Problem 67 - University Calculus: Early Transcendentals

Question

If the graphs of two differentiable functions \(f(x)\) and \(g(x)\) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, the graphs are identical as they have the same derivative and start at the same point.

Step 1: Understanding the Problem

We are given that two functions, \(f(x)\) and \(g(x)\), are differentiable and start at the same point, meaning \(f(a) = g(a)\). Additionally, they have the same rate of change at every point, which implies that their derivatives are equal: \(f'(x) = g'(x)\) for all \(x\) in their domain. The question asks if the graphs of these functions must be identical.

Step 2: Implication of Equal Derivatives

If two functions share the same derivative across their entire domain, it indicates they differ by only a constant. This is due to the Mean Value Theorem, which states that if \(f'(x) = g'(x)\), then \(f(x) - g(x) = C\), where \(C\) is a constant. This means \(f(x)\) and \(g(x)\) have the same rate of change but can differ by a constant shift along the y-axis.

Step 3: Considering the Starting Point

Given that \(f(x)\) and \(g(x)\) start at the same point, \(f(a) = g(a)\), this means that their difference at this specific point is zero, \(f(a) - g(a) = 0\). Since \(f(x) - g(x) = C\) and we know \(C=0\) (from the condition \(f(a) = g(a)\)), it can be concluded that \(f(x) = g(x)\) for all \(x\) in their domain.

Step 4: Conclusion

Since \(f(x)\) and \(g(x)\) have identical derivatives and start at the same point, the graphs of \(f(x)\) and \(g(x)\) must be identical. This is because any constant difference \(C\) is eliminated by the fact that \(f(x)\) and \(g(x)\) begin at the same point and thus \(C=0\).

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus that helps to connect the idea of derivatives to the average rate of change over an interval. When you have two differentiable functions that start from the same point and have identical derivatives, the Mean Value Theorem can give us deeper insights.

This theorem states that for any function "h" that is continuous on a closed interval \([a,b]\) and differentiable on the open interval \(a,b\), there is at least one point \(c\) in \(a,b\) where the derivative \(h'(c)\) is equal to the average rate of change, or slope, of the function over that interval. Mathematically, it can be represented as:

\[\frac{h(b) - h(a)}{b - a} = h'(c)\]
In simpler terms, if two functions have the same derivative throughout the entirety of their domains, such as our functions \(f(x)\) and \(g(x)\), then any difference between them must be constant across the interval, provided they meet at an initial point. This deduction further validates the conclusion that if \(f(x)\) and \(g(x)\) begin at the same place and grow at the same rate (same derivative), then their graphs cannot differ by anything more than a constant shift, which in our specific case must be zero.

Rate of Change

The rate of change of a function essentially tells us how the value of the function changes as its input changes. This concept is represented by the derivative of the function, \(f'(x)\), which is the instantaneous rate of change.

When two functions like \(f(x)\) and \(g(x)\) share the same derivative, it means they are "sloping" or changing at the same rate at each point along their respective domains.

• The derivative itself is a function that provides this rate at every point \(x\).
• If \(f'(x) = g'(x)\) for all points \(x\), both functions increase, decrease, or remain constant at equivalent rates regardless of their starting values.

Thus, knowing that the rate of change is identical is crucial in determining the nature of the functions. It tells us that both functions will maintain the same "angle" of climb or fall at any given moment, which lays down the groundwork for predicting their behavior over the entirety of their domains.

Equal Derivatives

When two functions, say \(f(x)\) and \(g(x)\), have equal derivatives \(f'(x) = g'(x)\) over a domain, an important characteristic emerges. This equality implies that the functions are essentially the same except for a potential constant vertical shift.

This concept is rooted in the properties of derivatives, as derivatives provide information about the slope or rate of change of a function. If one function is always changing at the same rate as another, it suggests that one is merely a shifted version of the other. Specifically:

• They will have identical slopes at every point, meaning any derived or integral-based analysis between them will yield consistent differences that are not dependent on "x".
• We can deduce that \(f(x) = g(x) + C\), where \(C\) is a constant that represents any vertical translation between the graphs.

Given that, if two such functions start at the same point, such as \(f(a) = g(a)\), then \(C\) must be zero because there is no vertical shift needed to align them at this initial point. This leads us to conclude that the functions are, in fact, identical over their domain.