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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7.41
Chapter 3, Problem 3.7.41

Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?

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Step 1: To find the points where the curve crosses the x-axis, set y = 0 in the equation of the curve x² + xy + y² = 7. This simplifies to x² = 7. Solve for x to find the x-coordinates of the points where the curve intersects the x-axis.
Step 2: The solutions to x² = 7 are x = √7 and x = -√7. Therefore, the points where the curve crosses the x-axis are (√7, 0) and (-√7, 0).
Step 3: To find the slope of the tangent lines at these points, we need to find the derivative of the curve with respect to x. Use implicit differentiation on the equation x² + xy + y² = 7. Differentiate both sides with respect to x.
Step 4: The derivative of the left side with respect to x is 2x + y + x(dy/dx) + 2y(dy/dx). Set this equal to the derivative of the right side, which is 0, and solve for dy/dx to find the slope of the tangent line.
Step 5: Substitute y = 0 and the x-values √7 and -√7 into the derivative equation to find the slope at each point. Show that the slopes are equal, indicating that the tangent lines are parallel. The common slope is the value of dy/dx at these points.

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Key Concepts

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

Finding Points of Intersection

To find where the curve intersects the x-axis, set y = 0 in the equation x² + xy + y² = 7. This simplifies to x² = 7, giving the points of intersection as (√7, 0) and (-√7, 0). These are the points where the curve crosses the x-axis.
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Implicit Differentiation

Implicit differentiation is used to find the derivative of a function defined implicitly, such as x² + xy + y² = 7. By differentiating both sides with respect to x, and treating y as a function of x, we can find dy/dx, which represents the slope of the tangent line at any point on the curve.
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Finding The Implicit Derivative

Parallel Lines and Slopes

Two lines are parallel if they have the same slope. After finding dy/dx using implicit differentiation, evaluate it at the points of intersection (√7, 0) and (-√7, 0). If the slopes at these points are equal, the tangent lines are parallel. The common slope is the value of dy/dx at these points.
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Slopes of Tangent Lines
Related Practice
Textbook Question

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

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x + √xy = 6, (4, 1)

Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = (2√x)/(3(1 + √x))

Textbook Question

In Exercises 19–22, find the slope of the curve at the point indicated.


y = x³ − 2x + 7, x = −2

Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


xy² − 4x³/² − y = 0

Textbook Question

One-Sided Derivatives


Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

Textbook Question

Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹

where m is a positive integer.