#Maple Lab 1 for Multivariable Calculus, Math251 (22-24), Fall 2020, taught by Dr. Z.
#Edit this file with answers, rename it MapleLab1FirstLast.txt and
#email DrZcalc3@gmail.com by Sunday Oct. 18, 2020

#As usual, Let a[i] be the i-th digit of your RUID, if it is zero, make it 1.

#DO THE QUESTIONS IN A MAPLE WORKSHEET, then (except for the pictures) copy-and-paste (or just enter) the answers in this .txt file


#NOTE: A quick way to compute the intergal of a function f of the variable t from t=a to t=b numerically in Maple, is the line

evalf(Int(f,t=a..b)):

#NOT

evalf(int(f,t=a..b)):

That is much slower.

#Start  a Maple session and put the line

with(plots):

#Question 1. Find the length of the curve  (in 2D)

#r(t)=<t^a[1]+a[3]*t^a[4],t^a[5]+a[1]*t^a[7]> from t=0 to t=1 doing it in TWO ways

#By using the formula int(|r'(t)|,t=a..b)

Ans. to Q1 (The first way):



2. Use if r(t)=<x(t),y(t)>

Do

#A
plot([x(t), y(t),t=0..1],scaling=constrained);

#B
plot([t, 0,t=0..1],scaling=constrained);

#[Added Oct. 14, 2020: The previous version had a comma missing, spotted by Rachel Balji, who won a dollar]

(e.g. if r(t)=<1+t,1+t^3> : plot([1+t, 1+t^3,t=0..1],scaling=constrained);

Save the picture as a .jpeg on your computer, PRINT IT OUT. Take a piece of thread and fit it on the curve. Then strech it out.
Also do the same for #B (to find out the unit length).
Then divide A by B and compare to the


Ans. to Q2 (The second way): 

How different are there?
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Question 2: A good way to numerically solve a system of equations in Maple is with the command

fsolve(SetOfEquations, SetOfUnknowns)

#For example, to solve the set of equations  {24*y^2-11*x=0, 9*x^2-11*y=0} you do

fsolve({24*y^2-11*x, 9*x^2-11*y},{x,y});

Let f:=a[5]*x^3-a[3]*x*y+ a[8]*y^3

Use the material in section 14.7 (Lecture 10) To find the critical point (or points) and then use the second partial derivatives
to find the value of the discriminant at this (those) points and decide whether it is a local max, local min, or saddle point.

Ans. to Question 2: The critical point is (are): ___________________________

The value of the discriminant, D, is: ________________________________

The value of fxx at that point is _______________________________

Hence it is a:

Now confirm it graphically by plotting with the Maple command

plot3d(f,x=-3..3,y=-3..3);

If possible, take a screen-shot and Email it to me as
NicePicFirstLast.jpeg

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Question 3:

Let f(x,y,z)= (5+x^a[1]+y^a[3]+z^a[8])^(1/3)


A. What is f(1,1,1)? __________________________________

B. Find the Linearization of f(x,y,z) at (1,1,1), call it L(x,y,z). _________________________

C.  What is L(1.1,0.95,0.97) ?  _________________________

D. What is f(1.1,0.95,0.97) ?  _________________________

E. What is the ratio? Are there close?__________________________________________________

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Question 4:

USING IMPLICIT DIFFERENTIATION with MAPLE Find dz/dx and dz/dy if
z(x,y) is given in terms of the RELATION

a[1]*x^4*z^5+ a[2]*x^3*y*z^2+ a[3]*(1+x^2*y)*z+ a[5]=0


(Hint, when you set it up replace z by z(x,y), then use the diff( ..., x) and diff( ..., y). then use the solve command
(NOT fsolve) to solve for z(x,y), getting an expression in x,y,z.

Ans to 4: _________________________________________________


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