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Chapter 2: Problem 62

Show that the graphs of the two equations \(y=x\) and \(y=1 / x\) have tangent lines that are perpendicular to each other at their point of intersection.

Short Answer

Expert verified
The steps analyzing the graphs of the two equations \(y=x\) and \(y=1 / x\) show that they indeed do have tangent lines that are perpendicular to each other at their points of intersection which are (1,1) and (-1,-1).

Step by step solution

01

Find Point of Intersection

Set the two functions equal to each other to find the point where they intersect: \(x=1 / x\). When you solve this equation for \(x\), you get \(x= \pm 1\), hence the point of intersection is at \(x=1\) and \(x=-1\). Plugging these values into any of the given equations (since at the intersection points they are equal), at \(x=1\), \(y=1\) and at \(x=-1\), \(y=-1\). Hence the points of intersections are \((1,1)\) and \((-1,-1)\).
02

Find Derivatives of the Given Functions

The derivative of \(y=x\) is \(y' = 1\). The derivative of \(y=1 / x\) is \(y' = -1 / x^2\)
03

Evaluate the Derivatives at the Points of Intersection

At the point \((1,1)\), the derivative of \(y=x\) is \(1\) and the derivative of \(y=1 / x\) is \(-1\). At the point \((-1,-1)\), the derivative of \(y=x\) is \(1\) and the derivative of \(y=1 / x\) is \(-1\) as well.
04

Check If Derivatives (slopes) Are Negative Reciprocals

Each time, the derivative of the first function at the intersection point is the negative reciprocal of the derivative of the second function at that point. So the tangent lines at the points of intersection are indeed perpendicular to each other.

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Most popular questions from this chapter

Find the second derivative of the function. \(g(x)=\sqrt{x}+e^{x} \ln x\)

In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(s(t)=\frac{(4-2 t) \sqrt{1+t}}{3},\left(0, \frac{4}{3}\right)\)

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1,2] . (c) Find the instantaneous velocities when \(t=1\) and \(t=2\). (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact.

In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(f(x)=\sqrt{x}(2-x)^{2}, \quad(4,8)\)

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).
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