Math 225: Calculus III     Assignment 3

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I affirm that the solutions presented in this assignment are entirely my own work.

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Instructions

This assignment contains problems on vector functions, including their derivatives and integrals,
from Chapter 12 in Finney & Thomas and Chapter 2 in the Lecture Notes. Remember, % stands for the previous output. For example,

r[t_] = t  i  + t^2 j + t^3 k

D[%,t]

Integrate[r[t],t]

% /. t->4

This last line shows how to substitute a value for a variable in an expression: 
    expression  /.  variable -> value
The double symbol /. applies the substitution that follows to the expression on the left. The substitution itself uses another double symbol -> (looks like an arrow). This example also shows that D[] and Integrate[] work correctly with vector functions.

You can combine plots with the Show[] command. First save the plots with names and list the names in Show[]:
    p1 = Plot3D[...]
    p2 = Plot3D[...]
    Show[p1,p2]

To get a 3D line segment representing a vector from p to q, use v1 = Vector[p,q]  and include v1 in a Show[] command. See the notebooks AdvancedCalculusDemo, and TangentsNormals in the Notebook folder or the back of the Lecture Notes.

Sometimes Mathematica  cannot solve an equation like f[x]==0 exactly. In such a situation you can obtain a numerical solution with NSolve[f[x]==0,x]. To find one solution near a point a, it is better to use FindRoot[f[x],{x,a}] where a is a number that tells Mathematica  where to start looking. Plotting f[x] can help you find good value for a.

Be sure to type in comments explaining what you are doing.

Remember to uncheck Show In/Out Names in the Kernel 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 f[t_] = (t^3-t)i + (3 - E^(2t))j + Log[t]k and
let g[t_] = t^2 i - t j + Sin[Pi t]k.
a) Compute f[t].g[t] and Cross[f[t],g[t]].
b) Find the derivative of f[t].g[t] at t == 1.
c) Find the integral of Cross[f[t],g[t]] from t == 1 to t == 2.

Solution

Problem 2

Let r[t_] = Sin[t^2] i + Cos[t^2] j + t k
a) Find a vector that is tangent to the curve at the point
    p = {P1,0,Sqrt[Pi/2]}.
b) Find the equation of the line tangent to the curve r[t] at p.
c) Use ParametricPlot3D[] to plot the curve and the tanget line near p, then 
    combine the plots with Show[].

Solution

Problem 3

Solve the initial value problem r''[t] == E^(2t) i - Cos[t]j + t k,
r'[0] == j + k, and r[0] == i.
a) Step one: let v[t] be integral of r''[t] with a constant vector added to the result. Then solve the equation v[0] == j + k for the unknown constants. This v[t] is now r'[t].
b) Step two: let r[t] be integral of v[t] with another constant vector added to the result. Then solve the equation r[0] == i for the unknown constants. This gives the final answer r[t].

Solution

Problem 4

The position of a particle in the plane at time t is
r[t_] = 1/(t^3+1) i - 1/(t^2+1) j.
a) Find the particle's speed s[t] at any time t >= 0. Plot s[t] as a function of t.
b) Find the particle's maximum speed for t >= 0.

Solution

Problem 5

Let r[t_] = t i + (1+t)^3 j + (1-t)^3 k.
a) Find the unit tangent vector T[t], the unit normal vector n[t] (remember, the letter N is reserved for the function N[] in Mathematica), and the binormal vector
B[t_] = Cross[T[t],n[t]] .
b) Plot T[1], n[1] and B[1] along with the plot of r[t] near t == 1.

Solution