Textbook Question
Business and Economics
62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.58
Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C
Verified step by step guidance1
Identify the given antiderivative formula: \(\int (3x + 5)^{-2} \, dx = -\frac{(3x + 5)^{-1}}{3} + C\).
To verify this formula, differentiate the right-hand side expression \(-\frac{(3x + 5)^{-1}}{3} + C\) with respect to \(x\).
Apply the chain rule for differentiation: if \(f(x) = (3x + 5)^{-1}\), then \(f'(x) = -1 \cdot (3x + 5)^{-2} \cdot 3\) because the derivative of the inside function \(3x + 5\) is 3.
Multiply the derivative by the constant factor \(-\frac{1}{3}\) outside the function, simplifying the expression step-by-step.
Confirm that the resulting derivative simplifies exactly to the original integrand \((3x + 5)^{-2}\), which verifies the antiderivative formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives and Indefinite Integrals
An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C, since differentiation of a constant is zero.
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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule helps integrate functions with variables raised to a power, including negative exponents, by increasing the exponent by one and dividing by the new exponent.
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Verification by Differentiation
To verify an antiderivative formula, differentiate the proposed antiderivative and check if it equals the original integrand. This process confirms the correctness of the integral by applying the fundamental theorem of calculus in reverse.
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Related Practice
Textbook Question
A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(sin2x − csc²x)dx
Textbook Question
Finding Extrema from Graphs
In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
f(x) = |x|, −1 < x < 2
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
h(x) = ³√x, −1 ≤ x ≤ 8
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(1 + cos 4t)/2 dt