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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 3.8.46a
Chapter 3, Problem 3.8.46a

45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)

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First, substitute the given point (1, 1) into the equation of the curve x³ + y³ = 2xy to verify if it satisfies the equation.
Substitute x = 1 and y = 1 into the equation: (1)³ + (1)³ = 2(1)(1).
Calculate the left side of the equation: 1 + 1 = 2.
Calculate the right side of the equation: 2 * 1 * 1 = 2.
Since both sides of the equation are equal, the point (1, 1) lies on the curve.

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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 differentiate equations where the dependent and independent variables are not explicitly separated. In the context of the equation x³ + y³ = 2xy, we differentiate both sides with respect to x, treating y as a function of x. This allows us to find the derivative dy/dx, which is essential for determining the slope of the tangent line at a specific point.
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Finding The Implicit Derivative

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 is equal to the derivative of the function at that point. To find the equation of the tangent line, we use the point-slope form, which requires the slope (found via differentiation) and the coordinates of the point on the curve.
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Slopes of Tangent Lines

Verifying Points on Curves

Verifying that a point lies on a curve involves substituting the coordinates of the point into the equation of the curve. If the left-hand side equals the right-hand side after substitution, the point is confirmed to be on the curve. In this case, substituting (1, 1) into the equation x³ + y³ = 2xy helps establish that the point is indeed on the curve before proceeding to find the tangent line.
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Summary of Curve Sketching
Related Practice
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

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.

a. Find dr/dh for a cone with a lateral surface area of A=1500π.

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

4x³ =y²(4−x); x=2 (cissoid of Diocles)

1
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Textbook Question

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

tan xy = x+y; (0,0)

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

a. Use implicit differentiation to find dy/dx.