query 101

#$&*

course Mth 277

2-27 7

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises.

query_10_1

*********************************************

Question: Find the domain of F(t) X G(t) when F(t) = t^2 i - (t+2)j + (t-1)k and G(t) = (1/(t+2))i + (t-5)j + sqrt(t) k.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

= [t^2 i - (t+2)j + (t-1)k ] X [(1/(t+2))i + (t-5)j + sqrt(t) k]

i j k

t^2 -(t+2) (t+1)

(1/(t+2)) (t-5) `sqrt(t)

= [-`sqrt(t)(t+2) - (t-5)(t+1)]i + [((t+1)/(t+2)) - t^2`sqrt(t)]j + [t^2(t-5) + (t+2)/(t+2)]k

D: t >= 0

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: Describe the graph of G(t) = (sin t)i + (cos t)j + (4/3)k

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

This is a graph of left circular helix, noticing that for t = 0 init pt would be (0, 1, 0).

For t = `pi/2, coord is (1, 0, 2`pi/3), which is moving counterclockwise and gains altitude of

2`pi/3.

For t =`pi, coord is (0, -1, 4`pi/3),which is moving counterclockwise again and new altitude of

4`pi/3.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: Given F(t)= (t)i - 5(e^t)j +(t^3)k, G(t) = ti - (1/t)k and H(t) = (t*sin t)i + (e^-t)j, find H(t) dot [G(t) X F(t)]

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

G(t) X F(t)

= [ti - (1/t)k] X [(t)i - 5(e^t)j +(t^3)k]

i j k

t 0 -(1/t)

t -5(e^t) t^3

= [ 0 - (5(e^t))/t ]i + [-t/t - t^4]j + [-5t(e^t) - 0]k

G(t) X F(t) = (-(5e^t)/t)i - (1 + t^4)j - (-5te^t)k

H(t) DOT [G(t) X F(t)]

= [(t*sin t)i + (e^-t)j] DOT [(-(5e^t)/t)i - (1 + t^4)j - (-5te^t)k]

-(5e^t)/t)*(t*sin(t)) + (e^-t)* (1 + t^4) + 0*(-5te^t)k

= -5e^t sin(t) + (e^-t + e^-t t^4)

H(t) DOT [G(t) X F(t)] = -5e^t sin(t) + e^-t + e^-t t^4

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: Find a vector function F whose graph is the curve given by the equation x/5 = (y-3)/6 = (z+2)/4.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

I’m not being able to figure this out????

This is a straight line through the point (0, 3, -2), which is parallel to the vector 5 `i + 6 `j + 4 `k.

To see this:

If (x, y, z) is on the line through the point (0, 3, -2), which is parallel to the vector 5 `i + 6 `j + 4 `k, then

(x - 0) `i + (y - 3) `j + (z +2 ) `k

is parallel to

5 `i + 6 `j + 4 `k,

meaning that the components of the two vectors are all in the same proportion.

The proportion of the x components is (x - 0) / 5.
The proportion of the y components is (y - 3) / 6.
The proportion of the z components is (z + 2) / 4.

Setting these proportions equal we get

x/5 = (y-3)/6 = (z+2)/4.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

*********************************************

Question: Find the limit as t -> 2 of ((t^4-2)/(t-2))i + ((t^2-4)/(t^2-2t))j + ((t^2 + 3)e^(t-2))k.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

??????I’m having trouble with this problem. It seems that this vector function is not continuous at the point that the limit is trying to be taken. l’Hospital’s rule can apply to the j component and get a limit of 2 as t 2 and for k we get limit = 7 as t 2. Just not sure what to do for i component???????????????

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

@& The limit doesn't care a bit what happens at the limiting point.

However if the limit doesn't exist, or approaches infinity, then the function is not continuous at the point.

If the limit does exist, but is not equal to the value of the function at the point, then again the function is not continuous at the point.

In this cae the expression is

(t^4 - 2) / (t - 2).

As t -> 2, the numerator approaches 14, while the denominator approaches zero.

As you approach from the right both numerator and denominator are positive, and as you approach from the left they are both negative. So the limit is +infinity.

*@

*********************************************

Question: How many revolutions are made by the circular helix R(t) = (sin t)i + (cos t)j + (3/4)tk in a vertical distance of 12 units.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

approx. 1.5

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

@& In a vertical distance of 12 units, 3/4 t would change by 12, so t would change by 16. That would be more like 5 pi, or 2.5 cycles of the sine and cosine function, leading to about 2.5 revolutions of the helix..
*@

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

query 101

@& In a vertical distance of 12 units, 3/4 t would change by 12, so t would change by 16. That would be more like 5 pi, or 2.5 cycles of the sine and cosine function, leading to about 2.5 revolutions of the helix.. *@

&#Good work. See my notes and let me know if you have questions. &#

@& In a vertical distance of 12 units, 3/4 t would change by 12, so t would change by 16. That would be more like 5 pi, or 2.5 cycles of the sine and cosine function, leading to about 2.5 revolutions of the helix..
*@