### Objective

The objective of this activity is to determine the relationship between Euler's Method and Riemann Sums approximations for scenarios involving accumlated change.

### Euler's Method

A boat departs from a dock on an island, heading towards open water. The boat speeds up at first, but over time slows down. After about 4 hours, the boat begins to quickly return to the dock but comes to an abrupt stop when it crashes into some rocks.

The rate at which the boat travels from the dock over the first 4.5 hours, in miles per hour, is

$\frac{dB}{dt}=18t\sin(0.79t)$.

Use Euler's Method with $\Delta t=0.5$ and $\frac{dB}{dt}=18t\sin(0.79t)$ to estimate how far the boat is from the dock.

Euler's method can be completed by hand or using technology. A script can be implemented in Python app.  After 4.5 hours, the boat is 89.0668 miles away from the dock.

### Interpreting the Area Under a Curve

The graph of $\frac{dB}{dt}$ is shown below. 1. Identify the key features of the graph of $\frac{dB}{dt}$. What do these features represent in the context of this scenario?
2. Location Description Context
t = 0 to around t = 2.5 The graph is positive and increasing The boat is moving away from the dock and accelerating.
Around t = 2.5 to around t = 4 The graph is positive but decreasing The boat is still moving away from the dock but is slowing down.
Around t = 4 to t = 4.5 The graph is negative and decreasing The boat is heading back towards the dock and speeding up.
t = 4.5 and on The graph has a horizontal line of y = 0 The boat is stopped. It hit the rocks.
3. On the graph, sketch left-side rectangles with $\Delta t=0.5$. Consider the units of the rectangles. What does the area under the curve represent in the context of this problem?
4. Left-side rectangles can be drawn by hand or using technology. A script can be implemented in Python app.  The width of each rectangle is $\Delta t$. The height of each rectangle is $\frac{\Delta B}{\Delta t}$. Thus the units for the area are $\frac{\Delta B}{\Delta t}*\Delta t = \Delta B$. In other words, the area of the rectangles represent the change in the boat's distance from the dock.

5. Determine the area of each rectangle and sum the areas to approximate the area under the curve of $\frac{dB}{dt}$ on the interval [0,4.5]. Compare your answer to your Euler's Method work.
6. Rectangle Width Height Area
1 0.5 0 0
2 0.5 3.463 1.732
3 0.5 12.786 6.393
4 0.5 25.015 12.508
5 0.5 35.998 18
6 0.5 41.374 20.687
7 0.5 37.653 18.827
8 0.5 23.169 11.585
9 0.5 -1.325 -0.663 The change in the boat's distance from the dock is 89.067 miles. That is, the boat is 89.067 miles away from the dock. This is the same value obtained from Euler's Method with $\Delta t=0.5$.

7. What was the total distance traveled by the boat over the time interval $0 \leq t \leq 4.5$?
8. To determine the total distance traveled, we consider the absolute value of the height of each rectangle. The boat traveled a total distance of 90.392 miles before hitting the rocks.

### Using the Particular Solution

From the same dock on the island, a jet ski heads towards open waters. Initially, it travels at a rate of 10 miles per hour. However, the jet ski malfunctions and continues to accelerate. Thirty minutes later, it is traveling at a rate of 60 miles per hour and needs to be rescued by the Coast Guard. The Coast Guard is stationed on the mainland 20 miles from the island.

The speed of the jet ski increases linearly and can be modeled by the differential equation

$\frac{dJ}{dt}=100t+10$ 1. Use left-side rectangles with $\Delta t=0.1$ hours to estimate how far the jet ski is from the dock after 30 minutes.
2. Left-side rectangles can be drawn by hand or using technology. A script can be implemented in Python app.  The jet ski is approximately 15 miles away from the dock after 30 minutes.

3. Let J(t) represent the distance that the jet ski is from the mainland. Note that at t = 0, J(t) = 20. Solve the differential equation $\frac{dJ}{dt}=100t+10$ by inspection and determine the particular solution appropriate to the given conditions.
4. The general solutions for $\frac{dJ}{dt}=100t+10$ is $J(t)=50t^2+10t+c$. Using the initial conditions (0,20)

$20=50(0)^2+10(0)+c$

$20=c$

We have the particular solution $J(t)=50t^2+10t+20$.

5. Use the equation you found to evaluate J(0.5) and J(0). Then calculate J(0.5) - J(0). What does this value represent in the context of this scenario?
6. $J(0.5)=37.5$

$J(0)=20$

$J(0.5)-J(0)=37.5-20=17.5$

J(0) represents the distance that the jet ski is from the mainland at time 0. This is when the jet ski is at the dock. J(0.5) represents the distance that the jet ski is from the mainland at time 0.5. The difference between these two values represents how far the jet ski has traveled from the dock over the time interval.

7. Compare your answers from questions 1 and 3. How are they related?
8. The answers are similar. In question 1, we are estimated the change in distance using left-side rectangles. However, because the graph of the differential equation is increasing, the left-side rectangles are an underestimate. In question 3, we have the solution function. We can find the actual values of J(t) and the difference between two values. Thus question 3 is giving the actual distance that the jet ski is from the dock at the end of the time interval.