Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.5.75b

Capitolo 3, Problema 3.5.75b

Use a graphing utility to plot the curve and the tangent line.
y = cos x / 1−cos x; x = π/3

Guida verificata passo dopo passo

First, understand the function y = \( \frac{\cos x}{1 - \cos x} \). This is a rational function where the numerator is \( \cos x \) and the denominator is \( 1 - \cos x \).

Next, find the derivative of the function to determine the slope of the tangent line at \( x = \frac{\pi}{3} \). Use the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = \cos x \) and \( v = 1 - \cos x \).

Calculate \( u' \) and \( v' \). For \( u = \cos x \), \( u' = -\sin x \). For \( v = 1 - \cos x \), \( v' = \sin x \). Substitute these into the quotient rule formula.

Evaluate the derivative at \( x = \frac{\pi}{3} \) to find the slope of the tangent line. Substitute \( x = \frac{\pi}{3} \) into the derivative expression you obtained.

Finally, use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in the previous step, and \( (x_1, y_1) \) is the point on the curve at \( x = \frac{\pi}{3} \). Plot the curve and the tangent line using a graphing utility.

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Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of the function at that specific location. To find the equation of the tangent line, one typically needs the derivative of the function evaluated at the point of tangency.

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Percorso guidato

05:13

Slopes of Tangent Lines

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve at any given point, which is essential for analyzing the behavior of the function.

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Percorso guidato

05:44

Derivatives

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and their properties. It can plot curves, compute derivatives, and display tangent lines, making it easier to analyze complex functions. Using a graphing utility helps in understanding the relationship between a function and its tangent line, especially at specific points like x = π/3.

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Percorso guidato

06:15

Graphing The Derivative

Pratica correlata

Domanda del libro di testo

62–65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x) = (x−1) sin^−1 x on [−1,1]

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Domanda del libro di testo

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

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

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

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Domanda del libro di testo

{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

389

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Domanda del libro di testo

Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².

b. Evaluate this derivative when r=30 and h=40.

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Domanda del libro di testo

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=e^−x tan^−1 x on [0,∞)