Calculus III – MATLAB Project 1

Project Overview

This repository contains symbolic and graphical MATLAB scripts for Calculus III Project 1.
The purpose of this project is to apply multivariable calculus concepts—partial derivatives, Lagrange multipliers, and triple integrals—using MATLAB’s symbolic math and 3D visualization tools.

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Contents

File	Description
MATLABCodeCalcIII.pdf	Main MATLAB script containing Problems 1–5
FinalProject1.pdf	Project instructions and grading rubric (provided by instructor)
CalcIIIMatlabProject.docx	File with documented problems and code
README.md	Project overview and documentation (this file)

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Problem Summary

Problem 1 – Plane Relationships

Use dot and cross products to determine whether pairs of planes are parallel, perpendicular, or neither.
Normal vectors are extracted directly from the plane equations.

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Problem 2 – Partial Derivatives

Define
[ z = 7y^3 e^{3xy^2} + x^3y ]

Compute the following symbolically using diff:
[ z_x, ; z_y, ; z_{yyx}, ; z_{xxy}, ; z_{yyxyx} ]

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Problem 3 – Lagrange Multipliers

Minimize
[ f(x, y, z) = xy + 2xz + 2yz ]
subject to
[ xyz = 4000 \cdot 7^3 ]

Use MATLAB’s solve to find critical points.
The real minimum occurs at:
[ (x, y, z) = (140, 140, 70), \quad f = 58{,}800 ]

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Problem 4 – Surfaces and Intersection

Plot the following:

• Paraboloid: ( z = 49 - 9x^2 - 9y^2 )
• Cylinder: ( x^2 + y^2 = \frac{49}{3} )

Plot their intersection curve using ezsurf and ezplot3.

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Problem 5 – Volume of a Region

Find the volume inside the cylinder ( r = 7 ) and below the cone ( z = 14 - r ).
Evaluate in polar coordinates: [ V = \int_{0}^{2\pi} \int_{0}^{7} (14 - r)r ,dr,d\theta = \frac{1372}{3}\pi \approx 1436.755 ]

Plot the cone and cylinder surfaces to visualize the region.

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MATLAB Commands Used

Only commands from the Matlab for Calculus III handout are permitted.

Category	Commands
Symbolic math	syms, diff, solve, int
Algebra & vectors	dot, cross, subs, simplify
3-D graphics	ezsurf, ezplot3, hold on, axis equal
Display & workspace	disp, clc, clear, close all

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How to Run

1. Open Project1.m in MATLAB.
2. Press Run.
3. For each problem section:
   • Results are displayed in the Command Window.
   • 3-D plots open in separate figure windows.
4. Copy numerical results and screenshots into the project report.

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Report Guidelines

Section	Required Content
Title Page	Project title, student name, date
Problems 1–5	Each MATLAB output and graph clearly labeled
Hand Calculations	Include where required (e.g., Problems 3 & 5)
Discussion	Short explanation of results and geometric interpretation

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Submission Instructions

Push the following to your GitHub repository:

• Project1.m
• README.md
• Screenshots (optional, in a /figures/ folder)

Submit your GitHub repository link according to the course submission procedure.

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Notes

• Only commands listed in the instructor’s MATLAB handout are allowed.
• Ignore complex (“i”) solutions; use the first real solution.
• Check by hand whether the critical point in Problem 3 is a minimum or maximum.
• Round numerical results only where appropriate (e.g., ( V \approx 1436.755 )).

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Author: Ihor Holubets
Course: Calculus III
Instructor: Lia Vas
Date: [Insert submission date]