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)
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 3.8.65b
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 the derivative of the function that defines the curve. The derivative will give us the slope of the tangent line at any point on the curve.
Set the derivative equal to zero and solve for the variable. This is because a horizontal tangent line has a slope of zero.
Identify the points on the curve where the derivative is zero. These points are where the curve may have horizontal tangent lines.
Verify if these points are indeed on the curve by substituting them back into the original equation of the curve.
Discuss the behavior of the curve at these points to confirm if they correspond to horizontal tangent lines, considering the context of the problem and any constraints given.
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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 indicates that the curve is steeply increasing or decreasing, and it can signify a cusp or a vertical asymptote. Identifying vertical tangents helps in understanding the nature of the curve's behavior at specific 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. This means that the rate of change of the function is momentarily flat, and it is essential for finding critical points in optimization problems. Analyzing horizontal tangents is key to understanding the overall shape and turning points of a curve.
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05:13Slopes of Tangent Lines
Related Practice
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
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
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Textbook Question
31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).
b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.
s(t)= −16t²+100t
Textbook Question
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.
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Textbook Question
{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.