45–50. Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve.(x²+y²)²=25/4 xy²; (1, 2)

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

문제 3.8.50a

3장, 문제 3.8.50a

45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
(x²+y²)²=25/4 xy²; (1, 2)

검증된 단계별 안내

First, substitute the given point (1, 2) into the equation of the curve \((x^2 + y^2)^2 = \frac{25}{4}xy^2\).

Calculate \(x^2 + y^2\) by substituting \(x = 1\) and \(y = 2\) into the expression, resulting in \(1^2 + 2^2 = 1 + 4 = 5\).

Substitute \(x = 1\) and \(y = 2\) into the right side of the equation \(\frac{25}{4}xy^2\), which becomes \(\frac{25}{4} \times 1 \times 2^2 = \frac{25}{4} \times 4 = 25\).

Now, substitute the calculated values into the original equation: \((5)^2 = 25\) and \(25 = 25\).

Since both sides of the equation are equal, the point (1, 2) lies on the curve.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we differentiate both sides of the equation (x² + y²)² = 25/4 xy² with respect to x, treating y as a function of x. This allows us to find the slope of the tangent line at a specific point on the curve.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The slope of the tangent line can be found using the derivative of the function at that point. The equation of the tangent line can then be expressed in point-slope form, using the coordinates of the point and the calculated slope.

Verifying Points on Curves

To verify that a point lies on a curve defined by an equation, we substitute the coordinates of the point into the equation. 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, 2) into the equation (x² + y²)² = 25/4 xy² will determine if the point lies on the curve.

45–50. Tangent lines Carry out the following steps. <IMAGE>a. Verify that the given point lies on the curve.(x²+y²)²=25/4 xy²; (1, 2)