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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 4.7.9c
Chapter 4, Problem 4.7.9c

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/3)x⁻⁴ᐟ³

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1
Identify the function to find the antiderivative of: \(-\frac{1}{3} x^{-\frac{4}{3}}\).
Recall the power rule for antiderivatives: For \(f(x) = x^n\), an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule by adding 1 to the exponent: \(-\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}\).
Divide the original coefficient by the new exponent: Multiply \(-\frac{1}{3}\) by \(\frac{1}{-\frac{1}{3}}\) which is the reciprocal of the new exponent.
Write the antiderivative as \(F(x) = -\frac{1}{3} \times \frac{x^{-\frac{1}{3}}}{-\frac{1}{3}} + C\), then simplify the expression and add the constant of integration \(C\).

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Key Concepts

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

Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using basic integration rules. The result includes a constant of integration since differentiation of a constant is zero.
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating functions with variable exponents, including negative and fractional powers, by increasing the exponent by one and dividing by the new exponent.
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Power Rule for Indefinite Integrals

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.
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Finding Differentials
Related Practice
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.

√x + 1/√x

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.

−π csc (πx/2) cot (πx/2)

1
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Textbook Question

Finding displacement from an antiderivative of velocity

a. Suppose that the velocity of a body moving along the s-axis is

ds/dt = v = 9.8t − 3.

iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.

Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = 1− 4/x², x ≠ 0

Textbook Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

Textbook Question

Dependence on Initial Point

8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations

c. x_0=2