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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.47
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
Verified step by step guidance1
Start by rewriting the integral to separate the terms inside the integral: \(\int \frac{1 + \cos 4t}{2} \, dt = \int \frac{1}{2} \, dt + \int \frac{\cos 4t}{2} \, dt\).
Integrate the first term \(\int \frac{1}{2} \, dt\) which is a constant multiple of \(dt\). Recall that \(\int a \, dt = at\) for constant \(a\).
For the second term \(\int \frac{\cos 4t}{2} \, dt\), factor out the constant \(\frac{1}{2}\) to get \(\frac{1}{2} \int \cos 4t \, dt\).
Use the substitution rule or recall the integral formula \(\int \cos (kt) \, dt = \frac{1}{k} \sin (kt) + C\). Apply this with \(k=4\) to find the antiderivative of \(\cos 4t\).
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative.
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2mKey Concepts
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Indefinite Integral and Antiderivative
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, providing a family of functions whose derivative equals the integrand.
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Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like cosine involves using known integral formulas, such as ∫cos(ax) dx = (1/a) sin(ax) + C. Recognizing and applying these formulas simplifies finding antiderivatives.
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Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral and helps adjust any initial guesses.
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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 Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
Textbook Question
Identifying Extrema
In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.
Textbook Question
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
2. y=x^4/4-2x^2+4
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
77. y' = cot(θ/2), for 0 < θ < 2π