Textbook Question
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
b. (f^-1)'(6)
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 3.8.82b
79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
Verified step by step guidance1
First, understand the given equation: \((x^2 + y^2)^2 = \frac{25}{3} (x^2 - y^2)\). This is the equation of a lemniscate of Bernoulli, a type of figure-eight curve.
To find the tangent line at the point \((x_0, y_0) = (2, -1)\), we need to differentiate the given equation implicitly with respect to \(x\). This involves using the chain rule and product rule.
After differentiating, substitute \(x = 2\) and \(y = -1\) into the derivative to find the slope of the tangent line at this point.
Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope found in the previous step, and \((x_1, y_1) = (2, -1)\), to write the equation of the tangent line.
The normal line is perpendicular to the tangent line. Therefore, its slope is the negative reciprocal of the tangent line's slope. Use the point-slope form again with this new slope to find 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. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative. For the given curve, the tangent line can be determined by calculating the derivative of the implicit function at the specified point (2, -1).
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05:13
Slopes of Tangent Lines
Normal Line
The normal line is perpendicular to the tangent line at a given point on a curve. Its slope is the negative reciprocal of the slope of the tangent line. To graph the normal line, one must first find the slope of the tangent line at the point of interest and then use this slope to determine the equation of the normal line, which will intersect the curve at the same point.
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05:13Slopes of Tangent Lines
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y. In this case, the equation (x²+y²)² = 25/3 (x²-y²) defines y as a function of x. By differentiating both sides with respect to x and applying the chain rule, one can find dy/dx, which is essential for determining the slopes of the tangent and normal lines.
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05:14Finding The Implicit Derivative
Related Practice
Textbook Question
A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>
b. Which runner has the greater average angular velocity?
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Textbook Question
Derivatives using tables Let and . Use the table to compute the following derivatives.
<IMAGE>
b.
Textbook Question
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counter example.
b. d²/dx² (sin x) = sin x.
Textbook Question
{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.
b. Graph and interpret the gas mileage m(g)/g.