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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.5.86b
Capitolo 3, Problema 3.5.86b

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.
b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

Guida verificata passo dopo passo
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Understand the problem: We need to evaluate the limit of s(x)/x as x approaches 0, where s(x) is the sine function with x in degrees.
Convert degrees to radians: Since the standard sine function sin(x) uses radians, convert x degrees to radians using the formula x radians = (π/180) * x degrees.
Express s(x) in terms of sin(x): Since s(x) is the sine of x in degrees, we can write s(x) = sin((π/180) * x).
Set up the limit expression: Substitute s(x) with sin((π/180) * x) in the limit expression, giving us lim x→0 sin((π/180) * x) / x.
Simplify the limit expression: Factor out (π/180) from the sine argument, resulting in lim x→0 (π/180) * sin(x * (π/180)) / x. Use the known limit lim x→0 sin(x)/x = 1 to evaluate the expression.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this context, we are interested in the limit of the function s(x)/x as x approaches 0. Understanding limits is crucial for evaluating the continuity and behavior of functions at specific points.
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One-Sided Limits

Sine Function

The sine function, denoted as sin(x), is a periodic function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. In this problem, s(x) represents the sine function evaluated in degrees, which is essential for correctly computing the limit as x approaches 0. The distinction between radians and degrees is vital for accurate calculations.
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Percorso guidato
5:53
Graph of Sine and Cosine Function

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. When evaluating lim x→0 s(x)/x, both the numerator and denominator approach 0, creating an indeterminate form. Applying L'Hôpital's Rule involves differentiating the numerator and denominator, allowing for the limit to be computed more easily.
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Power Rules
Pratica correlata
Domanda del libro di testo

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

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Domanda del libro di testo

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x³+y³=2xy; (1, 1)

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Domanda del libro di testo

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

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Domanda del libro di testo

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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Domanda del libro di testo

58–59. Carry out the following steps.

b. Find the slope of the curve at the given point.

xy^5/2+x^3/2y=12; (4, 1)

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Domanda del libro di testo

45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x⁴-x²y+y⁴=1; (−1, 1)

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