MATH 209 Lab Solutions Richard M. Slevinsky1 Exam Study Guide – Practice Questions with Verified Answers. GRADED A+. Latest 2026/2027 Update

MATH 209 Lab Solutions

Richard M. Slevinsky1

Exam Study Guide – Practice
Questions with Verified
Answers. GRADED A+.
Latest 2026/2027 Update




1

,Contents
1 Multivariable Functions and Limits 3

2 Partial Derivatives 7

3 Directional Derivatives and Gradients 16

4 Maximum and Minimum Problems 20

5 Lagrange Multipliers 27

6 Double Integrals 37

7 Polar Coordinates, Centre of Mass, Surface Area 42

8 Triple Integrals 45

9 Line Integrals 49

10 Green’s Theorem 54

11 Surface Integrals 56

12 Stokes’ Theorem and the Divergence Theorem 61




2

,Lab1


Multivariable Functions and Limits
Problems

1. Find the domain and range of the following functions and sketch their graphs:

(a) z = f(x,y) = px2 + y2;
The domain is {(x,y) : x ∈ R,y ∈ R}.
The range is {z : z ∈ R,z ≥ 0}.

(b) z = f(x,y) = 3x2;
The domain is {(x,y) : x ∈ R,y ∈ R}.
The range is {z : z ∈ R,z ≥ 0}.


(c) x = f(y,z) = p4 − y2 − z2.
The domain is {(y,z) : y ∈ R,z ∈ R,y2 + z2 ≤ 4}.
The range is {x : x ∈ R,0 ≤ x ≤ 2}.




2. Find each of the following limits, or show that the limit does not exist:


;
Let y = mx2. Then we have:




.



3

, But this is different for every value m we choose, so the limit DNE.

;
Let y = mx. Then we have:




.
Clearly, it works for any line y = mx. However, how can we be sure? Switch to polar coordinates:




,


.
Sometimes it is as easy as plugging in the values!



.

Exercises

1. Find the domain and range and sketch the graph for each of the following functions:

(a) z = f(x,y) = 2y;
The domain is {(x,y) : x ∈ R,y ∈ R}.
The range is {z : z ∈ R}.

(b) y = f(x,z) = 3x2 + z;
The domain is {(x,y) : x ∈ R,z ∈ R}.
The range is {y : y ∈ R}.


(c) z = f(x,y) = p12 − x2 − y2.
The domain is {(x,y) : x ∈ R,y ∈√R,x2 + y2 ≤ 12}.

The range is {z : z ∈ R,0 ≤ z ≤ 12}.

2. Find the limit, or show that the limit does not exist:




4