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MAT 201, Lab #4, Spring 2016 Name _____________________________________________ Directions: The labs for this course will require the use of MatLab (mostly). Instructions on how to access MatLab can be found under a separate document posted in the course. Use the listed example code to complete the following exercises. Print the resulting MatLab worksheet. Circle and number the solutions to the problems on the printout. Or, copy and paste the contents of the MatLab worksheet into a file, note the problem numbers in the code, to submit the problems electronically. Example Code File: 201lab_ex4.pdf The code file begins by declaring variables. You will need to declare any variables you intend to use before you use them. The file declares the variables: ๐ฅ, ๐ฆ, ๐ง. Then we define a function ๐(๐ฅ, ๐ฆ) = sin(๐ฅ๐ฆ) + cos(๐ฅ๐ฆ 2 ) and take the partial derivatives with respect to both ๐ฅ and ๐ฆ. If we want to compute a derivative like ๐๐ฅ๐ฆ๐ฆ , that code follows with embedded derivative commands. Recall the inside command is the one that gets executed first. 2 4 Integration works similarly. To compute the double integral ∫1 ∫−3 ๐ฅ๐ฆ๐๐ฆ๐๐ฅ , we define the function, and use embedded integration commands. Or we can create integrals with variable limits of integration, which follows in the code. You can do this in one stage or two, with the intermediate stage being a function. You will need to save the result using the define function code. We will use this when we find potential functions. The last bit of code creates a gradient vector ∇โ for โ(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ๐ฆ๐ง + 2 cos ๐ฅ − 3๐ง 2 − arcsin ๐ฆ, and then expresses it as a column vector. Use these examples to complete the following: 1. Find the indicated partial derivatives. a. ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ 2 ๐ฆ + arctan(2๐ง) , ๐๐ฅ , ๐๐ง , ๐๐ฆ๐ง b. ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ๐ ๐ฆ๐ง − cos(๐ฅ ln ๐ฆ) , ๐๐ฅ๐ฆ , ๐๐ง๐ฆ๐ฅ 2. Find the values of the double integrals. 4 2 a. ∫0 ∫−1 cos 2(๐ฅ) sin(๐ฆ) ๐๐ฆ๐๐ฅ 1 ๐ฅ2 b. ∫−1 ∫0 ๐ฅ 2 ๐ฆ 3 ๐๐ฆ๐๐ฅ 3. For the function ๐(๐ฅ, ๐ฆ, ๐ง), and the vector field ๐นโ (๐ฅ, ๐ฆ, ๐ง) = (๐ฅ + ๐ฆ)๐ฬ + (tan ๐ฆ + ๐ฅ๐ง)๐ฬ + (๐ง 2 − 1)๐ฬ, find the following: a. ∇๐ b. ∇ × ๐นโ c. ∇ โ ๐นโ d. ∇2 ๐ To submit your solutions, copy and paste the graphs and the code into a file for printing or to submit by email. Mark the sections of your code that correspond to which problems, and label your graphs.