Textbook Question
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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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.8.64b
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
Verified step by step guidance1
To determine if the curve has any horizontal tangent lines, we need to find where the derivative of the function is equal to zero. Horizontal tangent lines occur at points where the slope of the curve is zero.
First, identify the function that describes the curve. Let's denote this function as f(x).
Next, compute the derivative of the function, f'(x), using differentiation rules. This derivative represents the slope of the tangent line at any point x on the curve.
Set the derivative f'(x) equal to zero and solve for x. This will give you the x-values where the slope of the tangent line is zero, indicating potential horizontal tangent lines.
Finally, verify these x-values by substituting them back into the original function f(x) to find the corresponding y-values. This will confirm the points on the curve where horizontal tangent lines occur.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Lines
Tangent lines are straight lines that touch a curve at a single point without crossing it. The slope of the tangent line at a point on the curve represents the instantaneous rate of change of the function at that point. Understanding tangent lines is crucial for analyzing the behavior of curves, particularly in determining where they are increasing or decreasing.
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Slopes of Tangent Lines
Vertical Tangent Lines
A vertical tangent line occurs when the slope of the tangent approaches infinity, which typically happens when the derivative of the function is undefined at that point. This can indicate a cusp or a vertical asymptote in the curve. Identifying vertical tangents is important for understanding the limits and behavior of the function near those points.
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05:13Slopes of Tangent Lines
Horizontal Tangent Lines
A horizontal tangent line occurs when the slope of the tangent line is zero, indicating that the function has a local maximum or minimum at that point. To find horizontal tangents, one must set the derivative of the function equal to zero and solve for the corresponding x-values. This concept is essential for analyzing critical points and the overall shape of the curve.
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Related Practice
Textbook Question
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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Textbook Question
{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.
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Textbook Question
Use a graphing utility to plot the curve and the tangent line.
y = cos x / 1−cos x; x = π/3
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Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x⁴+y⁴ = 2;(1,−1)
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Textbook Question
Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x2 + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.
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