Textbook Question
Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.86
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
(y - x)² = 2x + 4, (6, 2)
Verified step by step guidance1
First, recognize that the given equation \((y - x)^2 = 2x + 4\) is implicitly defined. To find the tangent line, we need to differentiate this equation with respect to \(x\).
Apply implicit differentiation to \((y - x)^2 = 2x + 4\). Differentiate both sides with respect to \(x\), remembering to use the chain rule for the left side: \(2(y - x)(\frac{dy}{dx} - 1) = 2\).
Solve the differentiated equation for \(\frac{dy}{dx}\) to find the slope of the tangent line at any point \((x, y)\). Substitute \(x = 6\) and \(y = 2\) into the differentiated equation to find the specific slope at the point \((6, 2)\).
Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope found in the previous step and \((x_1, y_1)\) is the point \((6, 2)\), to write the equation of the tangent line.
To find the normal line, use the fact that the slope of the normal line is the negative reciprocal of the slope of the tangent line. Use the point-slope form again with this new slope to write the equation of the normal line at the point \((6, 2)\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of functions that are not explicitly solved for one variable in terms of another. In this problem, the equation (y - x)² = 2x + 4 involves both x and y, requiring implicit differentiation to find dy/dx, which is essential for determining the slope of the tangent line at the given point.
Recommended video:
05:14
Finding The Implicit Derivative
Tangent Line
The tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Its slope is equal to the derivative of the curve at that point. For the equation (y - x)² = 2x + 4 at the point (6, 2), the tangent line can be found using the derivative obtained through implicit differentiation.
Recommended video:
05:13Slopes of Tangent Lines
Normal Line
The normal line to a curve at a given point is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the slope of the tangent line. Once the slope of the tangent line is determined, the normal line can be calculated, providing insight into the geometric properties of the curve at the specified point.
Recommended video:
05:13Slopes of Tangent Lines
Related Practice
Textbook Question
Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is
a. perpendicular to the line y = 1 - (x/24).
_
b. parallel to the line y = √2 - 12x.
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)
Textbook Question
In Exercises 41–58, find dy/dt.
y = tan²(sin³(t))
Textbook Question
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx
Textbook Question
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
" style="" width="340">