Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 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

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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Textbook Question

Use the graph of g in the figure to do the following. <IMAGE>

b. Find the values of x in (-2,2) at which g is not differentiable.

Textbook Question

Vertical tangent lines

b. Does the curve have any horizontal tangent lines? Explain.

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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)

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. d/dx(tan^−1 x) =sec² x

Textbook Question

The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)

Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\lim_{x \to 0} \frac{\sqrt{4 + \sin (x)} - 2}{x}$