Witch of Agnesi Let y(x²+4)=8 (see figure). b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Ch. 3 - Derivatives

Problem 3.8.63b

Chapter 3, Problem 3.8.63b

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

Verified step by step guidance

First, rewrite the given equation y(x² + 4) = 8 in terms of y: y = 8 / (x² + 4). This represents the Witch of Agnesi curve.

To find the tangent lines, we need to determine the derivative of y with respect to x. Use the quotient rule for differentiation: if y = u/v, then y' = (u'v - uv') / v².

Apply the quotient rule to y = 8 / (x² + 4). Here, u = 8 and v = x² + 4. Calculate u' = 0 and v' = 2x. Substitute these into the quotient rule formula to find y'.

Set y = 1 in the original equation to find the x-values where the curve has y = 1. Solve 1(x² + 4) = 8 for x, which simplifies to x² + 4 = 8, leading to x² = 4. Solve for x to find x = ±2.

Substitute x = ±2 into the derivative y' to find the slope of the tangent lines at these points. Use the point-slope form of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point of tangency, to write the equations of the tangent lines.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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, since y is expressed in terms of x², we differentiate both sides of the equation with respect to x, applying the chain rule where necessary. This allows us to find dy/dx, which is essential for determining the slope of the tangent lines.

Tangent Line Equation

The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope at that point. To find the tangent lines when y=1, we first need to determine the corresponding x-values and then calculate the slope using the derivative obtained from implicit differentiation.

Witch of Agnesi

The Witch of Agnesi is a specific type of curve defined by the equation y(x² + 4) = 8, which can be rearranged to express y in terms of x. This curve is notable in calculus for its symmetrical properties and its applications in probability and statistics. Understanding its shape and behavior is crucial for analyzing tangent lines and their equations.

Related Practice

Textbook Question

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{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

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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)

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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

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

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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.

x⁴-x²y+y⁴=1; (−1, 1)