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Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.9.33
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3/x⁴ + 2 - 3/x²)
Guida verificata passo dopo passo1
Step 1: Break down the integral into separate terms. The given integral is ∫(3/x⁴ + 2 - 3/x²) dx. Rewrite it as ∫(3/x⁴) dx + ∫2 dx - ∫(3/x²) dx.
Step 2: Apply the power rule for integration to each term. Recall that the power rule states ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. For terms involving x raised to negative powers, adjust the exponent accordingly.
Step 3: For the first term, ∫(3/x⁴) dx, rewrite it as ∫3x⁻⁴ dx. Using the power rule, integrate to get (3 * x⁻³)/-3 = -x⁻³.
Step 4: For the second term, ∫2 dx, integrate the constant to get 2x.
Step 5: For the third term, ∫(3/x²) dx, rewrite it as ∫3x⁻² dx. Using the power rule, integrate to get (3 * x⁻¹)/-1 = -3x⁻¹. Combine all results and add the constant of integration, C.
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Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function whose derivative matches the given function.
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05:04
Introduction to Indefinite Integrals
Integration Techniques
To solve integrals, various techniques can be employed, such as substitution, integration by parts, and recognizing standard forms. In this case, the integrand consists of rational functions, which can often be integrated term by term. Understanding how to manipulate and simplify expressions is crucial for effectively applying these techniques.
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06:18Integration by Parts for Definite Integrals
Verification by Differentiation
After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative. This process ensures that the derivative of the obtained function returns to the original integrand. This step is a critical part of the integration process, confirming the correctness of the integration performed.
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05:53Finding Differentials
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Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).
f'(x) = 10 sin 2x on [-2π, 2π]
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Domanda del libro di testo
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((2 + 3 cos y)/sin² y)dy
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Domanda del libro di testo
Cylinder and cones (Putnam Exam 1938) Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?
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49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.
ƒ(x) = 1/3 x³ - 2x² - 5x + 2
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Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = x³ - (3/2)x² - 36x
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