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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.5.86b
Chapter 3, Problem 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.

Verified step by step guidance
1
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.
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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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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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Related Practice
Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 8x; a = −3

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Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x2 - 5x + 1; P(1,-7)

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Textbook Question

Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.

a. At what rate is the area of the triangle changing when the legs are 5 m long?

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Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines

b. Graph the tangent and normal lines on the given graph.

(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)

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Textbook Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 3t⁴; a= -2, 2

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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.

(x²+y²)²=25/4 xy²; (1, 2)

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