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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 63
Chapter 3, Problem 63

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=3x−4; P(1, −1)

Verified step by step guidance
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Step 1: Find the derivative of the function y = 3x - 4 to determine the slope of the tangent line. The derivative of y with respect to x is dy/dx = 3.
Step 2: Evaluate the derivative at the given point P(1, -1) to find the slope of the tangent line at that point. Since the derivative is constant, the slope of the tangent line at P is 3.
Step 3: Determine the slope of the line perpendicular to the tangent line. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the normal line is -1/3.
Step 4: Use the point-slope form of a line equation to find the equation of the normal line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point. Substitute m = -1/3 and the point P(1, -1) into the equation.
Step 5: Simplify the equation from Step 4 to obtain the final equation 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 can be found using the derivative of the function. In this case, finding the tangent line to the curve involves calculating the derivative and evaluating it at the specified point.
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Slopes of Tangent Lines

Normal Line

A normal line is a line that is perpendicular to a tangent line at a given point on a curve. The slope of the normal line is the negative reciprocal of the slope of the tangent line. This relationship is crucial for determining the equation of the normal line, as it allows us to use the slope of the tangent line to find the slope of the normal line and subsequently write its equation.
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Slopes of Tangent Lines

Equation of a Line

The equation of a line can be expressed in various forms, with the slope-intercept form (y = mx + b) being one of the most common. Here, 'm' represents the slope and 'b' the y-intercept. To find the equation of the normal line, we need the slope from the previous step and the coordinates of the point where the line intersects the curve, allowing us to substitute these values into the line equation format.
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Equations of Tangent Lines
Related Practice
Textbook Question

Calculate the derivative of the following functions.

y = (e^x / x+1)⁸

Textbook Question

Calculate the derivative of the following functions.

y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Textbook Question

Find an equation of the line tangent to the given curve at a.

y = 2x2 / (3x - 1); a = 1

Textbook Question

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = 2x2 / (3x - 1); a = 1

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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 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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