Textbook Question
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)
4. Applications of Derivatives
Implicit Differentiation
5:51 minutesProblem 3.8.74
Textbook Question
73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>
Exercise 46
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Identify the given curve equation and the specific point P where you need to find the normal line. Let's denote the curve as y = f(x) and the point as P(a, b).
Calculate the derivative of the curve, f'(x), to find the slope of the tangent line at point P. This involves differentiating the function y = f(x) with respect to x.
Evaluate the derivative at the point P to find the slope of the tangent line. This is done by substituting x = a into f'(x) to get the slope m_tangent = f'(a).
Determine the slope of the normal line. Since the normal line is perpendicular to the tangent line, its slope m_normal is the negative reciprocal of the tangent slope: m_normal = -1/m_tangent.
Use the point-slope form of a line to write the equation of the normal line. The point-slope form is y - b = m_normal(x - a), where (a, b) is the point P and m_normal is the slope of the normal line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 represents the instantaneous rate of change of the function at that point, which is determined by the derivative of the function. Understanding tangent lines is crucial for finding normal lines, as the normal line is defined in relation to the tangent.
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Slopes of Tangent Lines
Normal Line
A normal line at a point on a curve is a line that is perpendicular to the tangent line at that same point. To find the equation of the normal line, one must first determine the slope of the tangent line and then use the negative reciprocal of that slope. This concept is essential for solving problems involving normal lines, as it directly relates to the geometry of the curve.
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Slope and Derivatives
The slope of a line is a measure of its steepness, calculated as the change in the y-coordinate divided by the change in the x-coordinate. In calculus, the derivative of a function at a point gives the slope of the tangent line to the curve at that point. This relationship is fundamental for determining both the tangent and normal lines, as it provides the necessary slopes to construct their equations.
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