Problem 60 - Thomas Calculus

Question

Parallel tangents Assume that \(f\) and \(g\) are differentiable on \([a, b]\) and that \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one point between \(a\) and \(b\) where the tangents to the graphs of \(f\) and \(g\) are parallel or the same line. Illustrate with a sketch.

Step-by-Step Solution

Answer

There is at least one point \( c \) in \((a, b)\) where \( f'(c) = g'(c) \).

Step 1: Understanding the Given

We are given two differentiable functions \( f \) and \( g \) on the interval \([a, b]\) such that \( f(a) = g(a) \) and \( f(b) = g(b) \). We need to show that there is at least one point within this interval where their tangents are parallel or identical.

Step 2: Define the Difference Function

Define a new function \( h(x) = f(x) - g(x) \). The function \( h(x) \) is differentiable on \([a, b]\) because both \( f \) and \( g \) are differentiable.

Step 3: Evaluate the Endpoints of the Difference Function

Since \( f(a) = g(a) \) and \( f(b) = g(b) \), this implies \( h(a) = f(a) - g(a) = 0 \) and \( h(b) = f(b) - g(b) = 0 \).

Step 4: Apply Rolle’s Theorem

According to Rolle's Theorem, if a function \( h(x) \) is continuous on \([a, b]\), differentiable on \((a, b)\), and \( h(a) = h(b) \), then there exists at least one \( c \) in \( (a, b) \) such that \( h'(c) = 0 \).

Step 5: Interpret the Derivative Condition

The derivative \( h'(x) = f'(x) - g'(x) \). From Rolle’s Theorem, we have \( h'(c) = 0 \), which means \( f'(c) - g'(c) = 0 \). In other words, \( f'(c) = g'(c) \). This means at point \( c \), the tangents to \( f \) and \( g \) are parallel or the same line.

Step 6: Conclusion and Sketch Suggestion

Our conclusion from the above steps is that there exists a point \( c \) in the interval \( (a, b) \) where the tangent lines to \( f \) and \( g \) are parallel. This can be illustrated by drawing the graphs of any two functions that meet the given conditions and observing the parallel tangents at some point between \( a \) and \( b \).

Key Concepts

Differentiable functions

Differentiable functions are an essential concept in calculus. A function is said to be differentiable on an interval if it has a derivative at each point within that interval. This means that there is a tangent to the curve at each point and the slope of this tangent is the derivative.Differentiability ensures smoothness and continuity of the function on its defined interval. For the given exercise, both functions, \( f(x) \) and \( g(x) \), are differentiable on the interval \([a, b]\).
This implies neither function has any sharp corners or discontinuities within this range. The differentiability of these functions is a core requirement for applying important theorems, such as Rolle's Theorem, which we will explore further.

Parallel tangents

Parallel tangents occur when two different functions have the same slope at corresponding points. In this exercise, our task is to show that the tangents to the functions \( f(x) \) and \( g(x) \) are parallel somewhere within the interval \([a, b]\). Two lines are parallel if their slopes are equal. Thus, if \( f'(c) = g'(c) \), where \( c \) is some point between \( a \) and \( b \), the tangents to \( f \) and \( g \) at \( c \) are parallel.This can happen due to the conditions \( f(a) = g(a) \) and \( f(b) = g(b) \), meaning at some point in between, their rate of change must coincide. Rolle’s Theorem provides the mathematical basis for finding such a point, ensuring that this situation arises under certain conditions.

Difference function

To determine where the tangents to \( f(x) \) and \( g(x) \) are parallel, we first introduce the difference function \( h(x) = f(x) - g(x) \). This function is essential as it captures the differences between \( f \) and \( g \) over the interval \([a, b]\). We know \( h(a) = 0 \) and \( h(b) = 0 \) because \( f(a) = g(a) \) and \( f(b) = g(b) \). Hence, \( h(x) \) satisfies the conditions of Rolle's Theorem, which means there is at least one point \( c \) where \( h'(c) = 0 \).The derivative \( h'(x) = f'(x) - g'(x) \) then gives us \( f'(c) = g'(c) \) at this point, determining that the tangents to both \( f \) and \( g \) are indeed parallel at \( c \). This difference function method provides a clear mathematical framework to prove the existence of such a point.

Endpoint evaluation

Evaluating the endpoints is crucial in applying Rolle's Theorem to our problem. The conditions specify that \( f(a) = g(a) \) and \( f(b) = g(b) \). These endpoint conditions ensure that the difference function \( h(x) = f(x) - g(x) \) equals zero at the boundaries, i.e., \( h(a) = 0 \) and \( h(b) = 0 \). By establishing these equalities, we set up the perfect scenario to apply Rolle's Theorem, which requires that the function be continuous and differentiable while also having equal values at the endpoints.As a result, the theorem guarantees that there is at least one point \( c \) within \((a, b)\) where \( h'(c) = 0 \). This endpoint evaluation is a key step in showing that the conditions for Rolle's Theorem are met, leading us directly to the conclusion about parallel tangents.