Calculus Directional Derivative?
OnAJourney asked:
A. Find the directional derivative for
h(x,y)=e^-(x^2+y^2), P90,0), v=i+j
A. Find the directional derivative for
h(x,y)=e^-(x^2+y^2), P90,0), v=i+j
B. Evaluate the iterated integral
S=Integral
S [2(on top) 0 (bottom)] S [2y (on top) y (bottom)] (10+2x^2+2y^2)dx dy
C. Use a double integral to find the volume of the solid with the following boundaries:
y=2, y=x, z=4-y^2, 0 is greater than x greater than 1
Filed Under: Cardio
A
To find the directional derivative calculate the gradient for the function h and than take the dot product of the gradient for the function h with the unit vector in the direction of v.
Hint: v is not a unit vector so you will have to find the unit vector in the direction of v using the following formula
Uv = v*(1/ |v|)
where |v| is the length or magnitude of the vector v.
Uv: stands for the unit vector in the direction of v.
For the gradient of h I got
grad h = – e^(-x^2-y^2), – e^(-x^2 – y^2)
check this answer. Once you have the expression for the gradient evaluate it at the point (90, 0). Then take the dot product of the gradient for h with the unit vector you find for the vector v and you should have the directional derivative.
Remember the dot product should give a scalar as a result.
B
For the iterated integral, evalute it as you would normally.
In this case you have to evaluate with respect to x (with the dx term) first since the limits of integration are variable (in terms of y) instead of a number. When you integrate with respect to x you treat the y terms as constants.
After you evaluate from y to 2y the whole expression should be in terms of y and then you are left with a single variable integral and you integrate in terms of y from 0 to 2.
C
For this problem Draw the region you are integrating (which is in R2, the xy-plane). Draw the lines y= x and y=2 which will give you a triangular region. This will help you determine your limits of integration. You are integrating the function 4-y^2. Since you are doing a double integral it should be of the form
SS 4-y^2 dx dy or SS 4-y^2 dy dx
depending on how you choose the limits of integration (Type 1 or Type 2 region)