{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.

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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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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Textbook Question

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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Textbook Question

Consider the following cost functions.

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