Day 19 Pre-class assignment

Goals for today's pre-class assignment

In this pre-class assignment, you are going to learn how to:

• Numerically integrate a function
• Numerically differentiate a function
• Get a sense of how the result depends on the step size you use.

Assignment instructions

Watch the videos below and complete the assigned programming problems.

Question 1: Write a function that uses the rectangle rule to integrates $f(x) = \sin(x)$ from $x_{beg}= 0$ to $x_{end} = \pi$ by taking $N_{step}$ equal-sized steps $\Delta x = \frac{x_{beg} - x_{end}}{N_{step}}$. Allow $N_{step}$ and the beginning and ending of the range to be defined by user-set parameters. For values of $N_{step} = 10, 100$, and $1000$, how close are you to the true answer? (In other words, calculate the error as defined above.)

Note 1: $\int_{0}^{\pi} \sin(x) dx = \left. -\cos(x) \right|_0^\pi = 2$

Note 2: The "error" is defined as $\epsilon = |\frac{I - T}{T}|$, where I is the integrated answer, T is the true (i.e., analytic) answer, and the vertical bars denote that you take the absolute value.

Question 2: Write a function that calculates the derivative of $f(x) = e^{-2x}$ at several points between -3.0 and 3.0, using two points that are a distance $\Delta x$ from the point, x, where we want the value of the derivative. Calculate the difference between this value and the answer to the analytic solution, $\frac{df}{dx} = -2 e^{-2x}$, for $\Delta x$ = 0.1, 0.01 and 0.001 (in other words, calculate the error as defined above).

Hint: use np.linspace() to create a range of values of x that are regularly-spaced, create functions that correspond to $f(x)$ and $\frac{df}{dx}$, and use numpy to calculate the derivatives and the error. Note that if x is a numpy array, a function f(x) that returns a value will also be a numpy array. In other words, the function:

def f(x):
    return np.exp(-2.0*x)

will return an array of values corresponding to the function $f(x)$ defined above if given an array of x values.

Question 3: For the two programs above, approximately how much does the error go down as you change the step size $\Delta x$ by factors of 10?

// put your answer here

Assignment wrapup

Please fill out the form that appears when you run the code below. You must completely fill this out in order to receive credit for the assignment!

Congratulations, you're done!

Submit this assignment by uploading it to the course Desire2Learn web page. Go to the "Pre-class assignments" folder, find the dropbox link for Day 19, and upload it there.

See you in class!