Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The graph of the function \(f\) defined by $$ f(t)=\left\\{\begin{array}{ll} t & \text { for } 0 \leq t<1 \\ 0 & \text { for } t=1 \end{array}\right. $$ is shown in Figure Ex. 12.1.6. It does not have a high point. Furthermore, it does not have a horizontal tangent; yet it satisfies all but one of the hypotheses of Rolle's Theorem. Which hypothesis of Rolle's Theorem does it not satisfy?

Step-by-Step Solution

Answer

The function does not satisfy the continuity hypothesis on [0, 1].

Step 1: Understand Rolle's Theorem

Rolle's Theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and if \( f(a) = f(b) \), then there exists at least one number \( c \) in \((a, b)\) such that \( f'(c) = 0 \). We need to check these conditions for the function \( f(t) \).

Step 2: Check Continuity on [0, 1]

The function \( f(t) = t \) for \(0 \leq t < 1\) and becomes \(0\) at \(t = 1\). We need to check if \( f \) is continuous on \([0, 1]\). Since at \( t = 1 \), \( f(t) \) abruptly changes from \( t \) to 0, there is a discontinuity at \( t = 1 \). Thus, the function is not continuous on the whole interval \([0, 1]\).

Step 3: Verify Differentiability on (0, 1)

For \(0 < t < 1\), the function \( f(t) = t \) is differentiable with derivative \( f'(t) = 1 \). However, the differentiability at \( t = 1 \) is irrelevant because we are concerned with the open interval \((0, 1)\) where differentiability holds.

Step 4: Examine Functional Values, f(0) and f(1)

Check if \( f(0) = f(1) \). We find that \( f(0) = 0 \) but \( f(1) = 0 \), thus they are equal. Nevertheless, due to discontinuity, Rolle's Theorem is not fully applicable.

Step 5: Identify the Unmet Hypothesis

Rolle's Theorem requires continuity on the entire closed interval \([a, b]\). Since \( f(t) \) is not continuous on \([0, 1]\), the hypothesis of continuity on \([a, b]\) is not satisfied.

Key Concepts

Continuity

Continuity is a fundamental concept in calculus that concerns the smoothness of a function over an interval. For a function to be continuous on a closed interval \([a, b]\), it must not have any breaks, jumps, or holes. Simply put, you should be able to draw the graph of a continuous function on that interval without lifting your pencil.

• An important requirement for Rolle's Theorem is that the function must be continuous on the entire closed interval in question.
• In the original exercise, the function \( f(t) \) is not continuous at \( t = 1 \), as it makes an abrupt transition from \( t \) to 0, leading to a discontinuity.

Understanding where discontinuities occur helps in analyzing whether any theorem or theorem condition applies to the function at hand. Recognizing continuous functions is crucial because many properties in calculus, like differentiability and integrability, depend on it.

Differentiability

Differentiability is the property of a function that allows it to have a derivative, meaning that it has well-defined tangent lines at each point within its domain. For a function to be differentiable, it must also be continuous, but not all continuous functions are differentiable.

• In the exercise, \( f(t) = t \) is differentiable on the open interval \(0, 1\).
• Despite being continuous on \(0 < t < 1\), differentiability isn't needed at the endpoints for Rolle’s Theorem; it just needs to exist in the open interval.

Differentiability implies a smooth curve with no sharp corners or cusps within the interval analyzed. Understanding whether a function is differentiable allows mathematicians to calculate derivatives, which are essential for finding slopes and rates of change.

Piecewise Functions

Piecewise functions are defined by different expressions depending on the interval of the input value. They provide a way to describe functions that behave differently across different parts of their domains. These functions can be continuous or discontinuous, and they require careful attention to how each piece connects at their endpoints.

• The function \( f(t) \) in the exercise is a classic example of a piecewise function with different expressions on defined intervals.
• At \( t = 1 \), \( f(t) \) switches from the linear form \( t \) to a constant value \( 0 \), resulting in discontinuity.

Piecewise functions are common in practice, modeling real-world scenarios where conditions change at specific points. Understanding them helps in predicting behavior and ensuring conditions like continuity and differentiability are evaluated correctly.

Calculus

Calculus is a vast area of mathematics dealing with rates of change and accumulation. It is built upon concepts like limits, derivatives, and integrals. Rolle’s Theorem is a specific result in calculus that connects these ideas by stating conditions under which a function’s derivative must be zero at some point in an interval.

• Rolle’s Theorem requires continuity and differentiability along with \( f(a) = f(b) \) on the interval \(a, b\).
• In calculus, failure to meet these conditions means the theorem is not applicable, as seen in the exercise due to lack of continuity.

Mastering calculus concepts allows students to understand not just theoretical mathematics but also provides tools for solving practical problems involving dynamic systems.