MATH2720 Multi-variable Calculus: Test Practice Pools

多变量微积分代写 Let z = (y2x2)2. Find all critical points of z. Identify whether they are local minimum, local maximum or saddle points.

Test 2 Practice Pool  多变量微积分代写

Disclaimer: There is no guarantee that your actual test questions will resemble these practice problems.

1. Let

(a)Finda ray approaching the origin along which f (x, y) = 1.

(b)Finda ray approaching the origin along which f (x, y) = 0.

(c)Whatcan we say about the limit of lim_{(x,y)→(0,0)} f (x, y)?

2. Evaluate ifpossible:

3. Evaluate each of the following limits or show that it does notexist:

4. Let

(a)Computef_x using the defifinition.

(b)Computef_y using the defifinition.

(c)Find^d f (t, t)|_t=0.

5.Findall first and second order partial derivatives of the following function and evaluate them at the given   多变量微积分代写

f (x, y, z) = ln(1 + e^xyz), (2, 0, −1).

6. Findall first and second order partial derivatives of the following function and evaluate them at the given

7. Letf be any differentiable function of one  Define z = f (x² + y²). Does the equation

always hold?

yz_x − xz_y = 0

8. Letw = f (x, y, t) with x and y depending on  Suppose that at some point (x, y) and at some time t, the partial derivatives f_x, f_y, f_t are equal  多变量微积分代写

to 2,3,and 5 respectively, while  what is ^dw ? Is it the same with f_t? why or why not?

9. Evaluatew_s and w_t given w = x² + y² + z², x = st, y = s cos t, z = s sint
10. Suppose x²y +^z = 1. Let w = sin x cos(2y). Find

11.Letf  Find the equation of tangent plane to the surface z = f (x, y) at (−1, 1, ¹ ). 多变量微积分代写

12. Consider the surface z = f (x, y) defined implicitly by the equation xyz²+ y²z³ = 3 + x². Use a 3–dimensional gradient vector to find the equation of the tangent plane to this surface at the point (1, 1, 2).
13. Findall horizontal planes that are tangent to the surface with equation z = xye−(x  +y  )/2.

14.Find the directional derivative of f (x, y, z) = e^xyzin the direction of (0, 1, 1) at (0, 1, 1).  多变量微积分代写

15. Let z = (y²x²)². Find all critical points of z. Identify whether they are local minimum, local maximum or saddle points.

16.: Find all critical points for f (x, y) = x(x²+ xy + y² 9). Also find out which of these points give local maximum values for f(x, y), which give local minimum values, and which give saddle points.

17.Use the Second Derivative Test to find all values of the constant c for whichthe function z = x² + cxy + y² has a saddle point at (0, 0).

18.Findthe maximum and minimum values of f (x, y) = xy − x³y²when (x,y) runs over the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.