Domanda del libro di testo
Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).
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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 4.9.51
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/2y)dy
Guida verificata passo dopo passo1
Step 1: Recognize the integral ∫ (1/2y) dy as a basic indefinite integral. The goal is to find the antiderivative of the function (1/2y) with respect to y.
Step 2: Rewrite the integrand for clarity. The constant 1/2 can be factored out of the integral: ∫ (1/2y) dy = (1/2) ∫ (1/y) dy.
Step 3: Recall the antiderivative rule for 1/y. The integral of 1/y with respect to y is ln|y|, where ln represents the natural logarithm and the absolute value ensures the domain is valid.
Step 4: Combine the constant factor with the antiderivative. The result of the integral becomes (1/2) ln|y| + C, where C is the constant of integration.
Step 5: Verify your result by differentiating (1/2) ln|y| + C with respect to y. The derivative of (1/2) ln|y| is (1/2)(1/y), which matches the original integrand, confirming the solution is correct.
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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 F(y) such that F'(y) equals the integrand.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration is a fundamental technique used to find indefinite integrals of polynomial functions. It states that the integral of y^n with respect to y is (y^(n+1))/(n+1) + C, where n is not equal to -1. This rule simplifies the process of integrating functions by allowing us to increase the exponent and divide by the new exponent.
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Percorso guidato
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
Verification by differentiation involves checking the correctness of an indefinite integral by differentiating the resulting function. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This step is crucial in calculus to ensure that the integration process has been performed accurately.
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Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 6x² - 135x
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Domanda del libro di testo
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)
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Domanda del libro di testo
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = ln (x² + 1)
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Domanda del libro di testo
Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.
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Domanda del libro di testo
{Use of Tech} Growth rate of spotted owlets The rate of growth (in g/week) of the body mass of Indian spotted owlets is modeled by the function r(t) = 10,147.9e⁻²·²ᵗ/(37.98e⁻²·² + 1), where t is the age (in weeks) of the owlets. What value of t > 0 maximizes r? What is the physical meaning of the maximum value?
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