Question

e 09, 201 (6) 2 points An equation for the level curve of f(z, y) = In(z+y) that passes through the point (0, e2) is A. z + y = e2 B. I+y e C. z+y 3. D. None of the above (7) 2 points The gradient of f(z,y, z) = ep at the point (-1,-1,2) is A. (2e2,e2,2e2). B. (-e,-e,2e2). C. (-2e2,-2e2, e) D. (-2e2,-e,-e) (8) 2 points Let f be a function defined and continuous, with continuous first partial derivative at the origin (0,0). A unit vector u for which the minimum value of Duf (0,0) is: af A. Vf(0,0) z,0),(0,0)), af B. 1 C. D. None of the above. edzdy (9) 2 points Given the double integral equivalent integral with , an order of integration reversed is: 4x - dydr A. e dydz B. Jz/4 cz/4 e dydz. C. 0 Jo c4/z dydz y D. e" 0 f(x,y, z)dV = 7, where the region R = {(z,y, 2) E R| 0 S (10) 2 points Given R 2, -22< 2}. Find the average value of f over the region R. IS1, -1 Sy A. 7/12 B. 12/5 C. 5/12 D. 12/7

Answer 1

Q.7.) Gradient of the given function at given point.

What is Gradient?

☆ If we want to know the direction of any object, we draw the normal vector from that we come to know that what is the direction of the object,

☆ And if we want to know the direction of the function we use Gradient concept,

☆ Calulating gradient of any function at any point means we calculating its normal at that point from which we come to know the direction of the function at that particular point.

Find -:

Gradient of f(x,y,z)

x, y, z)= e

☆ Gradient

а о У— дх k- ду де

f + ay

Of of +j k

+ay

Vf(eyi+eV*xz j+e*V*ry k)

At point (-1,-1, 2)

Vf ( -2e2 i - 2e2j e2 k)

Q.9.)

e-ydrdy 4у

Changing the order of integration.

☆ Changing the order of integration means,

As we can see in the question, it is in the form,

C2 C2 f(y)dxdy

we have to integrate first w.r.t dx and then integrate it with dy,

☆ but in this method we reverse it order by, first we integrate it with respect to dy then w.r.t dx.

☆ this method makes most of the integration easy to solve.

☆ For this we have to simply interchange the limit value, in the form

$\int_{x_1}^{x_2}\int_{y_1}^{y_2} f(y) dydx$

☆ Now, let's start solving the question

e-ydrdy 4у

☆ It's limit are,

☆ For changing the order, we first have to draw the limit on the graph and then change its order,

Sニ0 Froom this view Foam this view.

I.) In the below figure, left most figure shows the correct diagram of the limit,

II.) In second figure, the limit is taken first with x as dependent variable and y as independent variable, which is mention in the question.

☆ For reversing the order of the limit, we just have to choose y as dependent variable and x as independent variable.

III.) In rightmost figure, the limit is taken as y as dependent figure and x as independent variable.

☆ The integral will look like

$\int_{0}^{4} \int_{0}^{x/4} e^{-y^2} dydx$

☆ The answer which is in option is completely wrong, I had solve all 4 integral which is in the option and all are wrong the form and I had obtain the answer with the right concept.

☆☆ Please comment below if you have any doubt regarding this question before rating this answer.