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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.2b
Chapter 4, Problem 4.7.2b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁷

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1
Recall that an antiderivative of a function is another function whose derivative is the original function. For power functions, the antiderivative rule is: if \(f(x) = x^n\), then an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).
Identify the exponent in the given function \(x^7\). Here, \(n = 7\).
Apply the antiderivative formula by increasing the exponent by 1: \(7 + 1 = 8\).
Divide by the new exponent to get the antiderivative: \(\frac{x^8}{8} + C\).
Verify your answer by differentiating \(\frac{x^8}{8} + C\) and checking that the derivative is \(x^7\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is often expressed with an arbitrary constant, C, since differentiation of a constant is zero.
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Power Rule for Integration

The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) + C. This rule is fundamental for finding antiderivatives of polynomial functions like x⁷.
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Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps avoid mistakes in integration.
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Related Practice
Textbook Question

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

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Textbook Question

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫3(2x + 1)² dx = (2x + 1)³ + C

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Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


b. Show that the only local extreme value of f occurs at x = 2.

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

4 sec 3x tan 3x

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Textbook Question

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?