If she walks around the lake, thats a distance of r = 2 km so would take 2/6 = /3 hours.
If she rows directly across the lake, thats a distance of 2r = 4 km so would take 4/3 hours.
Or: she could row partway and then walk. Lets work that out next:
If s
In practice (like when youre developing a rule of thumb for drug dosages, for example, or car stopping
distances): you choose a convenient a, nd L(x) for x near a, and then graphically, numerically or algebraically
determine the range in which this approx
Well discover a beautiful solution for some indeterminate forms in Section 4.5.
4.3
Lecture 18: Indeterminate forms and lHospitals rule
Last time, we talked about the precise denition of a limit but then we jumped right to the theorem that all
our favouri
In this case, the dierence quotient will be dierent at x = 1, depending on whether h > 0 or h < 0.
Case h < 0:
(1 + h)2 1
2h + h2
f (1 + h) f (1)
=
=
=2+h
h
h
h
which is continuous at h = 0 so we conclude
lim
h0
f (1 + h) f (1)
= lim (2 + h) = 2
h
h0
wher
Midterm Test 3: November 17, 2004, Version A, Solutions.
1
cos(x2 ) dx .
1. (2 points) Using the Midpoint Rule with n = 4, approximate
0
A. 0.9089
B. 3.6356
C. 1.2295
D. 0.3074
E. 0.8437
F. 0.4609
1
12
32
52
72
[cos( 8 ) ) + cos( 8 ) ) + cos( 8 ) ) + cos(
MAT 1320B
Fall 2004, Professor: W. Burgess,
Midterm Test 2: October 27, 2004, the questions.
Version A
dy
1. (2 points) If x2 y 2 + 2xy = 3x , what is dx at (1, 1) ?
A. 1/2
B. 1/4
C. 0
D. 1/4
E. 1/2
dy
dx
2. (2 points) If y = xx+1 , what is
A. 2.7726
B. 5
MAT 1320
Test 1
Student Number:
Time: 80 min.
Only basic scientic calculators are permitted: non-programmable, non-graphing, no
dierentiation or integration capability. Notes or books are not permitted.
Work all problems in the space provided. Use the
MAT 1320
Test 2
Student Number:
Time: 80 min.
Only basic scientic calculators are permitted: non-programmable, non-graphing, no
dierentiation or integration capability. Notes or books are not permitted.
Work all problems in the space provided. Use the
Mmtwwwmwwwwmmwmmwmwrm
_. .WWWWW ; w
NIAT 1320B Fall 2016 October 5th! 11:30 Prof. Desjardins
TEST #1
Nlax = 15
Name:
Student Number:
a Time: 80 min.
0 No calculuton' are permitted.
0 There are 5 multiple choice questions worth 1 mark each and 3 pr
IVIAT 1320B Fall 2016 November 16th, 11:30 Prof. Desjardins
TEST #2
hilax = 15
Name: WW
Student Number:
Circle the DGD which you attend (this is where you will pick up your graded midterm):
Nazanin Nicole. Mona Alex Mona
10:00 (DGDl) 11:30 (DGD?) 13:0
University of Ottawa -
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Name:
_
Class:
MAT1320 - Fall 2016
Class #:
_
Section #:
_
Instructor: Benoit Dionne
Assignment: Assignment 6 (Part B) - Fall 2016
Question 1
University of Ottawa -
2016-10-31, 11)27 AM
Assignment Worksheet
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Name:
_
Class:
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Class #:
_
Section #:
_
Instructor: Benoit Dionne
Assignment: Assignment 5 (part A) - Fall 2016
Question
University of Ottawa -
2016-11-08, 10)52 AM
Assignment Worksheet
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11/8/16 - 10:52:07 AM EST
Name:
_
Class:
MAT1320 - Fall 2016
Class #:
_
Section #:
_
Instructor: Benoit Dionne
Assignment: Assignment 6 (Part A) - Fall 2016
Question 1
f (x) is dened on the entire domain of f
f (x) = 0 when 1 = ln(x) or x = e
so only one critical point: x = e
f (x) > 0 if x < e so increasing there
f (x) < 0 if x > e so decreasing there
conclude that theres a local maximum at x = e: (e, 1/e), which
3.9
Lecture 14: Additional techniques of integration
This material is from Section 5.7 and Appendix G of the textbook. We have already discussed trigonometric
integrals (that is, integrals of the form
sinn (x) cosm (x) dx.
If one of n or m is odd, you us
In this case, the dierence quotient will be dierent at x = 1, depending on whether h > 0 or h < 0.
Case h < 0:
(1 + h)2 1
2h + h2
f (1 + h) f (1)
=
=
=2+h
h
h
h
which is continuous at h = 0 so we conclude
lim
h0
f (1 + h) f (1)
= lim (2 + h) = 2
h
h0
wher
Chapter 2
The Fundamental Theorem of Calculus
We now proceed to outline the major themes of Calculus, a discussion which culminates in the Fundamental
Theorem of Calculus, which is a big part of our motivation for the rest of the course.
2.1
Lecture 3: Th
2.3
Lecture 5: The Fundamental Theorem
So far:
Weve discussed the tangent problem and the area problem
These led to the denition of the derivative and of the denite integral
Weve seen how to approximate the denite integral using Riemann sums, and to de
b
Example 2.29. If (x) is the linear density of a rod, then a (x) dx = m(b) m(a), where m(x) is the mass of a
length b of the rod, measured from any arbitrary starting point.
Example 2.30. Suppose g (x) = f (x) + c, for a constant c. Then g (x) = f (x). T
The number f (ti )(ti+1 ti ) = f (ti )t is the product of the velocity of the particle at time ti and time ti+1 ti .
If the velocity didnt change over that interval, then this number would be exactly the displacement of the particle
over that time interva
We cannot apply the exponential rule, because the base is not a constant. So its not xx (ln(x).
We have y = xx so ln(y ) = x ln(x). We can dierentiate both sides with respect to x, remembering to use the
chain rule when dierentiating ln(y ):
1
y
= ln(x)
Beware!
d 2x
e = e 2x .
dx
The function e2x is going up twice as fast as ex is, so its derivative should
be twice as steep. To make this precise, well use (soon!) the chain rule.
3.3
Lecture 8: Derivatives of Trigonometric Functions, plus product rule,
qu
3.6
3.6.1
Lecture 11: Implicit dierentiation, and specic antiderivatives
The remaining inverse trigonometric functions
We dont tend to worry about the rest, because they can be expressed in terms of the ones we know.
For example,
1
1
y = arccsc(x) x = csc
Chapter 1
MAT1320A: Some notes related to the rst
two lectures.
My apologies for inevitable typos; these notes are just intended to help supplement what we said in class and of
course everything is explained more clearly in the textbook, with nice example
3.4
Lecture 9: The chain rule and more derivatives
See [S, Ch 3.4 and parts of 3.7] for todays lecture.
Last time:
Derivatives of trigonometric functions
Product rule
Quotient rule
Some examples:
Example 3.29. Dierentiation a product of three functions
3.7
Lecture 12: The substitution method for integration
So far we can only calculate indenite integrals when the integrand is a function whose anti-derivative we already
know. This is a fairly small list (see [S, Table 1 in Chapter 5.3], for example).
Tod
3.9.3
A partial fractions example
These examples can be very long to do. Please see the textbook for more examples.
Example 3.99. Find
2x3 + 3x2 + 22x 6
dx
x3 + 3x2 + 12x + 10
Step 1: long division. We compute
2x3 + 3x2 + 22x 6
3x2 + 2x + 26
=2 3
x3 + 3x2