QUERY25

course MTH 272

05/09,1830

assignment #025025.

Applied Calculus II

05-09-2010

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17:30:49

Query problem 7.3.14 f(x+`dx,y) and [ f(x, y+`dy) - f(x,y) ] / `dy.

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RESPONSE -->

3xy + y^2

f(x+`dx,y)

3(x + 'dx)y + y^2

3xy + 3'dxy + y^2

y(3x + 3'dx + y)

[ f(x, y+`dy) - f(x,y) ] / `dy.

[(3x(y + 'dy) + y^2 + 2y'dy + 'dy^2) - (3xy + y^2) ] / 'dy

(3x'dy + 2y'dy + 'dy^2) / 'dy

'dy(3x + 2y + 'dy)/'dy

3x + 2y +'dy

confidence rating #$&* 2

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17:31:10

Give the expressions for f(x+`dx,y) and [ f(x, y+`dy) - f(x,y) ] / `dy.

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RESPONSE -->

f(x+`dx,y)

3(x + 'dx)y + y^2

3xy + 3'dxy + y^2

y(3x + 3'dx + y)

[ f(x, y+`dy) - f(x,y) ] / `dy.

[(3x(y + 'dy) + y^2 + 2y'dy + 'dy^2) - (3xy + y^2) ] / 'dy

(3x'dy + 2y'dy + 'dy^2) / 'dy

'dy(3x + 2y + 'dy)/'dy

3x + 2y +'dy

confidence rating #$&* 2

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17:31:38

If your expression for [ f(x, y+`dy) - f(x,y) ] / `dy is not simplified, give the simplified expression.

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RESPONSE -->

Expression is simplified (3x + 2y + 'dy).

confidence rating #$&* 2

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17:38:32

What is your interpretation of the expression [ f(x, y+`dy) - f(x,y) ] / `dy, and what is its significance?

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RESPONSE -->

The first value f(x, y + 'dy) includes the oringinal x value and the y value added to the change in y. The values subtracted from this, f(x,y), are the original values of x and y. So, theoretically, the only difference here should be the 'dy or the change in the y value.

(x, y + 'dy) - (x, y) ='dy

The numerator is then divided by a denominator that expresses the change in y value, 'dy. Being identical to the numberator, the value should be 1.

'dy/'dy =1

However, using the given equation of 3xy + y^2, our values become altered and differ by more than simply the change in y values. This is evident by the linear equation we receive as an answer here: 3x + 2y + 'dy.

confidence rating #$&* 2

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17:40:45

Query problem 7.3.18 domain of ln(x+y)

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RESPONSE -->

ln (x + y)

As ln (0) is undefined, x + y > 0 must be the domain.

The range must similarly be greater than 0.

confidence rating #$&* 1

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17:42:01

Give the domain of the given function and describe this region in the xy plane.

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RESPONSE -->

Domain must be greater than zero. Therefore, I believe that the graph of this function would lie above the xy plane without actually having any intercepts within it.

confidence rating #$&* 1

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17:43:09

Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

Number 18 involving the ln (x + y) was particularly challeging to me and I have little faith in my answers. Natural log and exponentials always present a stumbling block for me.

confidence rating #$&* 2

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That problem should have included a given solution. Not sure why it wasn't included:

The function is defined for all x, y such that x + y > 0, which is equivalent to y = -x. The domain is therefore expressed as the half-plane y > -x. It consists of all points above the line y = -x in the x-y plane.

Since the natural log function can take any value as x + y goes from 0 to infinity, the range of the function is all real numbers.

As we approach the line y = -x from points lying above the line, x + y approaches zero so ln(x+y) approached -infinity. So the surface defined by the function has a rapid dropoff whose depth exceeds all bounds as we approach y = -x.

Since ln(1) = 0, the graph intersects the xy plane where x + y = 1--i.e., on the line y = -x + 1, which lies 1 unit above the line y = -x. Between the line y = -x and y = -x + 1 the graph rises from unbounded negative values to 0.

The graph will reach altitude 1 when ln(x+y) = 1, i.e., when x + y = e. This will occur above the line y = -x + e, approximately y = -x + 2.718.

The graph will reach altitude 2 when ln(x+y) = 2, i.e., when x + y = e^2. This will occur above the line y = -x + e^2, approximately y = -x + 8.2.

The graph will reach altitude 3 when ln(x+y) = 3, i.e., when x + y = e^3. This will occur above the line y = -x + e^3, approximately y = -x + 22.

Note that the distances required to increase by 1 unit in altitude increase by greater and greater increments.

The graph will continue reaching greater and greater altitudes, but the spacing between integer altitudes will continue to spread out and the steepness of the graph will decrease fairly rapidly.

QUERY25

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