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 3.8.80b
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)
Verified step by step guidance1
First, understand the problem: We need to find the equations of the tangent and normal lines to the curve given by \(x^4 = 2x^2 + 2y^2\) at the point \((x_0, y_0) = (2, 2)\).
To find the tangent line, we need the derivative of the curve. Start by implicitly differentiating the equation \(x^4 = 2x^2 + 2y^2\) with respect to \(x\). Use the chain rule for \(y^2\) since \(y\) is a function of \(x\).
After differentiating, solve for \(\frac{dy}{dx}\), which represents the slope of the tangent line at any point \((x, y)\) on the curve.
Substitute \((x_0, y_0) = (2, 2)\) into the derivative to find the slope of the tangent line at this specific point.
Use the point-slope form of a line, \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in the previous step, to write the equation of the tangent line. The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal of the tangent line's slope. Use this to write the equation of the normal line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. It represents the instantaneous rate of change of the function at that point, which can be found using the derivative. For the curve defined by the equation, the slope of the tangent line can be calculated by differentiating the equation implicitly.
Recommended video:
05:13
Slopes of Tangent Lines
Normal Line
The normal line at a point on a curve is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the slope of the tangent line. Understanding the normal line is essential for visualizing how the curve behaves at that point, and it can also be derived from the tangent line's slope.
Recommended video:
05:13Slopes of Tangent Lines
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated. In this case, the equation x⁴ = 2x² + 2y² involves both x and y. By differentiating both sides with respect to x, we can find dy/dx, which is necessary to determine the slopes of the tangent and normal lines.
Recommended video:
05:14Finding The Implicit Derivative
Related Practice
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
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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Textbook Question
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,∞)
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Textbook Question
City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.
297
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Textbook Question
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) = (x−1) sin^−1 x on [−1,1]
243
views