Problem 73
Question
A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of the day on both days.
Step-by-Step Solution
Verified Answer
There is a time \( c \) where the monk is at the same place on both days.
1Step 1: Define the Functions
Let the position of the monk on the path be a function of time. Define \( f(t) \) as the monk's position from the monastery to the mountain top on the first day and \( g(t) \) as the position on the second day, returning from the mountain top to the monastery.
2Step 2: Determine the Interval
Both functions are defined on the time interval from 7:00 AM to 7:00 PM, which is equivalent to the time interval from \( t=0 \) to \( t=12 \) hours.
3Step 3: Establish the Points
On the first day, the monk's position starts at the monastery: \( f(0) = 0 \), and ends at the top: \( f(12) = D \) where \( D \) is the distance of the path. On the return trip, the position starts at the top: \( g(0) = D \), and ends at the monastery: \( g(12) = 0 \).
4Step 4: Construct the Difference Function
Define a new function \( h(t) = f(t) - g(t) \). The function \( h(t) \) represents the difference in position between the two trips at any time \( t \).
5Step 5: Evaluate Boundary Values of Difference Function
Calculate \( h(0) = f(0) - g(0) = 0 - D = -D \) and \( h(12) = f(12) - g(12) = D - 0 = D \).
6Step 6: Apply the Intermediate Value Theorem
Since \( h(t) \) is continuous (both \( f(t) \) and \( g(t) \) are continuous functions of time) and it changes from \(-D\) to \(D\), by the Intermediate Value Theorem, there must be some \( c \) in \([0, 12]\) such that \( h(c) = 0 \).
7Step 7: Conclude the Solution
At \( t = c \), the value of \( h(c) \) implies that \( f(c) = g(c) \), meaning there exists a point on the path where the monk is at the same spot at the same time on both days.
Key Concepts
Continuous FunctionsDifference FunctionTibetan Monk ProblemPosition as a Function of Time
Continuous Functions
A continuous function is one where small changes in the input produce small changes in the output. In other words, there are no abrupt jumps or gaps. For many practical contexts, like our problem with the Tibetan monk, this represents a smooth path with no interruptions. The monk's position during his travels can be represented as a continuous function of time because he walks steadily without teleporting or backtracking suddenly.
In mathematics, this property is crucial when applying the Intermediate Value Theorem. For the monk's position, both functions—one for each day of travel—are continuous. That means at no point does the monk disappear and reappear somewhere else. His journey is a smooth experience from the base of the mountain to the top, and then back again the next day.
In mathematics, this property is crucial when applying the Intermediate Value Theorem. For the monk's position, both functions—one for each day of travel—are continuous. That means at no point does the monk disappear and reappear somewhere else. His journey is a smooth experience from the base of the mountain to the top, and then back again the next day.
Difference Function
The difference function is defined by taking the difference between two functions pointwise. In this exercise, we define a difference function as \( h(t) = f(t) - g(t) \), where \( f(t) \) and \( g(t) \) are the position functions for the monk's ascent and descent. This new function \( h(t) \) provides insight into the difference in the monk's positions on the two days at any given time.
By analyzing \( h(t) \), we can compare the positions of the monk on the two days. Initially, at time \( t = 0 \), \( h(t) \) takes a value of \(-D\) (the negative full distance, indicating the monk starting from different points: bottom vs. top). At time \( t = 12 \) (at the end of the day), \( h(t) \) reaches \( D \), as the monk finishes at opposite points again. This change from negative to positive indicates that a point where \( h(t) \) equals zero must exist.
By analyzing \( h(t) \), we can compare the positions of the monk on the two days. Initially, at time \( t = 0 \), \( h(t) \) takes a value of \(-D\) (the negative full distance, indicating the monk starting from different points: bottom vs. top). At time \( t = 12 \) (at the end of the day), \( h(t) \) reaches \( D \), as the monk finishes at opposite points again. This change from negative to positive indicates that a point where \( h(t) \) equals zero must exist.
Tibetan Monk Problem
The Tibetan monk problem is a classic application of mathematical theorems, often used to illustrate the Intermediate Value Theorem. It involves two journeys of the monk on consecutive days, moving along the same path. The key question is about whether there exists a point in time when the monk occupies the same position on both days.
This scenario leverages the fact that both journeys (each a continuous function as earlier defined) truly intersect at some point. The search for this intersection boils down to understanding when the difference function \( h(t) \) equals zero, meaning the positions \( f(t) \) and \( g(t) \) coincide. It's a readily accessible demonstration that continuous functions allow us to make certain precise predictions about real-world phenomena based on mathematical principles.
This scenario leverages the fact that both journeys (each a continuous function as earlier defined) truly intersect at some point. The search for this intersection boils down to understanding when the difference function \( h(t) \) equals zero, meaning the positions \( f(t) \) and \( g(t) \) coincide. It's a readily accessible demonstration that continuous functions allow us to make certain precise predictions about real-world phenomena based on mathematical principles.
Position as a Function of Time
In mathematics, describing the position of an object (like our monk) as a function of time helps us model its movement. In this exercise, we view the monk's position during his journey to the mountain and his return as continuous functions of time, \( f(t) \) and \( g(t) \) respectively.
This concept reflects how we can track and predict where the monk is at any given moment along the path. Knowing that he moves steadily, we can confidently apply continuous mathematical functions to describe his journey. The Intermediate Value Theorem then aids us in determining that there is at least one moment when his position from both days matches exactly. This method of modeling positions as a function of time not only simplifies analysis but also provides a powerful tool for understanding dynamic systems.
This concept reflects how we can track and predict where the monk is at any given moment along the path. Knowing that he moves steadily, we can confidently apply continuous mathematical functions to describe his journey. The Intermediate Value Theorem then aids us in determining that there is at least one moment when his position from both days matches exactly. This method of modeling positions as a function of time not only simplifies analysis but also provides a powerful tool for understanding dynamic systems.
Other exercises in this chapter
Problem 71
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(a) Show that the absolute value function \( F(x) = | x | \) is continuous everywhere. (b) Prove that if \( f \) is a continuous function on an interval, then s
View solution Problem 78
Prove, using Definition 9, that \( \displaystyle \lim_{x \to \infty} x^3 = \infty \).
View solution Problem 80
Formulate a precise definition of $$ \lim_{x \to -\infty} f(x) = -\infty $$ Then use your definition to prove that $$ \lim_{x \to -\infty} (1 + x^3) = -\infty $
View solution