Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Nov. 3, 2008
1. A farmer has 3000 meters of fence, and needs to make a rectangular animalshed, with a partition
in the middle, parallel to one of the sides, using the same fencing material. What are the dimensions
of the rectang
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solution to the QUIZ for Dec. 3, 2009
By using Stokes Theorem, or otherwise, evaluate
C
F dr, where
F (x, y, z) = (yz + 2y + 3z)i + (xz + 2x + 4z)j + (xy + 3x + 4y)k ,
where C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Dec. 4, 2008
1. Find the area of the region enclosed between y = x2 2x and y = x 2.
Sol. of 1: We rst nd the points of intersection by setting them equal to each other
x2 2x = x 2 ,
means
x2 3x + 2 = 0
.
Factoring, gives
(x 1)(x
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #17.3 (2nd ed.) [The Divergence Theorem]
By Doron Zeilberger
Problem Type 17.3a: Use the Divergence Theorem to calculate the surface integral
where
S
F d S,
F(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z)k ,
where S is a surf
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #17.1a (2nd ed.) [Greens Theorem]
By Doron Zeilberger
Problem Type 17.1aa: Use Greens Theorem to evaluate the line integral along the given positively oriented curve.
P (x, y) dx + Q(x, y) dy
,
C
where C is a given curve.
Example Pr
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #15.5 (2nd ed.) [Change of Variables in Multiple Integrals]
By Doron Zeilberger
Problem Type 15.5a: Find the Jacobian of the transformation
x = g(u, v, w) ,
y = h(u, v, w) ,
z = k(u, v, w).
Example Problem 15.5a: Find the Jacobian o
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #16.4 (2nd ed.) [Parametrized Surfaces and Surface Integrals]
By Doron Zeilberger
Problem Type 16.4a: Find an equation of the tangent plane to the given parametric surface at
the specied point.
x = x(u, v) ,
y = y(u, v) ,
z = z(u, v
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #15.4b [Integrations in Cylindrical and Spherical Coordinates]
By Doron Zeilberger
Problem Type 15.4ba: Evaluate
F (x, y, z) dV
,
E
where E is a solid region descibed in terms of cylinders and other stu.
Example Problem 15.4ba: Eval
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #15.4a (2nd ed.) [Double Integrals in Polar Coordinates]
By Doron Zeilberger
Problem Type 15.4aa: Evaluate the integral
F (x, y) dA
,
D
where D is a region best described in polar coordinates,
D = cfw_ (r, )  , h1 () r h2 () .
Exa
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #14.8 (2nd ed.) [Lagrange Multipliers: Optimizing with a Constraint]
By Doron Zeilberger
Problem Type 14.8a: Use Lagrange multipliers to nd the maximum and minimum values of
the function subject to the given conditions.
f (x, y, z)
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #15.2 (2nd ed.) [Double Integrals over General Regions]
By Doron Zeilberger
Problem Type 15.2a: Evaluate the double integral
F (x, y) dA
;
D = cfw_ (x, y)  a x b, f (x) y g(x) .
F (x, y) dA
;
D = cfw_ (x, y)  a y b, f (y) x g(y)
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Zs Math251 Handout #15.1 (2nd ed.) [Integration in Several Variables]
By Doron Zeilberger
Problem Type 15.1a: Calculate the iterated integral
b
d
f (x, y) dx dy
a
.
c
Example Problem 15.1a: Calculate the iterated integral
2
4
(x +
1
y) dx dy
.
0
Steps
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Last Handout and SCC Problems : The Art of Checking Your Answer and Avoiding Nonsense
By Doron Zeilberger
Note: The exercises here are mandatory for members of the SCC II. There are strongly recommended to everyone!
Avoid Grammatically Incor
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 6 Handout:
Using the Laplace Transform to solve Systems of Linear Dierential Equations
By Doron Zeilberger
Systems of ODEs
When we have one dierential equation, with one unknown function y(t) and we use the Laplace
Transform method,
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 5 Handout: The Dirac Delta Function
By Doron Zeilberger
Important Denition: The Dirac Delta Function, (t) is a function that is zero everywhere
but at t = 0. Its shift, (t t0 ) is zero everywhere and innity at t = t0 .
