Problem 7 - Thomas Calculus

Question

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{3}{2} \sqrt{x} \quad\) b. \(\frac{1}{2 \sqrt{x}} \quad\) c. \(\sqrt{x}+\frac{1}{\sqrt{x}}\)

Step-by-Step Solution

Answer

The antiderivatives are: a) \(x^{3/2} + C\), b) \(x^{1/2} + C\), c) \(\frac{2}{3}x^{3/2} + 2x^{1/2} + C\).

Step 1: Understand the Problem

We need to find an antiderivative for three functions: \( \frac{3}{2} \sqrt{x}, \ \frac{1}{2 \sqrt{x}}, \ \text{and} \ \sqrt{x} + \frac{1}{\sqrt{x}} \). An antiderivative of a function is a function that reverses the process of differentiation.

Step 2: Transform Functions

Rewrite the terms using exponent rules to simplify differentiation and integration. \( \sqrt{x} = x^{1/2} \) and \( \frac{1}{\sqrt{x}} = x^{-1/2} \). Thus:1. \( \frac{3}{2} \sqrt{x} = \frac{3}{2} x^{1/2} \)2. \( \frac{1}{2 \sqrt{x}} = \frac{1}{2} x^{-1/2} \)3. \( \sqrt{x} + \frac{1}{\sqrt{x}} = x^{1/2} + x^{-1/2} \).

Step 3: Calculate the Antiderivative of \( \frac{3}{2} x^{1/2} \)

The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \). Apply it to \( \frac{3}{2} x^{1/2} \):\[ \int \frac{3}{2} x^{1/2} \, dx = \frac{3}{2} \cdot \frac{x^{3/2}}{3/2} + C = x^{3/2} + C \]

Step 4: Calculate the Antiderivative of \( \frac{1}{2} x^{-1/2} \)

The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \). Apply it to \( \frac{1}{2} x^{-1/2} \):\[ \int \frac{1}{2} x^{-1/2} \, dx = \frac{1}{2} \cdot \frac{x^{1/2}}{1/2} + C = x^{1/2} + C \]

Step 5: Calculate the Antiderivative of \( x^{1/2} + x^{-1/2} \)

Integrate each term separately:\[ \int (x^{1/2} + x^{-1/2}) \, dx = \int x^{1/2} \, dx + \int x^{-1/2} \, dx \]\[ = \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \]

Step 6: Check by Differentiation

Differentiate each found antiderivative to check correctness:1. Differentiate \( x^{3/2} + C \) to get \( \frac{3}{2} x^{1/2} \).2. Differentiate \( x^{1/2} + C \) to get \( \frac{1}{2} x^{-1/2} \).3. Differentiate \( \frac{2}{3}x^{3/2} + 2x^{1/2} + C \) to get \( x^{1/2} + x^{-1/2} \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, involving the process of finding a derivative. A derivative represents the rate at which a function changes as its input changes. It's like finding the slope of a curve at any given point. Differentiation is essentially the reverse process of integration, similar to how subtraction is the reverse of addition.

To differentiate a function like an antiderivative and check your work, you:

• Identify the power of each term. If you have a function in the form of \( x^n \), the derivative is found by bringing down the exponent \( n \) and subtracting 1 from it, resulting in \( nx^{n-1} \).
• Differentiate constants as zero because they do not change.
• Add up all the derivatives of individual terms to get the derivative of the whole function.

By applying differentiation, you ensure that the antiderivative you've found indeed corresponds to the original function by reversing what you just integrated.

Exponent Rules

Exponent rules are like the grammar of algebra, essential for simplifying expressions, especially those involving roots and powers. When dealing with antiderivatives or integrals, understanding these rules is crucial to modifying terms into a form that's easier to manage.

Some foundational exponent rules include:

• \( x^m \times x^n = x^{m+n} \) - when multiplying like bases, add the exponents.
• \( \frac{x^m}{x^n} = x^{m-n} \) - when dividing like bases, subtract the exponents.
• \( (x^m)^n = x^{m \times n} \) - when raising a power to another power, multiply the exponents.
• \( x^{1/n} = \sqrt[n]{x} \), notably \( x^{1/2} = \sqrt{x} \) - these signify roots as fractional exponents.

Using these rules enables you to transform expressions like \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \) into \( x^{1/2} \) and \( x^{-1/2} \) respectively, making it simpler to apply integration or differentiation techniques.

Integration

Integration, the process of determining an antiderivative, is key to solving problems where you need to find a function given its rate of change. Essentially, it reverses differentiation, accumulating the area under a curve represented by a function.

When you're integrating expressions like \( x^n \), remember these steps:

• Increase the exponent by one: \( n + 1 \).
• Divide the coefficient by the new exponent: \( \frac{x^{n+1}}{n+1} \).
• Don’t forget to add the integration constant \( C \) because antiderivatives are not unique. They differ by constants.

For example, when integrating \( \frac{3}{2} x^{1/2} \), convert it to \( x^{3/2} \) by following the rule and adjust constants accordingly, ensuring that each part of the expression aligns with integration principles. This provides clarity and accuracy in understanding how integral calculus brings functions back from their rate of change to their original form.