More generally now, given a multivariable function
f(a1(t), 93205), . . . ,$n(t), t), the total derivative with respect to
one of the variables (here, 15) is dened by
df_ 8fdac1 demg 8f darn 8f
cit82310115 taxBat manat 815
Example 4. Suppose :3 2 t)
5 Some Basic Solution Techniques for Higher-Order
5.1 The Method of Reduction of Order
Suppose we already somehow know one solution, ym), to the
secondorder linear DE:
Llyl = y + p($)y' + may =, 0,
We can always nd a second solution, 1 fungi?
8.8 The Dirac Delta
.Another function that is often seen in applications is an lM'nLlSL
function, representing an im behaviour at a
particular moment of time. To create this function, we rst dene:
Note that the area under 66
1.7 - Solutions to DES
Finally, after all of this setup, were ready to talk about solving these
DES. But what does solving Inean exactly?
Consider the equation 250 + 3 = 5. Using grade-school algebra, we
would use basic arithmetic to solve the equation.
WINTER 2014 MATH*1080 TEST 1
NO CALCULATORS ALLO D
(PRINT) WINTER 2014 MATH*1080
NO CALCULATORS ALLOWE
Surname: :2 (32 (m
Each question in this part has exactly one correct answer.
answer on this test sheet.
WINTER 2014 MATHEMATICS MATH*108O
Lab Quiz #2
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1. (4 marks) Each year, in a national park, the population of wolves increases by 10 individuals, but then
5% of this population is killed by p
WINTER 2014 ' MATHEMATICS MATH*1080 Lab Quiz #1
O CALCULATORS ALLOWED
1. (3 marks) Assume that for a 'ven set of nn- 10 data. points (x, y) that exhibits a linear pattern,
100, and Z :tiyi = 50. Use the formula.
WINTER 2014 MATHEMATICS MATH*1080 Lab Quiz #3
NI CALCULATORS ALLOWED
Surname: , Initials:
1. (2 marks) The gas in a sphercal balloon expands causing an increase in the radius of the balloon
from 1.0 cm to 1.04 cm. Us- differentials to
Example 1. Prove, using the formal denition of a limit, that
_ ., or"
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'th Q< N Gan.-
3Xl 4 N r
'7 v: +0 Ina/Q. M anon 'bn
(=3; -2 1 Udkad. We .
A2. Evaluate: lim .
m>2+ 2 SC
A3. Evaluate: li'm 111(400)
. . _ 7
what 18 cfw_135 f(:1:).
A4. Suppose that f is continuous at a: = 2 and f(2) = 4. Then,
._ 3 _
A5. If f' (2) =2 11113?) gigE, then what is the function f (as)? 'A6.
Warm-Up Quiz (Practice)
Math*1200, Fall 2014
Friday, September 19
Last Name: 8 0 LU T10 N3
0 Ensure that your name and ID Number are clearly printed on the front and back
of this quiz, and check to make sure that
Part A: Quick Answer 10 marks
Read each question carefully. Please ensure that you ll in your nal answers in the
boxes provided at the right side of the page. Each correct answer is worth 1 mark.
A1. True or False: w/x + = +
A2. If f(m) = x4 +
Term Test #1
Math*1200, Fall 2015
Saturday, October 17 at 9:30 am
Instructions for right now:
Ensure that all fields are clearly filled out on the first and second pages of the test.
There are eight pages total, including this p
WINTER 2010 MATHEMATICS MATH*2130 Test #1 - K3
THERE ARE 7 PAGES TO THIS TEST
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Give all approximate (decimal) answers to three signicant gures, unless stated otherwise in the
question. You may quote relevant results
4 LHpitals Rule
ln MATH*1200 we studied limits extensively. We return now to the
discussion of 5 limits and the notion of an indeterminate form.
4.1 Indeterminate Forms
Indeterminate forms refer to results such as _Q when we try to
j _u;i_ p_lt%_ iJLL
2.4 Derivatives and Antiderivatives Involving Arctrig Functions
Weve introduced six new functions. With new functions come new
d ark/05mg fwxotiovxS !
Example 8. Use implicit dierentiation and the Pythagorean
theorem to nd the derivative of the function y
@311 When we nd the v we dont need to worry about including
+0. Why not?
Example 2. Integrate: / tsecgt) dt -"-'- 5 f (39) fl cfw_M (2*) 0H7
Lel' u. = t - mt)
We can use the integration by parts formula creatively to nd some
1 An Introduction to Complex Numbers
We are familiar with quadratic functions of the form
f(a:) = @532 + 633+ c,
Where a, b, and c are PEAJ_ rim . Such functions, as we
saw in high school math, may be re
' -.-'. :1: :I.-. '
2 Inverse Functions
Suppose that f is a function with a (1m of X and 5.
Isaac. of Y. (We sometimes write f : X > Y.) Then, the
inverse function is a function f"1 : Y + X suc
Homework Problems for Math*1030 (F16).
MARCUS R. GARVIE
June 3, 2016
of Mathematics & Statistics, University of Guelph
We use the following abreviations:
d.p. = decimal places, e.g. to 3 d.p. is 3.141
s.f. = significant figures, e.g. to
We review some high school arithmetic and algebra.
Let a, b, c 2 R. Then
a + b = b + a (Commutative Law of Addition)
ab = ba (Commutative Law of Multiplication)
(a + b) + c = a + (b + c) (Asso