Finding Functions from Derivatives

In Exercises 31–36, find all possible functions with the given derivative.

a. y' = 1 / 2√x

검증된 단계별 안내

Step 1: Recognize that finding a function from its derivative involves integration. The given derivative is \( y' = \frac{1}{2\sqrt{x}} \).

Step 2: Rewrite the derivative in a form that is easier to integrate. Note that \( \frac{1}{2\sqrt{x}} \) can be expressed as \( \frac{1}{2}x^{-1/2} \).

Step 3: Integrate the expression \( \frac{1}{2}x^{-1/2} \) with respect to \( x \). Use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).

Step 4: Apply the power rule to \( \frac{1}{2}x^{-1/2} \). The integral becomes \( \frac{1}{2} \int x^{-1/2} \, dx = \frac{1}{2} \cdot \frac{x^{1/2}}{1/2} + C \). Simplify this expression.

Step 5: Simplify the result from the integration to find the general form of the function \( y(x) \). Don't forget to include the constant of integration \( C \), which represents all possible functions that have the given derivative.

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주요 개념

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Antiderivatives

Antiderivatives, or indefinite integrals, are functions that reverse the process of differentiation. To find a function from its derivative, you need to determine its antiderivative. This involves integrating the derivative function, which in this case is y' = 1 / 2√x, to find the original function y(x).

Integration Techniques

Integration techniques are methods used to find antiderivatives. For the derivative y' = 1 / 2√x, recognizing the form of the function is crucial. The expression 1 / 2√x can be rewritten as x^(-1/2), which is a power function. Applying the power rule for integration, you add 1 to the exponent and divide by the new exponent to find the antiderivative.

Constant of Integration

The constant of integration is an arbitrary constant added to the antiderivative of a function. When integrating a derivative to find the original function, the constant accounts for any vertical shifts in the graph of the function. Since differentiation eliminates constants, the antiderivative must include this constant to represent all possible original functions.

Finding Functions from DerivativesIn Exercises 31–36, find all possible functions with the given derivative.a. y' = 1 / 2√x