Problem 82 - Applied Calculus

Question

Find an antiderivative and use differentiation to check your answer. $$h(x)=\frac{x}{\sqrt{x}}+\frac{\sqrt{x}}{x}$$

Step-by-Step Solution

Answer

An antiderivative is \( H(x) = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \).

Step 1: Simplify the Expression

First, simplify the function \( h(x) = \frac{x}{\sqrt{x}} + \frac{\sqrt{x}}{x} \). This can be rewritten as \( h(x) = x^{1/2} + x^{-1/2} \) by expressing \( \frac{x}{\sqrt{x}} \) as \( x^{1/2} \) and \( \frac{\sqrt{x}}{x} \) as \( x^{-1/2} \).

Step 2: Find the Antiderivative

To find an antiderivative of \( h(x) = x^{1/2} + x^{-1/2} \), integrate each term separately:\[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2} \]\[ \int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} = 2x^{1/2} \]Thus, an antiderivative is \( H(x) = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \) where \( C \) is the constant of integration.

Step 3: Differentiate the Antiderivative

Differentiate \( H(x) = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \) to verify the antiderivative:\[ \frac{d}{dx} \left( \frac{2}{3}x^{3/2} \right) = x^{1/2} \]\[ \frac{d}{dx} \left( 2x^{1/2} \right) = x^{-1/2} \]Thus, the derivative \( H'(x) = x^{1/2} + x^{-1/2} \) and it matches \( h(x) \), confirming the antiderivative is correct.

Step 4: Conclusion

The antiderivative \( H(x) = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \) is confirmed correct as its derivative \( H'(x) = x^{1/2} + x^{-1/2} \) equals the original function \( h(x) \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that helps us determine the rate of change of a function. Essentially, it lets us find the derivative of a function, denoted as \( f'(x) \), which represents how the function's output changes with respect to its input. In simple terms, if you imagine driving a car, the derivative would tell you how fast your speed is changing at any given moment.

When finding an antiderivative, as we did with the function \( h(x) = \frac{x}{\sqrt{x}} + \frac{\sqrt{x}}{x} \), differentiation plays a critical role in verifying the correctness of our answer. We do this by differentiating the antiderivative found (in this case, \( H(x) = \frac{2}{3}x^{3/2} + 2x^{1/2} + C \)) and checking if we retrieve the original function \( h(x) \).

Differentiation involves the application of several rules, including:

• The Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• The Sum Rule: The derivative of a sum of functions is the sum of the derivatives.
• The Constant Rule: The derivative of a constant is zero.

This practice of checking antiderivatives ensures that our integration process was executed correctly and logically connects differentiation and integration as inverse processes.

Definite Integrals

Definite integrals are another key concept in calculus that contrasts with finding an antiderivative, which is essentially an indefinite integral. While antiderivatives seek a general form of a function, definite integrals compute the accumulated value, like finding the area under a curve, within specific limits. This gives the exact total of change over a certain interval.

Mathematically, if you have a function \( f(x) \) and you want to find the area under its curve from \( a \) to \( b \), you use the definite integral: \[ \int_{a}^{b} f(x) \, dx \]This expression finds the net area between the curve \( f(x) \) and the x-axis from \( x = a \) to \( x = b \). Unlike the antiderivative process, which results in a function plus a constant \( C \), a definite integral results in a numerical value that represents accumulated quantities such as areas, volumes, and more.

Some important facts about definite integrals include:

• The Fundamental Theorem of Calculus, which connects differentiation and integration, states that if \( F \) is an antiderivative of \( f \), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).
• They can account for the total positive (above the x-axis) and negative (below the x-axis) areas by their sign.
• When finding the area under a curve, the result is often used in real-world applications, like physics or economics, to compute quantities like distance, work done, or total accumulated value.

Integration

Integration is a pivotal concept in calculus, often described as the reverse process of differentiation. While differentiation breaks down a function to describe how it changes at any point, integration builds up to find a function’s accumulated value, like when we find an antiderivative.

The process of integration can be applied in two main ways:

• Indefinite Integration: This involves finding an antiderivative of a function, which results in a family of functions \( F(x) + C \), where \( C \) is the constant of integration.
• Definite Integration: This computes a specific numerical value that represents, for example, the total area under a curve, using set limits on the integral.

In our exercise, we simplified \( h(x) \) to find its antiderivative as part of indefinite integration, an important process that allows us to obtain a broader understanding of a function's behavior. The rules of integration mirror those of differentiation to allow us to compute antiderivatives efficiently:

• The Power Rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
• The Sum Rule: \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \).
• Substitution: Useful for more complex integrals, simplifies by changing variables.

Integration is not only essential for solving calculus problems but has a wide range of applications in science, engineering, and other analytical fields.