Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 49

Chapter 3, Problem 49

Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

Verified step by step guidance

First, identify the slope of the line y = 2x + 3. The slope is 2, as it is the coefficient of x.

Since we want the tangent lines to be perpendicular to y = 2x + 3, we need the negative reciprocal of 2, which is -1/2. This will be the slope of the tangent line.

Find the derivative of the function y = x/(x - 2) to determine the slope of the tangent line at any point (x, y). Use the quotient rule: if y = u/v, then y' = (v*u' - u*v')/v^2.

Apply the quotient rule: let u = x and v = x - 2. Then, u' = 1 and v' = 1. Substitute these into the quotient rule to find the derivative y'.

Set the derivative equal to -1/2 (the slope of the perpendicular line) and solve for x. This will give you the x-coordinates of the points where the tangent lines are perpendicular. Substitute these x-values back into the original function to find the corresponding y-coordinates.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Lines

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. To find the tangent line, we typically use the derivative of the function, which gives us the slope at any point on the curve.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m1, the other line must have a slope of m2 such that m1 * m2 = -1. In this context, we need to determine the slope of the given line (y = 2x + 3) and find the negative reciprocal to identify the slope of the tangent lines we are looking for.

Finding Derivatives

The derivative of a function provides a way to calculate the slope of the tangent line at any point on the graph. For the function y = x/(x - 2), we can use the quotient rule to find its derivative. This derivative will help us identify the points where the slope of the tangent line matches the required slope for perpendicularity to the given line.

Recommended video:

06:02

The Second Derivative Test: Finding Local Extrema

Related Practice

Textbook Question

a. Find an equation for the line that is tangent to the curve y = x³ − 6x² + 5x at the origin.

[Technology Exercise] b. Graph the curve and tangent line together. The tangent line intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

[Technology Exercise] c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent line simultaneously.

views

Textbook Question

In Exercises 51 and 52, find dp/dq.

q = (5p² + 2p)⁻³/²

Textbook Question

Find by implicit differentiation.

x²y² = 1

Textbook Question

Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.

Textbook Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

views

Textbook Question

Find by implicit differentiation.

x² + xy + y² - 5x = 2