Question

Using MatLab

2. Given the parametric equations x = t^3 - 3t, y = t^2-3:

(a) Find the points where the tangent line is horizontal or vertical (indicate which in a text line)

(b) Plot the curve parametrized by these equations to confirm.

(c) Note that the curve crosses itself at the origin. Find the equation of both tangent lines.

(d) Find the length of the loop in the graph and the area enclosed by the loop.

3. Use what you learned about vector functions and parametrized curves to make a smile emoji! The emoji consists of the following:

(a) FACE: circle of radius 2, centered at the origin.

(b) EYES: circles of radius 0.1 centered at (?1, 1) and (1, 1)

(c) SMILE: bottom half of a semicircle from (?1, 0) to (1, 0)

Answer 1

2

Script

clear all

close all

clc

syms t

x=t.^3-3*t;

y=t.^2-3;

a=diff(x);

b=diff(y);

equ1= a == 0;

equ2= b== 0;

solx1=solve(equ1,t);

solx2=solve(equ2,t);

pver=[subs(x,t,solx1) subs(y,t,solx1)]; % tangent line is vertical at (2,-2) and (-2,-2)

phor=[subs(x,t,solx2) subs(y,t,solx2)]; % tangent line is horizontal at (0,-3)

x1=subs(x,t,-3:0.1:3);

y1=subs(y,t,-3:0.1:3);

plot(x1,y1)

hold on

scatter(pver(:,1),pver(:,2))

scatter(phor(:,1),phor(:,2))

g=-20:20;

hort=-3;

plot(g,hort*ones(size(g)),'*r')

vert=[2 -2];

h=-3:6;

plot(vert(1)*ones(size(h)),h)

plot(vert(2)*ones(size(h)),h)

Result