## Exam 2 (MTH 252)-Spring 2023

Exam 2 will be on March 15 and covers Sections 14.1-14.7 , 15.1 and 15.2. You should review your class notes and examples discussed during our meetings. I will hold a review session for the exam on Tuesday (Mar. 14) from 4:00-5:30 pm on BAC 245. Below are some problems to practice for the exam. We will discuss some of these problems during the review session.

### 1.Find and sketch the domain of the following functions:  微积分考试代写

(a) f(x, y) =

(b) f(x, y) =

2.Sketch the graph of the following functions:

(a) f(x, y) = 1 y2.

(b) f(x, y) = x2 + (y 2)2.

(c) f(x, y) = sin x.

3.Sketch the level curves of the following functions for k = 0, ±1; if the level curves do not exist for some of these values explain your reasoning:

(a) f(x, y) =

(b) f(x, y) = ex + y.

(c) f(x, y) = yex

### 4.Find the limit if it exists or show that it does not exist:  微积分考试代写

(a) lim(x,y)(0,0)(x2 + y2 ) ln(x2 + y2 ).

(b) lim(x,y)(0,0)

(c) lim(x,y)(0,0)

(d) lim(x,y)(0,0)

5.Find the fifirst partial derivatives of the following functions:  微积分考试代写

(a) f(x, y) = y sec(x2 + y2 ).

(b) f(x, y) =

(c) p(u, v, w) = u arctan

6.Determine the set of points at which the function

is continuous.

### 7.If z = sin(x + sin t) show that

zxzxt = ztzxx.

8.For each equation, fifind dy/dx (assume that y is a function of x):

(a) arctan(x2y) = x + xy2 .

(b) cos(xy) = 1 + sin y.

9.For the following equation, fifind ∂z/∂x and ∂z/∂x (assume that z is a function of x and y):

sin(xyz) = x2 y2 z2 .

### 10.If z = y + f(x2− y2 ), show that

yzx + xzy = x.

11.Find the equations of the tangent plane and normal line to the following surfaces at the given point:

(a) xy + yz + zx = 3 at (1, 1, 1).

(b) z = 3x2 y2 + 2x at (1, 2, 1).

12.Find the points on

x2 + 4y2 z2 = 4

where the tangent plane is parallel to the plane 2x + 2y + z = 5.

### 13.Explain why the following functions are difffferentiable at the given point. Then fifind the linear approximation of the function at that point:  微积分考试代写

(a) f(x, y) = x2ey at (1, 0).

(b) f(x, y) = y + sin(x/y) at (0, 3).

14.Find the directional derivative of each function at the point P in the given direction :

(a) f(x, y) = x 2 e y at P(1, 0) and  =

(b) f(x, y, z) = xy xy2 z 2 at (2, 1, 1) and  =

15.Find the maximum and minimum rates of change of f at the given point and the direction in which they occur:

(a) f(x, y) = sin(xy) at P(1, 0).

(b) f(x, y, z) = x ln(yz) at (1, 2, 1).