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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.3c
Capitolo 4, Problema 4.7.3c

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⁻⁴ + 2x + 3

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1
Identify the function to find the antiderivative of: \(x^{-4} + 2x + 3\).
Recall the power rule for antiderivatives: for \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
Apply the power rule to each term separately: for \(x^{-4}\), the antiderivative is \(\frac{x^{-4+1}}{-4+1} = \frac{x^{-3}}{-3}\).
For the term \$2x$, rewrite it as \$2x^1$ and apply the power rule: \(2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2\).
For the constant term \(3\), recall that the antiderivative of a constant \(a\) is \(ax\), so the antiderivative is \$3x$.

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Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It is also called the indefinite integral and includes a constant of integration since differentiation removes constants. Finding antiderivatives involves reversing differentiation rules.
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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) plus a constant. This rule is essential for integrating polynomial terms like x⁻⁴ and 2x in the given function.
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Linearity of Integration

Integration is linear, meaning the antiderivative of a sum of functions equals the sum of their antiderivatives. This allows us to find the antiderivative of each term separately and then combine the results, simplifying the process for functions like x⁻⁴ + 2x + 3.
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Checking Antiderivative Formulas


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


∫tanθ sec²θ dθ = (1/2) sec²θ + C

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Theory and Examples


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


Let f(x) = |x³ − 9x|.


b. Does f'(-3) exist?

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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) = (x − 1)²(x + 2)

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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)

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105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?

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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

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