Important Form
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 3 Handout: Translation Theorems for the Laplace Transform
By Doron Zeilberger
Version of Sept. 15, 2011 (thanks to Joey Reichert).
Part A: Translation along the saxis s s a.
Important Formulas
If, as usual Lcfw_f (t) = F (s), then
L
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 4 Handout: Operational Properties for the Laplace Transform
By Doron Zeilberger
Important Formula
Once we know the Laplace transform F (s) of some function f (t) we can immediately gure out the
Laplace transform of tn f (t) for any p
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 2 Handout: The Inverse Laplace Transform and Derivatives
By Doron Zeilberger
Theory: The Laplace Transform is a dictionary that goes from functions of t (usually time) to
functions of s. It is often necessary to be able to translate
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Dr. Z.s Calc5 Lecture 1 Handout: Denition of the Laplace Transform
By Doron Zeilberger
Theory:
The Denition of the Laplace Transform.
Input: A function f (t) dened on the nonnegative real axis [0, ).
Output: Another function, of s, given by :
f (t)ets dt
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Pat Q. Student
AME 60611
28 August 2015
A
A
This is a sample le for the text formatter L TEX. I require you to use L TEX for the following reasons:
It produces the best output of text, gures, and equations of any program I have seen.
It is machineindep
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Sept. 21, 2009
1. Find the limit if it exists, or show that the limit does not exist.
lim
(x,y)(0,0)
2x
2x + 3y
.
Sol. We rst try to plugitin, and get 0/0, which is indeterminate.
We next try to prove that the limit does not ex
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Sept. 28, 2009
1. Find the directional derivative of the function f (x, y, z) = xy 2 z 3 at the point (2, 1, 1) in the
direction 2, 1, 1 .
Solution: We rst nd the gradient
f.
f = fx , fy , fz
.
Since fx = y 2 z 3 , fy = 2xyz 3 ,
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Sept. 24, 2009
1. Compute the partial derivatives with respect to x and y.
z = ln(x2 + y 3 ) .
Sol.
z
x ,
alias zx is obtained by treating x as the variable, and y as constant. By the chainrule:
zx =
Analogously,
chainrule:
z
y
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ of Sept. 18, 2008
1. Explain why x4 + x 3 has a real root in the open interval 1 < x < 2.
Solution:
We are going to use the Intermediate Value Theorem (IVT).
1) f (x) is continuous (since it is a polynomial).
2) Pluggingin at the en
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for SEPT. 17, 2009
1, Find the curvature for
r(t) = sin t i + cos tj + t k .
Solution: In conventional notation (that I nd easier)
r(t) = sin t , cos t , t
The formula for the curvature is
(t) =
.
r (t) r (t)
r (t)3
We have
r (t)
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solution to the QUIZ for Monday, Sept. 8, 2008
1. : (a) Draw the function
x + 2,
7,
f (x) = 2x + 2,
5,
3x 2,
if
if
if
if
if
x < 1;
x = 1;
1 < x < 2;
x = 2;
x > 2.
Solution. I am too lazy to use a graphing program to plot it, but let me tell you how to
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ for Sept. 15, 2008
1. Evaluate the limit if it exists:
lim
x2
2x + 5 3
x2
Solution: First plugin x = 2 and see what happens. You get 0/0. So this is indeterminate, and
we must simplify. The conjugate of 2x + 5 3 is 2x + 5 + 3 , so w
Birla Institute of Technology & Science, Pilani  Hyderabad
Elementary calculus
MATH 6661

Spring 2013
Solutions to the QUIZ of SEPT. 14, 2009
1. Find a parametric equation for the tangent line to the curve with the given parametric equation
at the specied point
x = cos t
,
y = sin t
,
z = t2 + 1 ;
(1, 0, 1)
Solution: First we write it in vector form
r(t)