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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.
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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.R.69
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3
Guida verificata passo dopo passo1
First, substitute x = 1 into the function to check if the limit results in an indeterminate form. Calculate the numerator: 1^4 - 1^3 - 3(1)^2 + 5(1) - 2 and the denominator: 1^3 + 1^2 - 5(1) + 3.
After substitution, if both the numerator and denominator evaluate to 0, the limit is in the indeterminate form 0/0, which means l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. The derivative of the numerator (x^4 - x^3 - 3x^2 + 5x - 2) is 4x^3 - 3x^2 - 6x + 5.
The derivative of the denominator (x^3 + x^2 - 5x + 3) is 3x^2 + 2x - 5.
Evaluate the limit of the new function formed by the derivatives as x approaches 1: lim_x→1 (4x^3 - 3x^2 - 6x + 5) / (3x^2 + 2x - 5). Substitute x = 1 into this expression to find the limit.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 1 involves determining the behavior of the given rational function near that point.
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex rational functions.
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Polynomial Functions
Polynomial functions are expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit problem, both the numerator and denominator are polynomials. Understanding their behavior, such as factoring or finding roots, is crucial for evaluating limits effectively.
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6:04Introduction to Polynomial Functions
Pratica correlata
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45–46. Linear approximation
a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?
ƒ(x) = x²⸍³ ; a =27; ƒ(29)
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60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_y→0⁺ (ln¹⁰ y) / √y
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Domanda del libro di testo
Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.
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Domanda del libro di testo
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
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Domanda del libro di testo
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
a . Give the approximate coordinates of the local maxima and minima of ƒ
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