60–62. {Use of Tech} Multiple tangent lines Complete the following steps. b. Graph the tangent lines on the given graph.4x³ =y²(4−x); x=2 (cissoid of Diocles)

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

문제 3.8.62b

3장, 문제 3.8.62b

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
4x³ =y²(4−x); x=2 (cissoid of Diocles)

검증된 단계별 안내

Identify the given equation of the curve: \(4x^3 = y^2(4-x)\). This is known as the cissoid of Diocles.

To find the tangent line at a specific point, we need to determine the derivative of the curve with respect to \(x\). Start by differentiating both sides of the equation implicitly with respect to \(x\).

Apply implicit differentiation: Differentiate \(4x^3\) to get \(12x^2\) and differentiate \(y^2(4-x)\) using the product rule, which gives \(2y \frac{dy}{dx} (4-x) - y^2\).

Set the derivatives equal: \(12x^2 = 2y \frac{dy}{dx} (4-x) - y^2\). Solve for \(\frac{dy}{dx}\) to find the slope of the tangent line.

Substitute \(x = 2\) into the derivative to find the slope at this point. Then, use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point on the curve, to write the equation of the tangent line.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Tangent Lines

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 represents the instantaneous rate of change of the function at that point, which can be found using the derivative. Understanding how to calculate and graph tangent lines is essential for analyzing the behavior of functions.

Derivatives

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line at any point on the curve. To find the derivative, various rules such as the power rule, product rule, and chain rule can be applied, depending on the function's form.

Graphing Techniques

Graphing techniques involve plotting points, lines, and curves on a coordinate plane to visually represent mathematical functions and their properties. When graphing tangent lines, it is important to accurately determine the point of tangency and the slope derived from the derivative. This visual representation aids in understanding the relationship between the function and its tangent lines.

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>b. Graph the tangent lines on the given graph.4x³ =y²(4−x); x=2 (cissoid of Diocles)