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

Problema 3.8.63b

Capitolo 3, Problema 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.

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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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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.

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05:14

Finding The Implicit Derivative

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.

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Percorso guidato

05:14

Equations of Tangent Lines

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.

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