Problem 30

Question

Suppose that you drive from St. Paul to Duluth and you average $50 \mathrm{mph}$. Explain why there must be a time during your trip at which your speed is exactly $50 \mathrm{mph}$.

Step-by-Step Solution

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Answer

By the Mean Value Theorem, your speed must be exactly 50 mph at least once.

1Step 1: Understanding the Mean Value Theorem (MVT)

The Mean Value Theorem states that if a function $f(x)$ is continuous over the interval $[a, b]$ and differentiable over $(a, b)$, then there exists at least one point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. In this context, consider your position as a function of time.

2Step 2: Defining the function

Define $s(t)$ as the position function, representing your distance from St. Paul at time $t$. Thus, the average speed over the entire trip is given by $\frac{s(b) - s(a)}{b - a}$, where $a$ and $b$ are the start and end times, respectively.

3Step 3: Applying the Mean Value Theorem

Given that you average $50 \mathrm{mph}$ over the trip, the MVT implies there is at least one time during your journey when $s'(c) = 50$, where $s'(t)$ represents the instantaneous speed at time $t$.

4Step 4: Conclusion

Therefore, according to the MVT, there must be at least one moment during your drive when your instantaneous speed is exactly equal to the average speed, which is $50 \mathrm{mph}$. This ensures that even if your speed varies throughout the trip, there is a point where it matches the average speed.

Key Concepts

Instantaneous SpeedAverage SpeedDifferentiable FunctionContinuous Function

Instantaneous Speed

Instantaneous speed is the speed of an object at a specific moment in time. Imagine you're taking a drive and glancing at your speedometer. The number it displays is your instantaneous speed.

It tells you exactly how fast you're going right then.
Not to be confused with average speed, which looks at the whole journey's speed.

To determine instantaneous speed, we consider the derivative of the position function, also known as velocity. In calculus, this is often denoted as $s'(t)$. For example, if you average a speed of 50 mph from St. Paul to Duluth, by applying the Mean Value Theorem, we learn that there was at least one moment during your trip when your instantaneous speed was exactly 50 mph. This matches the average across the entirety of your journey at that moment.

Average Speed

Average speed is like your journey's overall score. It measures how fast you went on average from start to finish, regardless of speed changes along the way.

Calculated as the total distance traveled divided by the total time taken.
For example, if a trip takes 2 hours and you covered 100 miles: $\frac{100}{2}=50$ mph.

This makes it easy to see the general pace of your trip. Even if your speed was up and down throughout the drive from St. Paul to Duluth, averaging 50 mph overall means combining all segments together equals driving a steady 50 mph the whole way. Thanks to the Mean Value Theorem, at least once, you hit this exact speed during your drive.

Differentiable Function

A differentiable function is a type of function that has a derivative at every point in its domain. This is crucial for applying the Mean Value Theorem as it ties back to determining instantaneous speed through differentiation.

To be differentiable, the graph of a function should be smooth and not have any sharp corners.
This ensures that the instantaneous rate of change (or speed) can be found at any given time.

In the context of your drive, the position function $s(t)$, which represents where you are at any time $t$, must be differentiable across the whole trip from St. Paul to Duluth so we can rely on principles like the Mean Value Theorem to make inferences about your speed at any moment.

Continuous Function

A function is continuous if it has no breaks or gaps for its entire length.

To visualize, imagine tracing a graph with your finger without lifting it from the paper. No jumps or holes mean it's continuous.
For your trip's position function $s(t)$, continuity means your journey is being tracked smoothly from start to finish.

This smoothness is necessary for applying the Mean Value Theorem, which helps us deduce there will be a point where your instantaneous speed equals the average speed. In simple terms, having no interruptions in your path ensures we can rely on a solid understanding of how speeds change over time without missing any crucial bits.

Problem 30

Other exercises in this chapter

Problem 29

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$

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Problem 30

Use Leibniz's rule to find $\frac{d y}{d x}$. $$ y=\int_{x^{2}}^{3} \frac{1}{1+t} d t, x>0 $$

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Problem 30

Use the algebraic rules for sums to evaluate each sum. Recall that $$\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$$ and $$\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$$ $$

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Problem 31

Use Leibniz's rule to find $\frac{d y}{d x}$. $$ y=\int_{x^{2}}^{1} \sec t d t,-1

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