Textbook Question
Finding Functions from Derivatives
Suppose that f'(x) = 2x for all x. Find f(2) if
b. f(1) = 0
197
views
4. Applications of Derivatives
Differentials
2:39 minutesProblem 34a
Textbook Question
Finding Functions from Derivatives
In Exercises 31–36, find all possible functions with the given derivative.
a. y' = 1 / 2√x
Verified step by step guidance1
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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).
Recommended video:
05:50
Antiderivatives
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.
Recommended video:
05:04Introduction to Indefinite Integrals
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.
Recommended video:
05:04Introduction to Indefinite Integrals
Related Videos
Related Practice