Problem 56 - 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

By Rolle's Theorem, there exists a point \(c\) where \(f'(c) = g'(c)\), hence the tangents are parallel or the same.

Step 1: Understanding the problem

We need to show that there is at least one point between \(a\) and \(b\) where the tangents to \(f\) and \(g\) are parallel. The conditions given are \(f(a) = g(a)\) and \(f(b) = g(b)\), which implies that both functions start and end at the same points on the interval \([a, b]\).

Step 2: Applying Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists some point \(c\in(a, b)\) where the derivative (slope of the tangent line) \(f'(c)\) equals the average rate of change over \([a, b]\). Apply this to both functions \(f\) and \(g\).

Step 3: Finding the points of parallel tangents

Since \(f(a) = g(a)\) and \(f(b) = g(b)\), the average rate of change of \(f\) over \([a, b]\) is zero, same as for \(g\). Therefore, there exist points \(c_1\) and \(c_2\) in \((a, b)\) such that \(f'(c_1) = 0\) and \(g'(c_2) = 0\). If \(f'(c_1) = g'(c_2)\) for some \(c\) between \(a\) and \(b\), the tangents are parallel at this point.

Step 4: Conclusion using Rolle's Theorem

Since \(f(a) = g(a)\) and \(f(b) = g(b)\), consider \(h(x) = f(x) - g(x)\). \(h(a) = h(b) = 0\), so by Rolle's Theorem, there exists some \(c\in(a, b)\) such that \(h'(c) = 0\), which implies \(f'(c) = g'(c)\). Hence, the tangents are parallel or the same.

Step 5: Illustration

Draw two curves on the same graph that start and end at the same points on the x-axis at \(a\) and \(b\). Show that at some point vanishing the derivative, indicating parallel tangents. Although a sketch can't be displayed here, imagine smoothly connecting these points, creating one or more places where the slopes would align.

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus that helps us understand when tangents to two curves can be parallel. The theorem states that if a continuous function is differentiable over an interval \([a, b]\) and has equal values at its endpoints, then there is at least one point \(c\) in the interval where the derivative is zero. This means the tangent to the curve at that point is horizontal.

To apply Rolle's Theorem to our problem, we consider the function \(h(x) = f(x) - g(x)\). Given that \(f(a) = g(a)\) and \(f(b) = g(b)\), it implies that \(h(a) = h(b) = 0\). According to Rolle's Theorem, there must exist a point \(c\) within the interval \([a, b]\) where \(h'(c) = 0\).

• This translates to \(f'(c) = g'(c)\),
• Therefore, the tangents of the functions \(f\) and \(g\) at this point are indeed parallel.

Understanding Rolle's Theorem can greatly enhance our ability to identify such intersections of conditions in calculus.

Differentiability

Differentiability is a core concept in calculus, describing a function with a derivative at each point in its domain. When a function is differentiable on an interval, it means we can draw a tangent line at any point within that interval, giving us a clear slope.

For the given problem, both functions \(f\) and \(g\) are differentiable on the interval \([a, b]\).

• This implies the existence of derivatives \(f'(x)\) and \(g'(x)\) at each point \(x\) within \((a, b)\).
• The differentiability ensures that the conditions required to apply the Mean Value Theorem, and therefore Rolle's Theorem, are satisfied.

Thus, differentiability ensures that the functions' behavior between \(a\) and \(b\) is smooth, providing the foundation for analysing their tangent line slopes.

Parallel Tangents

The idea of parallel tangents is visually compelling and finds its basis in calculus. For two curves, having parallel tangents at a point means they share the same slope at that point, making their tangent lines parallel with respect to the x-axis.

Using Rolle's Theorem and differentiability, we concluded that \(f'(c) = g'(c)\) for some \(c\) in \((a, b)\).

• This result indicates that the tangents to the graphs of \(f\) and \(g\) are not only parallel, but they can also coincide or embody the same line at this point.
• Graphically, this is comparable to plotting the two curves and finding that at least one tangent ties their paths around \(c\), proving they share the same direction at a certain moment between \(a\) and \(b\).

The concept of parallel tangents serves as an elegant blend of algebraic reasoning and geometric intuition, illustrating the dynamic relationships between two differentiable functions.