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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.8.82b
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 guidance
1
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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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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Slopes 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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Finding The Implicit Derivative
Related Practice
Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).

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

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

cos y = x; (0, π/2)

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} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t - 10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

b. Find dx/dt and interpret the meaning of this derivative.  

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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.