course Mth 272
......!!!!!!!!...................................
16:42:50
Query problem 7.3.14 f(x+`dx,y) and [ f(x, y+`dy) - f(x,y) ] / `dy.
......!!!!!!!!...................................
RESPONSE -->
confidence assessment:
.................................................
......!!!!!!!!...................................
16:43:05
Give the expressions for f(x+`dx,y) and [ f(x, y+`dy) - f(x,y) ] / `dy.
......!!!!!!!!...................................
RESPONSE -->
a) f(x+`dx,y) = 3(x+dx) + y^2 = 3x + 3dx + y^2
b) [ f(x, y+`dy) - f(x,y) ] / `dy = [3x(y + dy) - 3xy + y^2]/dy = [3xy + 3xdy - 3xy + y^2]/dy = [3x + y^2]/dy
confidence assessment: 2
Good. More detail:
** If f(x,y) = 3 x y + y^2 then
[ f(x, y+`dy) - f(x,y) ] / `dy = [ 3 x (y + `dy) + (y + `dy)^2 - ( 3 x y + y^2) ] / `dy.
Simplifying the numerator we get
[ 3 x y + 3 x `dy + y^2 + 2 y `dy + `dy^2 - 3 x y - y^2 ] / `dy, or
[ 3 x `dy + 2 y `dy + `dy^2 ] / `dy.
Dividing the terms of the numerator by the denominator we have
3 x + 2 y + `dy.
Interpretation:
The expression 3 x + 2 y + `dy represents the change in f due to a small change `dy in the y value, divided by the change in the y value. This is the average rate of change of f with respect to y, over the y interval from y to `dy. This is therefore the average value of the partial derivative with respect to y.
As `dy -> 0 this expression gives us 3 x + 2 y.
The numerator of the expression [ f(x, y+`dy) - f(x,y) ] / `dy is the difference in f near the point (x, y) due to a change `dy in y, divided by that change `dy. That is, [ f(x, y+`dy) - f(x,y) ] / `dy is the average rate at which f(x, y) changes near (x,y) with respect to a change in y.
The limiting value 3 x + 2 y of this expression is the rate at which the function changes with respect to a change in y. This is the definition of the partial derivative of the function with respect to y. **
.................................................
......!!!!!!!!...................................
16:43:18
If your expression for [ f(x, y+`dy) - f(x,y) ] / `dy is not simplified, give the simplified expression.
......!!!!!!!!...................................
RESPONSE -->
confidence assessment:
.................................................
......!!!!!!!!...................................
16:43:25
What is your interpretation of the expression [ f(x, y+`dy) - f(x,y) ] / `dy, and what is its significance?
......!!!!!!!!...................................
RESPONSE -->
confidence assessment:
.................................................
......!!!!!!!!...................................
16:43:28
Query problem 7.3.18 domain of ln(x+y)
......!!!!!!!!...................................
RESPONSE -->
confidence assessment:
.................................................
......!!!!!!!!...................................
16:43:38
Give the domain of the given function and describe this region in the xy plane.
......!!!!!!!!...................................
RESPONSE -->
The domain is the set of all points in the xy plane except those on the x and y axis.
confidence assessment: 2
The natural log is defined only for numbers greater than 0.
** 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. **
.................................................
......!!!!!!!!...................................
16:43:41
Query Add comments on any surprises or insights you experienced as a result of this assignment.
......!!!!!!!!...................................
RESPONSE -->
confidence assessment:
.................................................
&#Good work. See my notes and let me know if you have questions. &#