A. Lines
Parametric Eq. of a Line: x = x1 + a(t) , y = y1 + b(t) , z = z1 + c(t)
Symmetric Eq. of a Line: x x1 = y y1 = z z1
a
b
c
1. Find line L that passes through point (x1,y1,z1) & is parallel to
Parametic: x = x1 + a(t) , y = y1 + b(t) , z = z1 + c(
MATH 420 FINAL EXAM DECEMBER 2013 PROF. CAGINALP
No books, notes, calculators, etc. Explain all of your reasoning.
Each problem is worth 20 points.
1. Explain how the exponential and logarithm functions can be used to define b x for
all x in R .
2. a Let
Solutions. MATH 0420 EXAM 2 NOVEMBER 22, 2013
No books, notes, etc. Explain all reasoning.
1. (20 points) Using the limit definition of derivative prove that
d 5x 2 3 10x.
dx
You can use basic theorems on limits (e.g., sum of limits) and do not have to us
MATH 0420 info for the nal exam
Kiumars Kaveh
December 5, 2014
The nal is accumulative but the emphasis is more on Chapter 5 and
Sections 6.1 and 6.2.
There will be 5-6 questions in the test.
You are required to know/study the examples in the sections
SOLUTIONS MATH 420 EXAM 2 NOVEMBER 5, 2012 PROF. CAGINALP
No books, notes, calculators, etc. Write all of your reasoning.
1. Show that the function
fx :
x 2 sin
1
x2
if x 0
if x 0
0
has the following properties:
a Using the differentiation rules (do not n
MATH 0420 info for midterm 2
Kiumars Kaveh
November 12, 2014
The midterm covers: Sections 3.5, 4.1-4.4 and 5.1-5.2
There will be 4-5 questions in the test.
The emphasis of the test is on Chapters 4 and 5 but of course you still
need to know material fr
Math 0420 Homework 3
Guoqing Liu
September 23, 2012
Exercise 3.3.1 Find an example of a discontinuous function f : [0, 1] R where the
intermediate value theorem fails.
Solution: Dene the following function:
cfw_
1
x
if x [0, 2 ]
f (x) :=
1
x + 1 if x ( 2
Calc 3 F2013, Optimization
Optimization
Critical points, local max/min, Lagrange multipliers, absolute max/min
Types of problems
There are 3 types of optimization problems in Calc 3, and they require 3 dierent strategies.
I. Local max/min.
Problem types:
Lon Capa basics
Logging in
Lon Capa can be accessed from any computer with internet connection.
Go to http:/homework.math.pitt.edu/ or http:/homework.phyast.pitt.edu/ (you can also
Google Lon Capa Pitt and click on the rst link)
If youre logging in for th
Calculus 3 Quiz 3 (25 min) NAME:
Problem 1. Find parametric equations of the tangent line to r(t) = (sin(3t),cos(3t),et) at the
point (0, —1,e7'/3).
/ u . ,1. ,
r M? 1 3a,; 2M, 32-» e * >
nW/ﬁw”! meg/"‘9" ’3 a/cnbéth'tqr ([email protected] 4 ,4 4 #01274 _/
Calculus 3 Quiz 2 (25 min) NAME:
Problem 1. Find the plane through the points P(1,1,1)J Q(1,2,3), R(0, 1,0).
‘1 PM a P/azza we need FF”? ’5’
'1 W W P/ahe and q we W/ Vtm‘ﬁﬂ-‘s
I (/1. W415“? (9% +0090Mﬂ-I" v50 7l/7c’: paw;
"Mw‘ ( > ’Liwa! FD/V"g;
: 4,
Calculus 3 Quiz 1 (20 min) NAME:
Problem 1. Find two unit vectors orthogonal to both i+j + k and 2i + k.
(A yea/pg ggmfpmr/ # {My f/l’eh Mmmﬁcﬂaj; £34,002,”
,4 m a
‘l w .- Iq ‘ ,1 , ,1
[(114145” AA J K/ [II]; i/ﬂyi/sl/fk/fl :
44/42,; K {WM} —; 52 z i I
Calculus 3 Quiz 5 (20 min) NAME:
Problem 1. Find the directional derivative Duf(—6,4), where f(:v, y) : sin(2m + 3y) and u =
éb/gi—j).
44W_menn_7
/’ 11 .
" 4’13 ‘— J ' law/ : “:5 2 t sa £7 Ar 4 WW4"
H :‘ e Z J i 2 j A? l 0/ I 44,29}? K
V4 far [Wt meal p
Calc 3 F2013, Flux integrals
Flux integrals
Flux integral is an integral of a vector eld over a surface.
F dS =
S
F (ru rv ) dA
D
General strategy
First determine how many dierent surfaces are there, and break the integral into sum of integrals
over each
Department of Mathematics
University of Pittsburgh
MATH 0420 (Theoretical Calculus)
Midterm 1, Fall 2014
Instructor: Kiumars Kaveh
Last Name:
Student Number:
First Name:
TIME ALLOWED: 75 MINUTES. TOTAL MARKS: 100
NO AIDS ALLOWED. WRITE SOLUTIONS ON THE SP