Textbook Question
Find y'' for the following functions.
y = tan x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.2.64
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= √x; P(4, 2)
Verified step by step guidance1
Step 1: Find the derivative of the function y = \(\sqrt{x}\). The derivative, y', represents the slope of the tangent line at any point x. Use the power rule for derivatives: if y = x^{1/2}, then y' = \(\frac{1}{2}\)x^{-1/2}.
Step 2: Evaluate the derivative at the given point P(4, 2) to find the slope of the tangent line. Substitute x = 4 into the derivative y' = \(\frac{1}{2}\)x^{-1/2} to find the slope at x = 4.
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. If the slope of the tangent line is m, then the slope of the normal line is -1/m.
Step 4: Use the point-slope form of a line to write the equation of the normal line. The point-slope form is 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(4, 2).
Step 5: Substitute the slope of the normal line and the coordinates of point P into the point-slope form equation to find the 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.
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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 once the slope of the tangent is determined.
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05:13Slopes of Tangent Lines
Finding the Derivative
Finding the derivative of a function is a fundamental concept in calculus that allows us to determine the slope of the tangent line at any point on the curve. For the function y = √x, the derivative can be calculated using power rules, which will then be evaluated at the specific point P(4, 2) to find the slope needed for both the tangent and normal lines.
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06:02The Second Derivative Test: Finding Local Extrema
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Textbook Question
Find y'' for the following functions.
y = x sin x