University:
Whitman College
Course:
MATH-225 | Calculus III
Academic year:

2022
Views:

322

Pages:

2
Author:

Jaslyn Pineda

Calculus Lab, Week 3 Our goal this week is to answer some mathematical questions using Maple, then write up our solutions in LaTeX. For the lab, write a LaTeX report with three sections (one section for each question below). Before next week’s lab, upload the tex and PDF files to a new CLEO folder for the week. You do not have to turn in the Maple worksheet. There’s a template LaTeX file you can use on the class website to get started. 1 Exercise: Riemann Sums In this exercise, the goal is to get into the Maple help and figure out how to get the Riemann sum for a given function (and its plot). Your “answer” to this section should, in a narrative form: • Discuss the relationship between the definite integral and the limit of the Riemann sum. Here’s an example of some LaTeX- Change it to suit your answer: \lim_{n\rightarrow \infty} \sum_{i=1}^n f\left( \frac{...}{...}\right)... You might look in a Calculus textbook to find the notation used for the Riemann sum using right endpoints. • Look up the help file for the Maple command: RiemannSum. Use the example to get a graph and estimate for the integral of the following function on the following interval: f (x) = x + sin(x) − π ≤ x ≤ 2π You should use 25 rectangles, right endpoints for the heights. Color in the rectangles using a nice pink color and set the transparency to 0.5 (you’ll see this in the help file). • In your solution, insert the figure obtained from Maple in getting the Riemann sum (you can keep the text that is output as well). 2 Exercise: Find the Pattern In this exercise, we want to see if there is a pattern to the integral: Z 1 dx (x + p)(x + q) where p, q are integers. Try a few first to see if you can find a pattern- For example, Z Z Z 1 1 1 dx dx dx (x − 3)(x + 2) (x + 2)(x + 6) (x − 3)2 You might want to look at two cases- One where p 6= q, and one where p = q. In your LaTeX file, give the problem statement and your solution write up. 1 3 Exercise: More on Plotting In Maple, plot the function f given below, together with its derivative and antiderivative. Define your antiderivative as: Z t F (t) = f (x) dx. −1 −t2 0 Plot f (t) = e , with f (t) and F (t) on the same axis, −2 ≤ x ≤ 4. Label the three functions using a legend. Label the x−axis as “Time”, and include the end result in a figure in your LaTeX document. 2

Calculus Lab, Week 3

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