︠62de3e44-46c6-48be-ae2d-dd675a97d372︠ # Iterate Euler's method for 1000 steps (time 0 to 100 in steps of 0.1) # Use the change equation 0.2*x*(1-x/100) # Store N values in a list called "N" # Initial value is 10 counts = srange(0, 1000) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [10] # Define initial value for i in counts: # x_new = x_old + changevec(x_old)*delta_t # N[-1] Pulls out the last entry from the list N. (The last entry = N_current) # changevec(N[-1]) takes the last entry of the list N (N_current) as an input into the changevec equation. This outputs the change vector # Finally, we multiply the change vector by 0.1 and add to x_current. N_new = N[-1] + changevec(N[-1])*0.1 N.append(N_new) # The final line takes the newly calculated N value and appends it (glues it on) to the end of the list # So, e.g. if N was 4 elements long when you stared the loop [?, ?, ?, ?], it's now 5 elements long: [?, ?, ?, ?, N_new] # Next time we start the loop, N_new (this step's output) will become N_current (next step's input) ︡9bd5dede-4950-49ba-834a-2d1801896a7b︡{"done":true}︡ ︠cca4f510-5633-4e9a-b3e2-e70a14e2e9c0︠ N # Show final list of N values ︡51f32787-76ba-401a-a51f-879fd59c1b11︡{"stdout":"[10, 10.1800000000000, 10.3628735200000, 10.5486531608817, 10.7373714073976, 10.9290606065975, 11.1237529455809, 11.3214804285737, 11.5222748533262, 11.7261677868336, 11.9331905403773, 12.1433741438903, 12.3567493196484, 12.5733464552916, 12.7931955761805, 13.0163263170940, 13.2427678932773, 13.4725490708478, 13.7056981365714, 13.9422428670207, 14.1822104971285, 14.4256276881541, 14.6725204950778, 14.9229143334436, 15.1768339456718, 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Then turn them into functions once you know they work. ## Turn this script into a function by "wrapping" it in a def statement and a return statement. ## Inputs: none (this tells us what we need to put in the parentheses in the def statement) ## Outputs: N list (this tells us what we need to put in the return statement) def eulers(): counts = srange(0, 1000) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [] # Define initial value for i in counts: N_new = N[-1] + changevec(N[-1])*0.1 N.append(N_new) return N ︡1859d92f-6eb8-4e79-8df4-62aa5ff49803︡{"done":true}︡ ︠9e0f039f-7c72-4ec6-80d3-60d20434556d︠ eulers() ︡eed23bc0-2e84-4fa9-afa2-f587750df673︡ ︠9f250605-920c-4a4e-88b9-35163323b0e6s︠ ## Add initial condition variable and step size as an input def eulers(initial, stepsize): counts = srange(0, 100/stepsize) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [initial] # Define initial value for i in counts: N_new = N[-1] + changevec(N[-1])*stepsize N.append(N_new) return N ︡fe132170-e870-40f4-a4ff-98feccb9cd86︡{"done":true}︡ ︠144838a8-646f-45b8-a9e9-bc9f62a841e6︠ eulers(initial = 10, stepsize = .2) ︡c11e3f8d-9795-4cfe-8593-b00bb85e5529︡ ︠14cb4a27-b3c3-4ada-b6dc-3f84974bd11c︠ # Hints for # 15 # Replace x limits with a variable! plot(16*x^2, (x, -5, 5))+point([1,16], color = "red", size = 50) ︡75aa7cb3-c5fc-4a46-80f6-5e3127546ece︡{"file":{"filename":"/home/user/.sage/temp/project-16c4fcef-446d-437c-8ffd-4671a6e90c3b/192/tmp_bNk7Y7.svg","show":true,"text":null,"uuid":"57845e8c-c346-447e-973a-7d866a4fb598"},"once":false}︡{"done":true}︡ ︠6d8ed38d-2746-47d0-97aa-ac8b1d8f3ecas︠ ## Hints for #16 # 16 ## Eventually, you will have to define an interactive function that inputs the x2 value of interest ## For now, pick a single x_2 value and get your plot working for that value only ## Then, once your code works for one x_2 value, you can get the rest working. ## ## SUGGESTED STEPS: ## 1. Define the function you want to plot, f(x)=16*x^2, in your code ## 2. Plot the function ## 3. The point you're focused on (x_1, y_1) is [1,16]. Add that point to the plot and color it red. ## 4. Next, you need to add the secant line to the plot. Let's think about how to do this. ## BEFORE YOU WRITE ANY MORE CODE, figure out the equation for the secant line and write it in a comment. ## --> What is the slope of the secant line (hint - slope = average rate of change from x1 to x2) ## --> Once you are able to calculate the slope, you can write down the secant line equation in point-slope form ## 5. Above your plot code, add a line of code that calculates the slope, given your current value of x2. Store this to a variable called "slope" or "mm" or something like that. ## 6. Use your calculated slope to add the secant line to your existing plot. Choose a color other than blue. Look at your comments above if you forget the equation. ## 7. Add the point [x2, f(x2)] to your plot. Choose the same color as your secant line. ## 8. FINALLY: Now that your script works, turn it into a function that takes "x2" as an input (aka, turn x2 into a variable, and add that variable name as an input into the function. ## 9. Turn this function into an interact ︡f0ca6498-de08-47d8-8dec-6143d843f594︡{"done":true}︡