7.4.1Follow those slopes!

EULER’S METHOD

Nounie is troubled. She is trying to find the $y$-value when $x = 3$, given the differential equation $\frac { d y } { d x }= -0.1x + 0.2y$ with the initial condition $(0, 1)$. After setting up the integral, she realizes she cannot separate the variables!

1. Though Nounie does not know the particular solution, she does have the ability to write the equation of the tangent line at $x = 0$. Nounie decides to use that tangent line to find an approximate $y$-value at $x = 3$. What is this value?

2. After looking at Nounie’s answer, her best friend, Kristen, comes up with a plan to find an even better approximation. Kristen’s method involves many steps. Instead of approximating a $y$-value at $x = 3$, Kristen finds an approximate $y$-value at $x = 1$. Next, she writes the equation of a new tangent line at this point, and uses it to approximate the $y$-value at $x = 2$. She proceeds with this method until she reaches $x = 3$. Use Kristen’s method to approximate the $y$-value at $x = 3$.   

The way Kristen approximated a particular solution of a differential equation in problem 7-172 is called Euler’s Method. This method can be thought of as a “chain of tangent lines” that determines a series of points distributed at regular intervals of $x$. Kristen used the graph at right to organize her work in problem 7-172. In your own words, explain how Kristen found each coordinate point on her solution graph.

Use Euler’s Method to approximate the $y$-value when $x = 4$ for $\frac { d y } { d x }= − 0.1x + 0.2y$, with an initial point of $(0, 4)$ and $Δx = 1$. Draw a total of $4$ segments. Compare your results to the solution curve found in problem

APPROXIMATE VS. EXACT SOLUTIONS

1. Using $\frac { d y } { d x }= −5x$ with $Δx = 0.1$ and an initial point $(1, 4)$, calculate the first four approximate points or the solution curve. Draw in the segments on a graph.   

2. Write the actual solution equation using the initial condition. Sketch the function on the same set of axes as in part (a).

3. Describe the relationship between the graph of the actual solution and the approximate solution as $x$ increases. How can you get a better approximation of the graph of the solution curve?

Given a differential equation, describe a general method to approximate the point on the solution curve $(x_n, y_n)$. Let $Δx = h$ and let $m_{n-1}$ be the slope at the point$(x_{n-1}, y_{n-1})$.

Use Euler’s Method starting at $(0, 1)$ with $Δx = 0.2$ to estimate $y(1)$ given that $y^\prime = y$. Without determining the function, determine if your estimate for $y(1)$ is an overestimate or an underestimate. Use $y^{\prime\prime}$ to justify your answer.

Suppose $\frac { d y } { d x }= 2x + y - 1$ and $y = h(x)$ is a solution to the differential equation given $h(0) = p$.  If $h(1) ≈ 2$, determine the value of $p$ using Euler’s Method and $Δx = 1$. 

In general, when will Euler’s Method produce an overestimate and when will it produce an underestimate of an actual $y$-value? When will it be exact? Use the second derivative and concavity to justify your answer.

Without actually using Euler’s Method, determine if it will produce an underestimate or an overestimate to the solution of $\frac { d y } { d x }= 3x^2$ based on the second derivative at $x = 3$, $x = 0$, and $x = -3$. Homework Help ✎

Newton High is located at the intersection of a north-south road and an east-west road. The calculus students are looking out the class window in disbelief at two students, Shant and Narek, who are rounding the corner on pogo sticks! The students see Shant traveling toward the school from the east. Narek is traveling north away from the school. Shant is traveling at $10$ ft/min at the moment he is $30$ ft from the school. At the same time, Narek is $40$ ft from the school moving at $8$ ft/min. Determine whether the distance between the two students is increasing or decreasing, and at what rate. 7-181 HW eTool. Homework Help ✎

Determine if each of the following integrals converges or diverges. If the integral converges, state the value. Homework Help ✎

Etube is tending to the orange trees on his lot. Unfortunately, the trees are becoming infected and dying. The rate at which the trees are being infected per month is given by $3\sqrt{N}$ where $N$ is the number of living trees. Homework Help ✎

1. If at $t = 0$ there are $3000$ living trees, set up the appropriate differential equation to represent the number of living trees $N$ as a function of $t$, where $t$ is measured in months.

2. Solve the initial value problem to determine the number of living trees after $t$ months.

3. How many trees will be infected at $t = 5$ months?

Evaluate. Homework Help ✎

1. $\frac { d } { d x } \int ( 3 x ^ { 2 } + \operatorname { sin } ( x ^ { 2 } ) ) d x$

2. $\int \frac { d } { d x } ( 3 x ^ { 2 } + \operatorname { sin } ( x ^ { 2 } ) ) d x$

3. $\frac { d } { d x } \int _ { - 1 } ^ { \operatorname { cos } ( x ) } ( 3 x ^ { 2 } + \operatorname { sin } ( x ^ { 2 } ) ) d x$

4. $\frac { d } { d x } \int _ { \operatorname { ln } ( x ) } ^ { \operatorname { cos } ( x ) } ( 3 x ^ { 2 } + \operatorname { sin } ( x ^ { 2 } ) ) d x$

No calculator! Suppose $\frac { d y } { d x } = \frac { 3 x ^ { 2 } } { e ^ { y } }$. Homework Help ✎

1. Write the particular solution of this differential equation containing the point $(0, 3)$.

2. If you have not already done so, solve your equation for $y$. Confirm that your solution is correct by substituting into the differential equation.

3. State the domain and range of your equation.

Multiple Choice: The area of the region bounded above by the curve $y = \arctan(x)$ and below by the curve $y = x^2 + 3x$ is approximately: Homework Help ✎