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Calculus of Multiple Variables - A Tutorial with Problems, Solutions, MCQ Quiz - Part I : Functions of several variables , Limits and Continuity

Functions of several variables , Limits and Continuity

After understanding this topic you might also be interested in trying out the MCQ Quiz here.

Important points to remember :

Homogenous functions : A multivariate(more than 1 variable) is homogenous of degree k  if , each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by tk.

                                          f (tx1, tx2,…..,txn ) =  tkf(x1,x2,.......xn)

Scalar field : A scalar field associates every point in space with a scalar quantity. It gives  a single value of some variable for every point in  space. Potential fields and Temperature, Pressure fields are scalar fields.

Limits : Limit of a function - emphasizing that limit is unique and independent of the path. Introducing the ways in which a limit can be expressed.

Continuity : A function z=f(x,y)  is said to be continuous at a point (x0,y0) if
1. f(x,y)  is defined at the point f(x0,y0)
2. Limit exists at (x0,y0)
3. Limit is equal to function at that point.

Partial Derivative : The derivative of a function of several variables with respect to one of the independent variables keeping all the other independent variables constant is called partial derivative of the function with respect to that variable . ∂f / ∂x  is the symbol used
for partial derivative with respect  to x keeping all other variables constant.

Example :
zx=2x  partial derivative with respect to x , keeping y constant
zy=2y partial derivative with respect to y  , keeping x constant

Gradient : Gradient of a scalar field is a vector field that points in the direction of  the greatest rate of increase of the scalar field and its magnitude is the rate of increase  

Directional derivative :Let Φ(x, y, z) be a scalar point function defined over some region R of space. At some point P(x, y, z) ,the rate of change of a function Φ , in a specified direction, is the directional derivative  of Φ at  P in that direction.
Total Differential : In case of multivariate functions total differential is the total change  taking all variables into consideration.

Tangent Planes and Normal : Finding the tangent plane and normal for a function.

Critical points :  A critical point of a function is a point  where the partial derivatives  of first order is equal to zero or not defined.  Critical points are points in the xy-plane where the tangent plane is horizontal.

Saddle points : Points at which the derivative is defined and equal to zero  but not extrema (maxima or minima) . The second derivative test can be used to find out if the point is a maxima, minima or saddle point.

Local or Relative Maximum and Minimum : Maximum and Minimum  values  of a function, together called the  extremum, are the largest and smallest value respectively that the function takes at a point  within a given neighborhood .

Global or Absolute Maximum and Minimum :  The largest and smallest  value respectively that the function takes at a point on the entire domain  of the function.
A global minimum is also a local minimum but a local minimum is not a global minimum.
A global maximum is also a local maximum but a local maximum is not a global maximum.

Second derivative Test :
Finding when the point is a a SADDLE POINT or a MAXIMUM or a MINIMUM and when the test is inconclusive.

Here are some of the problems solved in this tutorial.

Easy Problems :

1. Find the level curves of the function z = 2− x − y2   for z = 1, 0,−1 .

2. Find a general form for the level curves z = k for the function                            
                       z = (x2 + y)/(x + y2)

3. Find the partial derivatives with respect to x and y of f(x, y) =4eysin x

4. Find the partial derivatives of f(x, y) = x3y2 − 5x + 7y3.  

5.Find all the second order partial derivatives of  f(x, y) = x3y2 − 5x + 7y3.

6. Suppose that the temperature in degrees Celsius on a metal plate in the xy plane is given by   T(x, y) = 4 + 2x2 + y3  where x and y are measured in feet. What is the change of temperature with respect  to distance, measured in feet, if we start moving from the point (3, 2) in the direction  of the positive y-axis.    

7. Calculate lim (x2 − y2)/(x2 + y2)   as you approach the origin
(a) along the x-axis; (b) along the line y=3x; (c) along the parabola y=5x2;

8.  Show that lim (x4-y4)/(x4+y4)   does not exist as you approach the origin.

9. Determine where the function (x4-y4)/(x4+y4) is continuous

10. Determine where the function f(x, y) =x-y+1     is continuous .

11.  Find the gradient of f(x, y) = sin3(x2y)

12. Find the directional derivative of f(x, y) = 2x2 + xy − y2    at p =(3,−2) in the direction a = i − j

13. Find the directional derivative of f(x, y, z) = x3y − y2z2    at p =(−2, 1, 3) in the direction a = i − 2j + 2k

14. Suppose that the temperature T(x, y, z) of a ball of some material
centered at the origin is given by the function
T(x, y, z) =100/(10 + x2 + y2 + z2)
Starting at the point (1, 1, 1), in what direction must you move to obtain the greatest
increase in temperature?

15. Find the stationary points of the function  f(x, y) = x2 − 7xy +12y2 − y

16. Find the stationary points of f(x, y) = x3 + y3 − 6xy

17. Classify the stationary points of  f(x, y) = x3 + y3 − 6xy

18. Let w = x4y2, x = sint, y = t2 . Find dw/dt

19. Find and classify the stationary points of  f(x, y) = xy

Difficult Problems :

20.A rectangular metal tank with an open top is to hold 256 cubic feet of liquid. What are the dimensions of the tank that requires the least material to build the tank?

21. Show the function
f(x,y)  = (x+y)sin(1/x+y) , x+y != 0
           = 0                          ,x+y=0   

22. If u=ln(x3+y3+z3-3xyz) then show that  ux+uy+uz=3(x+y+z)-1

23. If xy+yx=c, find dy/dx

24.  Determine if the domain of the following function is connected

25. Show that the function is discontinuous at the given point
 f(x,y)=(x2+xy+x+y)/(x+y) ,   (x,y) != (2,-2)
          =4                                         ,   (x,y)=(2,-2)

26. Show that the limit lim tan-1y/x  does not  exist as (x,y) approaches (0,1).

27. If  f=f(x,y,z), x=u+v+w, y=uv+uw+wv, z=uvw.
Show that   ufu+vfv+wfw=xfx+2yfy+3zfz

28. Find and classify all the stationary points of f(x, y) = e−(x^2+y^2−4y)

Complete Tutorial

MCQ Quiz #1

Companion MCQ Quiz for Multiple Variable Calculus (Functions, Limits, Continuity)- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz: Multivariate Calculus- Functions of Several Variables

In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables :