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

Functions of several variables ,Theorems and Coordinates

After you learn about this topic, you might benefit from these MCQ Quizzes:

Important points to remember :

Euler’s theorem : If a homogeneous function f(x,y) of degree n exists then

1. x(∂f/∂x) + y(∂f/∂y) =nf(x,y)

2. x^{2}(∂^{2}^{2}^{2}y^{2}(∂^{2}^{2}= n(n-1)f(x,y)

Taylor’s Theorem : Taylor’s expansion of one variable can be extended to functions of two variables . If h,k are small

f(x+h,y+k) =f(x,y) +(h ∂/∂x+k ∂/∂y)f+(h ∂/∂x+k ∂/∂y)2f/2! +.......

Jacobians : The
Jacobian matrix is the matrix of all first-order partial derivatives of
a vector or scalar-valued function with respect to another vector.

Lagrange’s method of undetermined multipliers : Introducing how we use this to find the maximum or
minimum value for a function.

Polar Coordinates : It is
a two-dimensional coordinate system in which each point on a plane is
determined by a distance from a fixed point and an angle from a fixed
direction.

Cylindrical Coordinates : It is an extension of polar coordinates, but we extend it into the third dimension as well.

Spherical Coordinates : A
spherical coordinate system is a coordinate system for
three-dimensional space where the position of a point is specified by
three numbers

**Here are some of the problems in the tutorial which will help you apply the concepts introduced in the tutorials :**

Easy Problems :

1. Convert the equation r^{2}

2. If u=(x^{2}^{2}^{2}^{1/2}∂^{2}u/∂x^{2}∂^{2}∂y^{2}∂^{2}∂z^{2}

3. Convert the equation x^{2}^{2}

4. If z=log((x^{2}^{2}∂^{2}∂x∂y=∂^{2}∂y∂x

5. Convert the equation 2x^{2}^{2}^{2}

6. Convert the equation **ρsin φ** = 1 from spherical coordinates to Cartesian coordinates.

^{1/2}

8. Find the Taylor Polynomial of order 3 of f(x) = (x+1)^{1/2}

9. Find du/dx if u=x^{2}^{2}^{2}

Moderately difficult Problems :

10. If f(x,y)=0, show that d^{2}^{2}^{2}^{2}^{3} where p= ∂f/ ∂x , q= ∂f/ ∂y , r= ∂^{2} ∂x^{2} ∂^{2} ∂x ∂y , t= ∂^{2} / ∂y^{2}

11. Find the minimum value of x^{2}^{2}^{2}^{3}

12. Find the extreme values of f(x,y,z)=2x+3y+z such that x^{2}^{2}

13. Expand e^{x}

14. Find the shortest distance between the line y=10-2x and the ellipse(x2/4)+(y2/9) =1 .

15. Find the minimum distance from the point (3,4,15) to the cone x2+y2=4z2

17. Find the point on the plane 2x−3y+5z = 19 that is nearest to the origin.

Difficult Problems :

18.
Suppose the cost of manufacturing a particular type of box is such
that the base of the box costs three times as much per square foot as
the sides and top. Find the dimensions of the box that minimize the
cost for a given volume.

19. The cone z^{2}^{2}^{2}

20. Expand x^{2}

21. Expand e^{x}

22. Expand e^{xy}

23.
Let f(x,y) and g(x,y) be two homogeneous functions of degree
m and n respectively where m0. Let h=f+g . If x ∂h/∂x + y ∂h/∂y
=0, show that f=g for some scalar

24. If u(x,y)=cos^{-1}x ∂u/∂x + y ∂u/∂y= -1/2 cot u

25. In a plane triangle, find the maximum value of cosA cosB cosC

**Complete Tutorial (there is an MCQ Quiz after this):**

Multivariable Calculus Tutorial 2

**Companion MCQ Quiz #1 for this topic- test how much you know about the topic. ****Your score will be e-mailed to you at the address you provide.**

#### Google Spreadsheet Form

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**MCQ Quiz #2 for Multivariate Calculus- Tutorial 2**

Read the questions in the document below. Fill up your answers in the answer submission form below it. Your score will be emailed to you.

#### MCQ Quiz: Multivariate Calculus- Tutorial 2

Google Document

**Answer Submission Form for MCQ Quiz #2**

#### MCQ Quiz #2

Google Spreadsheet Form

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