Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.2.53
Chapter 4, Problem 4.2.53

Applications


A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

Verified step by step guidance
1
Recognize that this problem can be approached using the Mean Value Theorem (MVT) for integrals, which states that if a function is continuous on a closed interval, then there exists at least one point where the instantaneous rate of change (speed, in this case) equals the average rate of change over the interval.
Calculate the average speed of the marathoner over the entire race. The average speed is given by the total distance divided by the total time. Here, the total distance is 26.2 miles and the total time is 2.2 hours. Therefore, the average speed is \( \frac{26.2}{2.2} \) mph.
Apply the Mean Value Theorem for integrals. Since the initial and final speeds are zero, and the speed function is continuous, there must be at least one point in time where the speed equals the average speed calculated in the previous step.
Since the problem asks to show that the marathoner was running at exactly 11 mph at least twice, note that the average speed calculated is approximately 11.91 mph. This implies that the speed must have been exactly 11 mph at least twice, as the speed function must cross this value to reach the average speed.
Conclude that by the Intermediate Value Theorem, which states that for any value between the minimum and maximum of a continuous function, there must be at least one point where the function takes that value, the marathoner must have been running at exactly 11 mph at least twice during the race.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean Value Theorem

The Mean Value Theorem states that for a continuous function on a closed interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over the interval. In this context, it implies that the marathoner's speed must equal the average speed of 11 mph at least once during the race.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1

Average Speed Calculation

Average speed is calculated by dividing the total distance traveled by the total time taken. For the marathoner, the average speed is 26.2 miles divided by 2.2 hours, which equals 11.909 mph. This average speed helps in applying the Mean Value Theorem to find instances where the instantaneous speed matches this average.
Recommended video:
06:37
Average Value of a Function

Continuous and Differentiable Functions

A function is continuous if there are no breaks or jumps in its graph, and differentiable if it has a derivative at every point in its domain. The marathoner's speed function is assumed to be continuous and differentiable, allowing the application of the Mean Value Theorem to find specific speeds during the race.
Recommended video:
05:34
Intro to Continuity
Related Practice
Textbook Question

26. Constructing cylinders Compare the answers to the following two construction problems.

a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

" style="" width="350">

189
views
Textbook Question

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

134
views
Textbook Question

Theory and Examples


In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.


y = x¹¹ + x³ + x − 5

206
views
Textbook Question

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).

215
views
Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.


g(θ) = θ³ᐟ⁵, −32 ≤ θ ≤ 1

240
views
Textbook Question

Motion with constant acceleration The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is s = (a/2)t² + v₀t + s₀, where v₀ and s₀ are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem

Differential equation: d²s/dt² = a

Initial conditions: ds/dt = v₀ and s = s₀ when t=0.

52
views