Antiderivative for \(f(x)=\sin 2x\)

First, we use the antiderivative formula for the sine function, which is \(-\cos x + C\). However, we need to account for the 2 multiplying x in the sine function. This involves using the chain rule of derivatives in reverse, yielding: \[-\frac{1}{2}\cos 2x + C\] Here, C represents the constant of integration.

Step 2: Find Antiderivative for \(f(x)=\cos(x / 3)\)

We use the antiderivative formula for the cosine function, which is \(\sin x + C\). However, given that x is divided by 3 in the cosine function, again we need to use the chain rule. This gives us the antiderivative: \[3\sin\left(\frac{x}{3}\right) + C\] Here, C is the constant of integration.

Key Concepts

Chain Rule

The Chain Rule is a fundamental principle in calculus that helps us understand how to differentiate composite functions. When you have a function inside another function, like the sine of 2x or cosine of x/3, the Chain Rule allows you to handle the inner and outer functions separately. In the context of finding antiderivatives, we reverse the process of differentiation. This means that we need to "undo" the Chain Rule.

When you take the derivative of a function, the Chain Rule multiplies the derivative of the outer function by the derivative of the inner function. So, when finding antiderivatives, we divide by the derivative of the inner function.

• For \( \sin 2x \), the antiderivative involves reversing the process. You get \( -\cos 2x \) and then divide by the derivative of the inner function, which is 2.
• For \( \cos (x/3) \), it similarly involves reversing and gives \( 3\sin(x/3) \) by dividing by the derivative of \( x/3 \), which is \( 1/3 \).

Understanding this concept is crucial for correctly finding antiderivatives for composite functions.

Integration

Integration is the process of finding an antiderivative, or an integral, of a function. This is the reverse process of differentiation. While differentiation gives us the rate of change of a function, integration helps us find the original function from its rate of change.

The notation of integration involves the integral sign \(\int\), and the constant of integration \(C\) is always added to indefinite integrals.

• If you have \( \int \sin 2x \, dx \), it results in \( -\frac{1}{2} \cos 2x + C \).
• And for \( \int \cos(x/3) \, dx \), it results in \( 3\sin(x/3) + C \).

This constant \(C\) is important because it represents an entire family of functions that differ by constant amounts. In definite integrals, this constant is not present as the limits of integration account for it.

Sine and Cosine Functions

Sine and cosine functions are key in trigonometry and calculus. Their periodic properties and derivatives make them unique and very important in many fields. The sine function’s derivative is the cosine function, and vice versa. Understanding this relationship is crucial when dealing with their antiderivatives.

For finding the antiderivative:

• The formula for the antiderivative of \( \sin x \) is \( -\cos x + C \).
• For \( \cos x \), the antiderivative is \( \sin x + C \).

When additional constants are involved, like \( \sin 2x \) or \( \cos(x/3) \), using the Chain Rule allows you to correctly adjust these basic antiderivative formulas.