Math 225: Calculus III    Assignment 4

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Instructions

The problems in this assignment involve curves, acceleration, functions of 2 variables, their graphs, level curves, and derivatives (Chapter 12-13 in Finney and Thomas, Chapter 2-3 in the Lecture Notes). Before starting, you may want to review the notebooks Derivatives, and Exercises (in the Notebook folder or the back of the Lecture Notes). To make the graphs look smoother you can increase the number of plotted points with the optional optional argument PlotPoints. For example,
Plot3D[ x^2 - y^2, {x,-2,2}, {y,-2,2}, PlotPoints->30]
or
ContourPlot[ x^2 - y^2, {x,-2,2}, {y,-2,2},
   PlotPoints->30]

The function Limit[] only finds limits of functions of one variable. In problem 4, you should substitute different expressions for y,
then take the limit as x -> 0. For example,
 z = Sin[x^3y^2]/(x^5 + y^5)
 z1 = z /. {y -> 2x}
 Limit[z1, x->0]
 
Be sure to type in comments explaining what you are doing.

Remember to uncheck Show In/Out Names in the File menu and close this group before printing.

Initialization

<<Calculus`Math225`

i = {1,0,0}; j = {0,1,0}; k = {0,0,1};

Problem 1

Let r[t_] = (3t^5 - 5t^3) i + (t^4+4t) j + (t+1)^3 k.
a) Find the point(s) where the curve defined by r[t] is not smooth.
b) Plot the curve with ParametricPlot3D[] near the singular point(s).
c) Find the length of the Curve from t==-1 to t==1. (Use NIntegrate[]).

Solution

Problem 2

A particle's position is given by
r[t_] = (1 + Sin[t])i + (3 t^2)j + (t + t^3)k.
a) Find the acceleration, a[Pi], of the particle at time t == Pi.
b) Find the tangential and normal components of a[Pi], i.e., find NUMBERS u and v such that  a[Pi] = u T[Pi] + v n[Pi]  without calculating T[Pi] or n[Pi]! (See Section 12.4 in Finney &Thomas or 2.4 in the Lecture Notes).

Solution

Problem 3

Plot the graphs of the following functions, z = f[x,y], with Plot3D[] and plot several level curves of each function with ContourPlot[] (See Section 3.1 in the Lecture Notes and the examples in the Notebook directory.) Compare the graphs to the level curves.
a) z = x + y^2 - y x^3
b) z = z = (x^3 - y^3)/(x^2 + y^2)
c) z = E^(-x^2-y^2)Sin[x]

Solution

Problem 4

By considering different paths of approach, show that the limit of the function
f[x_,y_] = x^4 y^2/(x^6 + x^2 y^4 + x^3y^3)
as x and y both go to 0 does not exist. Graph the function near {0,0} and explain why this is so based on the appearance of the graph.

Solution

Problem 5

a) Find the partial derivatives of
    f[u_,v_] = u^2 Cos[u/v] with respect to u and v.

b) Find the second order partial derivatives of
    f[x_,y_] = x^3 y^2 - y/x^2 + E^(x y)

Solution