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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 65
Chapter 3, Problem 65

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y = 2/x; P(1, 2)

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Step 1: Find the derivative of the function y = \(\frac{2}{x}\) to determine the slope of the tangent line at point P(1, 2). The derivative, y', can be found using the power rule and the fact that \(\frac{d}{dx}\)(x^{-1}) = -x^{-2}.
Step 2: Evaluate the derivative at x = 1 to find the slope of the tangent line at point P(1, 2). Substitute x = 1 into the derivative to get the slope of the tangent line.
Step 3: Determine the slope of the line perpendicular to the tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent line.
Step 4: Use the point-slope form of a line equation, y - y_1 = m(x - x_1), where m is the slope of the normal line and (x_1, y_1) is the point P(1, 2), to write the equation of the normal line.
Step 5: Simplify the equation from Step 4 to get the final equation of the normal line in slope-intercept form, y = mx + b, or any other preferred form.

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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 can be found using the derivative of the function.
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Slopes of Tangent Lines

Normal Line

A normal line is a line that is perpendicular to the tangent line at a given point on a curve. Its slope is the negative reciprocal of the slope of the tangent line. This relationship is crucial for finding the equation of the normal line, as it allows us to determine its slope based on the tangent line's slope.
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Slopes of Tangent Lines

Finding Derivatives

Finding the derivative of a function is essential for determining the slope of the tangent line. For the function y = 2/x, we can use the power rule or quotient rule to compute the derivative. Evaluating this derivative at the point P(1, 2) gives us the slope needed to find both the tangent and normal lines.
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The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

Calculate the derivative of the following functions.

y = tan(xe^x)

Textbook Question

Calculate the derivative of the following functions.

y = (e^x / x+1)⁸

Textbook Question

Population growth Consider the following population functions.

b. What is the instantaneous growth rate at t=5?

p(t) = 600 (t²+3/t²+9)

Textbook Question

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

Textbook Question

Let f(x) = x2 - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 0.

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Textbook Question

Let f(x) = x2 - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 2.

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