Math 1502

Quiz #1  Jan 26  Tom Morley 8:00 am

Name: ______________________________________
Section: Circle One:
Write the name of your TA (1 point)

Open Book and Notes.  Carefully explain your proceedures and answers. Calculators allowed, but answers mush be exact.

Problem 1 (12 points)

M . Indurain needs to know the following : Consider the twisted cubic : r (t) = t i + t^2 j + t^3k, for 0 ≤ t ≤ 2.

Find the velocity (2 points) , acceleration (2 points) , speed (3 points)   and curvature (5 points) .

Ans

In[71]:=

Clear[r]

In[72]:=

r[t_] := {t, t^2, t^3}

In[73]:=

v = D[r[t], t]

Out[73]=

{1, 2 t, 3 t^2}

In[74]:=

a = D[v, t]

Out[74]=

{0, 2, 6 t}

In[75]:=

speed = Sqrt[v . v]

Out[75]=

(1 + 4 t^2 + 9 t^4)^(1/2)

There are several was to get the curvature .

In[77]:=

EuclidianLength[x_] := Sqrt[ x . x]

In[78]:=

EuclidianLength[Cross[v, a]]/EuclidianLength[v]^3

Out[78]=

(4 + 36 t^2 + 36 t^4)^(1/2)/(1 + 4 t^2 + 9 t^4)^(3/2)

Problem 2

Find the arc length between  0 and t for the curve : r (t) = e^t cos (t) i + e^tsin (t) j + 2  e^tk

In[79]:=

r[t_] = {E^t Cos[t], E^t Sin[t], 2 E^t}

Out[79]=

{^t Cos[t], ^t Sin[t], 2 ^t}

In[80]:=

v = D[r[t], t]

Out[80]=

{^t Cos[t] - ^t Sin[t], ^t Cos[t] + ^t Sin[t], 2 ^t}

In[83]:=

speed = (EuclidianLength[v]//Simplify)

Out[83]=

6^(1/2) ^(2 t)^(1/2)

In[84]:=

∫_0^t speed t<br />

Out[84]=

6^(1/2) (-1 + ^(2 t)^(1/2))

Problem 3 (9 points)

Find the tangential component of the acceleration of the curve

r (t) = t i   + t^2 j + t^4k

You do not have to find T or N .

In[85]:=

r[t_] := {t, t^2, t^3}

In[86]:=

v = D[r[t], t]

Out[86]=

{1, 2 t, 3 t^2}

In[87]:=

speed = EuclidianLength[v]

Out[87]=

(1 + 4 t^2 + 9 t^4)^(1/2)

In[89]:=

D[speed, t]

Out[89]=

(8 t + 36 t^3)/(2 (1 + 4 t^2 + 9 t^4)^(1/2))

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