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Chapter 4: Problem 8

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. \begin{equation} \text { a. }\frac{4}{3} \sqrt[3]{x} \quad \text { b. } \frac{1}{3 \sqrt[3]{x}} \quad \text { c. } \sqrt[3]{x}+\frac{1}{\sqrt[3]{x}} \end{equation}

Short Answer

Expert verified
Antiderivatives are: (a) \( x^{4/3} + C \), (b) \( \frac{1}{2}x^{2/3} + C \), (c) \( \frac{3}{4} x^{4/3} + \frac{3}{2} x^{2/3} + C \).

Step by step solution

01

Rewrite Using Power Rule

Identify each term in the expressions and rewrite the cube roots using the power rule: - (a) \( \frac{4}{3} \sqrt[3]{x} = \frac{4}{3}x^{1/3} \)- (b) \( \frac{1}{3 \sqrt[3]{x}} = \frac{1}{3}x^{-1/3} \)- (c) \( \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}} = x^{1/3} + x^{-1/3} \).
02

Find Antiderivatives

Apply the antiderivative (integration) rule for powers: - For \( ax^n, \) the antiderivative is \( \frac{a}{n+1}x^{n+1} + C \):- (a) \( \int \frac{4}{3}x^{1/3} \, dx = \frac{4}{3} \times \frac{3}{4} x^{4/3} + C = x^{4/3} + C \)- (b) \( \int \frac{1}{3}x^{-1/3} \, dx = \frac{1}{3} \times \frac{3}{2} x^{2/3} + C = \frac{1}{2}x^{2/3} + C \)- (c) Decompose into parts: \( \int x^{1/3} \, dx + \int x^{-1/3} \, dx = \frac{3}{4} x^{4/3} + \frac{3}{2} x^{2/3} + C \).
03

Verify by Differentiation

Differentiate each antiderivative to confirm:- (a) Differentiate \( x^{4/3} + C \) to obtain \( \frac{4}{3}x^{1/3} \), which matches the original.- (b) Differentiate \( \frac{1}{2}x^{2/3} + C \) to retrieve \( \frac{1}{3}x^{-1/3} \), consistent with the original.- (c) Differentiate \( \frac{3}{4} x^{4/3} + \frac{3}{2} x^{2/3} + C \) to get \( x^{1/3} + x^{-1/3} \), confirming correctness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a key concept in calculus used for simplifying the process of finding derivatives and antiderivatives. When dealing with expressions involving powers of a variable, the power rule makes it straightforward to find their derivatives or integrals. In our context:
  • To differentiate polynomials like \( x^n \), the power rule states: Move the exponent in front, and subtract one from the exponent, giving \( \frac{d}{dx} [x^n] = nx^{n-1} \).
  • For integration, which is finding the antiderivative, the power rule is reversed. We add 1 to the exponent and divide by the new exponent: \( \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \).
In this exercise, we transformed terms like \( \sqrt[3]{x} \) into fractional powers, \( x^{1/3} \) to utilize the power rule effectively.
Integration
Integration is the process of finding the antiderivative, or the inverse of differentiation. In simpler terms, it's about finding a function whose derivative corresponds to a given function. Integration is crucial for calculating areas, volumes, and solving various other mathematical problems involving accumulation and summation of quantities.
  • The definite integral of a function represents the area under the curve between two points, while the indefinite integral is more general, adding a constant \( C \) to the solution.
  • In this exercise, we apply the integration power rule to terms like \( \frac{4}{3} x^{1/3} \) and others. For example, transforming \( \frac{4}{3} x^{1/3} \) by integrating gives \( x^{4/3} + C \).
Through integration, we find that each of these expressions, though starting differently, can be handled similarly by following systematic rules.
Differentiation
Differentiation is about figuring out how a function changes. It's essentially finding the rate of change or the slope of the function at any point. It's the opposite of integration, making it a powerful tool in calculus for verifying results.
  • When you differentiate a function, you obtain its derivative, which informs you of the function's behavior, identifying any increasing or decreasing periods.
  • In our exercise, after finding the antiderivatives of given expressions, we utilize differentiation to verify our work. For example, differentiating \( x^{4/3} + C \) should yield \( \frac{4}{3} x^{1/3} \), confirming that we did the integration correctly.
  • This check ensures the antiderivatives we found are accurate, thus reaffirming the correctness of our calculations.
Differentiation and integration are interconnected techniques, each validating and complementing the other in calculus.

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Most popular questions from this chapter

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