Problem 35 - Thomas Calculus

Question

Find all possible functions with the given derivative. \begin{equation}\quad \text { a. }y^{\prime}=\sin 2 t \quad \text { b. } y^{\prime}=\cos \frac{t}{2} \quad \text { c. } y^{\prime}=\sin 2 t+\cos \frac{t}{2}\end{equation}

Step-by-Step Solution

Answer

a. \( y = -\frac{1}{2}\cos 2t + C \); b. \( y = 2\sin \frac{t}{2} + C \); c. \( y = -\frac{1}{2}\cos 2t + 2\sin \frac{t}{2} + C \).

Step 1: Understanding the Problem

We need to find functions whose derivatives match the provided expressions. The expressions given are derivatives, so finding their antiderivatives will give us the original functions.

Step 2a: Find Antiderivative for \( y' = \sin 2t \)

To find the function \( y \) whose derivative is \( \sin 2t \), we integrate \( \sin 2t \). Using the integration rule for \( \sin(kx) \), we obtain:\[ \int \sin 2t \, dt = -\frac{1}{2}\cos 2t + C \]where \( C \) is the constant of integration.

Step 2b: Find Antiderivative for \( y' = \cos \frac{t}{2} \)

To find the function \( y \) whose derivative is \( \cos \frac{t}{2} \), we compute its integral. The integration rule for \( \cos(kx) \) is applied here:\[ \int \cos \frac{t}{2} \, dt = 2\sin \frac{t}{2} + C \]where \( C \) is the constant of integration.

Step 2c: Find Antiderivative for \( y' = \sin 2t + \cos \frac{t}{2} \)

For the function whose derivative is \( \sin 2t + \cos \frac{t}{2} \), we integrate each term separately and add the results:1. \( \int \sin 2t \, dt = -\frac{1}{2}\cos 2t + C_1 \)2. \( \int \cos \frac{t}{2} \, dt = 2\sin \frac{t}{2} + C_2 \)Adding these gives the solution:\[ y = -\frac{1}{2}\cos 2t + 2\sin \frac{t}{2} + C \]where \( C = C_1 + C_2 \), a constant.

Key Concepts

Integration

Integration is the mathematical process of finding a function when its derivative is given. It essentially reverses the process of differentiation. When you have a function's derivative, integration helps you find the original function that was differentiated.

• The Purpose of Integration: It allows us to find antiderivatives, which are functions whose derivatives give the original function.
• Integration Techniques: There are various methods to perform integration, such as substitution, integration by parts, and more simple rules for basic functions.
• Definite and Indefinite Integrals: An indefinite integral represents a function plus a constant of integration, often described as a family of functions. A definite integral, on the other hand, computes the area under the curve for a specific interval.

Whenever you integrate a function like the ones in this exercise, you're asking, "What function, when differentiated, will give us this result?" For the expressions given, applying trigonometric integration rules allows us to find the functions that correlate to each derivative.

Constant of Integration

The constant of integration is a vital component when finding antiderivatives. It represents the fact that there are infinitely many possible functions that could have the same derivative.

• Why We Need It: Given the derivative, the antiderivative can differ by a constant because the derivative of a constant is zero.
• How It Appears: When you integrate a function, you add a constant (often denoted as \(C\)) to the result to indicate that any of these infinite variations are part of the solution.
• In the Example: In the solved exercise, you'll notice how each integral has its own constant \(C\), reflecting the indefinite nature of finding antiderivatives.

Integrating the function exposes all potential original functions by accounting for variable starting values. This constant ensures that we represent all possible vertical shifts of the graph of the integrated function.

Trigonometric Functions

Trigonometric functions are fundamental in calculus because they often show up in real-world scenarios, such as waves and oscillations. Understanding their behavior, and how to integrate them, is crucial.

• Basic Trigonometric Functions: Sine (\(\sin(x)\)), Cosine (\(\cos(x)\)), and Tangent (\(\tan(x)\)) are the core trigonometric functions that you will encounter.
• Integrating Trigonometric Functions: Integration involving these functions often requires specific formulas, like integrating \(\sin(kx)\) and \(\cos(kx)\), which have known antiderivatives.
• Example Applications: In our problem set, you saw the integration for \(\sin(2t)\) and \(\cos\left(\frac{t}{2}\right)\). Recognizing patterns and applying the right integration rule is key to solving these problems.

Understanding and mastering the integration of trigonometric functions will not only help in finding antiderivatives for mathematical problems but also apply in analyzing physical and engineering systems that use these functions to describe waveforms and periodic behavior.