Problem 80 - Calculus of a Single Variable

Question

(a) Let \(f(x)=x^{2}\) and \(g(x)=-x^{3}+x^{2}+3 x+2 .\) Then \(f(-1)=g(-1)\) and \(f(2)=g(2)\) . Show that there is at least one value \(c\) in the interval \((-1,2)\) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c)) .\) Identify \(c .\) (b) Let \(f\) and \(g\) be differentiable functions on \([a, b]\) where \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one value \(c\) in the interval \((a, b)\) where the tangent line to \(f\) at \((c, f(c))\) is parallel to the tangent line to \(g\) at \((c, g(c))\).

Step-by-Step Solution

Answer

For part (a), there exists a value \(c = 1\), in the interval (-1,2), where the tangent lines to the functions \(f\) and \(g\) are parallel. For part (b), according to Rolle's theorem, there must be some \(c\) in the interval (a, b) where the tangent lines to \(f\) and \(g\) are parallel.

Step 1: Find the Derivatives of the Functions

First, we find the derivatives of given functions \(f(x)\) and \(g(x)\). If \(f(x) = x^2\), then \(f'(x) = 2x\). If \(g(x) = -x^3 + x^2 + 3x + 2\), then \(g'(x) = -3x^2 + 2x + 3.

Step 2: Apply Rolle's Theorem for Part (a)

We know that \(f(-1) = g(-1)\) and \(f(2) = g(2)\). Therefore, we consider the function \(h(x) = f(x) - g(x)\) which has \(h(-1) = h(2)\). Because \(h(x)\) is a differentiable function, according to the Rolle's theorem, there exists at least a \(c\) in (-1, 2) so that \(h'(c) = 0\). Since \(h'(x) = f'(x) - g'(x)\), it follows that there exists a \(c\) where \(f'(c) = g'(c)\), which means the tangent lines to \(f\) and \(g\) at the points (c, f(c)) and (c, g(c)) respectively are parallel.

Step 3: Find the Value of 'c' for Part (a)

To find the value of \(c\), set the derivatives equal to each other and solve for \(x\):2x = -3x^2 + 2x + 3.Solving this equation results in \(c = 1\), which is indeed in the interval (-1,2). Therefore, the tangent lines to \(f\) and \(g\) at the points (1, f(1)) and (1, g(1)) respectively are parallel.

Step 4: Apply the Concept to Differentiable Functions

Part (b) is fundamentally a conceptual question. It doesn't require computation as much as comprehension of the Rolle's theorem. The statement from part (b) fits the hypothesis of Rolle's theorem exactly. That is, \(f\) and \(g\) are both differentiable on the interval \([a, b]\), with \(f(a) = g(a)\) and \(f(b) = g(b)\). Therefore, there must be some \(c\) in (a, b), at which the derivative of \(f\) equals to the derivative of \(g\), meaning the tangent lines to \(f\) and \(g\) at the points (c, f(c)) and (c, g(c)) respectively are parallel.

Key Concepts

Derivatives of Functions

Understanding derivatives is crucial when analyzing the behavior of functions. A derivative represents how a function changes as its input changes—essentially, it shows the rate of change or slope of the function at any given point.

In the textbook exercise, we calculated the derivatives of the given functions, represented by the notations f'(x) for function f and g'(x) for function g. For a quadratic function like f(x) = x^2, the derivative is a linear function, f'(x) = 2x, indicating the slope of a tangent line to f at any point x. Similarly, g'(x) tells us the slope of the tangent line to g at any x.

Why are these derivatives important? Well, they are the key to predicting the behavior of functions and finding points where functions have certain characteristics, such as where their tangent lines are parallel, as featured in the given problem.

Tangent Lines

A tangent line to a curve at a given point is a straight line that just 'touches' the curve at that point. This line is a local linear approximation of the curve near that point, and its slope is equal to the value of the derivative of the curve's function at that point.

In the context of our problem, we were interested in finding where the tangent lines to the functions f(x) and g(x) are parallel. Two lines are parallel if they have the same slope, so we looked for a point c where f'(c) = g'(c). This equality of derivatives means that the tangent lines to the functions f and g at c are parallel. It's a powerful concept that connects the abstract idea of derivatives with the geometric understanding of curves and lines.

Differentiable Functions

A function is differentiable at a point if it has a derivative there; in other words, if the function's graph has a definite tangent line at that point. For a function to be differentiable on an interval, it must have a derivative at every point in that interval; the graph of the function must be smooth, without any breaks, sharp turns, or vertical tangents.

In the exercise we examined, Rolle's Theorem was employed to show that for two differentiable functions f and g that have the same value at the endpoints of an interval, there must be some point within that interval where their derivatives—and thus their tangent lines—are equal. This principle relies critically on the assumption of differentiability, which ensures continuous and smooth behavior within the interval, allowing us to apply Rolle's Theorem with confidence.