Problem 10 - University Calculus: Early Transcendentals

Question

Find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. a. \(\frac{1}{2} x^{-1 / 2}\) b. \(-\frac{1}{2} x^{-3 / 2}\) c. \(-\frac{3}{2} x^{-5 / 2}\)

Step-by-Step Solution

Answer

a. \( x^{1/2} + C \); b. \( x^{-1/2} + C \); c. \( x^{-3/2} + C \).

Step 1: Understand the Problem

We are asked to find the antiderivative of three given functions. An antiderivative is a function whose derivative equals the given function. Use the formula for the antiderivative of a power function: if \( f(x) = x^n \), then the antiderivative is \( F(x) = \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \) and \( C \) is the constant of integration.

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

For \( f(x) = \frac{1}{2} x^{-1/2} \), identify \( n = -1/2 \). Increase the exponent by 1 to get \( x^{(-1/2) + 1} = x^{1/2} \). The antiderivative is \( F(x) = \frac{1/2}{1/2} x^{1/2} + C = x^{1/2} + C \).

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

Differentiate \( F(x) = x^{1/2} + C \). The derivative is \( \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2} \), which matches \( f(x) = \frac{1}{2} x^{-1/2} \). This confirms the solution is correct.

Step 4: Find the Antiderivative of \( -\frac{1}{2} x^{-3/2} \)

For \( f(x) = -\frac{1}{2} x^{-3/2} \), identify \( n = -3/2 \). Increase the exponent by 1 to get \( x^{(-3/2) + 1} = x^{-1/2} \). The antiderivative is \( F(x) = \frac{-1/2}{-1/2} x^{-1/2} + C = x^{-1/2} + C \).

Step 5: Verify the Antiderivative of \( -\frac{1}{2} x^{-3/2} \)

Differentiate \( F(x) = x^{-1/2} + C \). The derivative is \( \frac{d}{dx}(x^{-1/2}) = -\frac{1}{2} x^{-3/2} \), which matches \( f(x) = -\frac{1}{2} x^{-3/2} \). This confirms the solution is correct.

Step 6: Find the Antiderivative of \( -\frac{3}{2} x^{-5/2} \)

For \( f(x) = -\frac{3}{2} x^{-5/2} \), identify \( n = -5/2 \). Increase the exponent by 1 to get \( x^{(-5/2) + 1} = x^{-3/2} \). The antiderivative is \( F(x) = \frac{-3/2}{-3/2} x^{-3/2} + C = x^{-3/2} + C \).

Step 7: Verify the Antiderivative of \( -\frac{3}{2} x^{-5/2} \)

Differentiate \( F(x) = x^{-3/2} + C \). The derivative is \( \frac{d}{dx}(x^{-3/2}) = -\frac{3}{2} x^{-5/2} \), which matches \( f(x) = -\frac{3}{2} x^{-5/2} \). This confirms the solution is correct.

Key Concepts

Differentiation

Differentiation is the process used to calculate a derivative. A derivative represents the rate at which a function is changing at any given point. When we differentiate a function, we essentially find a new function that gives the slope of the original function's graph at every point. This is particularly useful in physics for finding velocity from a position function, or economics for finding marginal cost from a total cost function.

In mathematical notation, if you have a function denoted as\(f(x)\), its derivative is denoted as \(f'(x)\) or \(\frac{d}{dx}f(x)\). When we solve for antiderivatives, we often use differentiation to verify our solution. If differentiating an antiderivative returns you to the original function, you've found the right antiderivative.

• For example, if you have an antiderivative function \(F(x)\), differentiating \(F(x)\) should yield the original function
• This validation step confirms the correctness of the antiderivative by realigning it with its derivative.

Constant of Integration

The constant of integration, often denoted as \(C\), is a crucial part of finding antiderivatives. When performing indefinite integration to find an antiderivative, there isn't just one correct answer. Instead, there's a family of functions that differ only by a constant. This is because the derivative of a constant is zero, leaving infinite possibilities for the antiderivative which differ by any constant value.

For instance, when you find the antiderivative of a function \(f(x) = x^n\), it could be any function \(F(x) = \frac{x^{n+1}}{n+1} + C\). Here, \(C\) represents an arbitrary constant that reflects all the constant possibilities that could have been present before differentiation.

• This leads to an infinite number of possible antiderivatives, all sharing the same slope but shifting vertically by \(C\).
• The constant of integration accounts for various shifts in the graph of a function.

Power Function Formula

The power function formula is a mathematical tool essential for finding antiderivatives of power functions. A power function is a function of the form \(f(x) = x^n\), and it follows the general formula for finding its antiderivative:\[F(x) = \frac{x^{n+1}}{n+1} + C\]Where \(neq -1\), and \(C\) is the constant of integration. This formula works under the condition that the exponent \(n\) is not equal to -1, since that would lead to division by zero. This formula is straightforward, but there are some nuanced steps involved in applying it.

When finding the antiderivative:

• Identify the exponent \(n\) of your power function.
• Increase the exponent by 1.
• Divide by the new exponent to get the antiderivative.
• Don’t forget to add \(C\), the constant of integration.

By following these steps, you can efficiently find the antiderivative of any power function, assuming \(n eq -1\). The formula simplifies integration and is a powerful tool in calculus.