Formula Sheet for Midterm 1
Derivatives:
d
1
(csc 1 x)
dx
| x | x2 1
d
1
(sin 1 x )
dx
1 x2
d
1
(tan 1 x)
dx
1 x2
Integrals:
x
2
1
1
x
dx tan 1 C
2
a
a
a
csc xdx ln | csc x cot x | C
secxdx ln | secx tanx | C
tan xdx ln | sec x | C
cot xdx ln | sin
Calculus II Midterm 1 from Winter 2006
Hey Everyone!
Heres last years Midterm 1! TRY IT! i.e. put away all your notes and textbooks,
and give yourself exactly 90 minutes. Well post up solutions next week!
4
1. (6 marks) Evaluate
2 x ln x dx
1
1
Calculus
Calculus II Midterm 1
Feb 8, 2007
Z
1. (6 marks) Evaluate
ln x
dx.
x2
Hi Everyone! Here's last year's midterm for you to try! A few
words of caution:
- last year, the midterm was written out of class so students
had more time and the midterm is longer tha
Calculus II Midterm 2
Mar 17, 2006
1
1. (5 marks) Consider the function f ( x) = x sin x + 4 shown below on the
interval [0, 2].
a) Find the area of the surface obtained by revolving f (x) around the x-axis,
on the interval [0, 2]. Set up the integral onl
Calculus 11 Midterm 2 w 2010 NHL? \JEQgion 2
1. (4 mark each; total 12 marks)
(a) Let f(a:, y, z) = easp(a: yz) + 3:2 sin(4y3 + ln ,2). Find fr): and fxy.
= 8,931+ 0.x MLQ\33+QM7)
L1 = C 1 (-1) x 0 an (Hm+0~2)- 131
(b) Calculate the slope of the tangent
HONOUR HOMEWORK:
Section 3.5: 53, 57 [If you feel you need more practice, # 45, 47,
49, 51 are all doable]
Section 3.6: 15, 29, 37, 45 [If you feel you need more practice, #
3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 31, 33, 35, 39, 41, 43, 47,
49 are al
MATHlOlOU: Chapter 1 cont. 1
FUNCTIONS AND MODELS cont.
Inverse Functions and L0 arithms Section 1.6 . 59 cont.
Recall: WWW sEE ammo par-Es Far. A qmu.
45wa cr- Hiscuem. rum.
Lets now apply this knowledge to nd the inverse trigonometric inctions.
Recall:
MATHl O 1 0: Chapter 4 cont. . . 1
APPLICATIONS OF DERIVATIVES cont
The Mean Value Theorem (Section 4.2 of Stewart, lg. 2802 cam.
Recall: Last class we discussed the Mean Value Theorem and Rolles Theorem.
The Mean Value Theorem: Let f be a function which
MATHI O 1 0: Chapter 5 cont. . . 1
INTEGRATION cont. . .
The Fundamental Theorem 0 Calculus Section 5.3 . 379
cant.
Recall: Last day, we found denite integrals using Riemann sums, which was time
consuming. . .we had discussed that the Fundamental Theorem
Students will know:
4.2: THE MEAN VALUE THEOREM
Use of MVT together with IVT to prove that an equation has only one root.
4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
Locating intervals on which a function is increasing/decreasing;
How to determine if
MATHl O 1 CU: Chapter 3 1
DIFFERENTIATION RULES
Derivatives of Polynomials and Exponential Functions3.1,gl 732
Recall: Last day, we learned how to nd the derivative 'om rst principles.
Now, lets consider some useful rules to help us do differentiation.
De
MATHl O l 0: Chapter 5 cont. . . l
INTEGRALS cont.
The Detmite Integral (Section 5.2 of Stewart, (2g. 3662 cont. .
Recall: Last day, we dened the approximation to the area under the curve as
A = limif(xi*)Ax
71)00 _
and we worked on evaluating the limit o
Syllabus for Calculus I, Fall 2009
Course Information
Course MATH1010U-001, MATH1010U-002, MATH1010U-026
numbe
r
Course Thursday, September 10, 2009 through Wednesday, December 9, 2009
date
Lectur
es
Section 001: UA1350
Mon 12:40 pm 2:00 pm
UA1350 Wed 11:
MATHl O l 0: Chapter 3 cont. . .
DIFFERENTIATION RULES cont
I licit Di erentiation Section 3.5 0 Stewart .207 cont.
Derivatives of Inverse T rigonometric Functions
Question: Earlier in the course, we studied inverse trigonometric inctions. How in the wo
MATHl O 1 0: Chapter 4 1
APPLICATIONS OF DERIVATIVES cont.
Max and Min Values Section 4.1 0 Stewart . 271 cont.
Recall: Last day we introduced the Extreme Value Theorem. _
EUI: .\F E is caMTl-Muws cu [dub] Tu-EL] 4% mg A #55:er Mg/MN
Question: What happ
Students will know:
2.5: CONTINUITY
the definitions of continuity and discontinuity at a point, continuity on an interval;
the concept of continuity from the left and right of a point;
how to determine where a function is continuous, using a graph and alg
MATHl O 1 0: Chapter 4 cont. . .
APPLICATIONS OF DERIVATIVES cont
Indeterminate Forms and L Hos ital 3 Rule 4.4 . 298
g and developed techniques
6
Recall: In the past, weve encountered limits of the form
to evaluate them.
0 KC 0 9! ' 9
Question: What abou
HONOUR HOMEWORK:
This might look like a lot, but several of the questions are very quick to
do.
Section 2.5: 3, 8, 41, 47, 61 [If you feel you need more practice,
# 5, 7, 9, 15, 17, 19, 35, 37, 39, 43, 45, 49, 51a, 53a, 63 are all
doable]
Section 2.6: 7,
MATHl O 1 0: Chapter 5 cont. . . 1
INTEGRATION cont.
Indenite Integrals and the Net Change Theorem (5.4, 2g. 3912
cont.
Agglications
Recall: Last day, we stated the following relationship between velocity and position of
an obj ect:
b
Mod: = s(b)s(a)
a
Th
MATHlOlO: Chapter 3 cont. and Chapter 4 l
DIFFERENTIATION RULES cont. . .
LinearA roximations and Di erentials Section 3.10 . 24 7
Recall: We rst started studying derivatives because we were interested in nding
tangent lines to curves.
2r
The tangent
MATHl O l 0: Chapter 4 cont. . . l
APPLICATIONS OF DERIVATIVES cont
Antiderivatives (Section 4.9 at Stewart, pg. 340)
So far, given a inction, we know how to nd a rate of change, but what if all we knew
was how a function was changing with time, and we wa
MATHl O l 0: Chapter 5 cont. . . l
INTEGRATION cont.
The Substitution Rule (Section 5.5 at Stewart, Qg. 4002 cont.
Question: What about denite integrals? Can we still do u-substitution for these?
We have two options:
1. Deal with the indenite integral for
Practice problems for your own review:
Again, this should be review from high school but make sure you get as
much practice as you need, as the basic skills developed here are a
foundation for the rest of the course. For even more practice, check out
http
MATHl O l 0: Chapter 3 cont. . . l
DIFFERENTIATION RULES cont
Rates 0 Chan e in the Natural and Social Sciences 3. 7 221
Recall: : f(x2) x )
Ax x_ xl
dy Ay
Also, lim
dx NHO Ax
is the average rate of change of y with respect to x
is the instantaneous rate
MATHl O l 0: Chapter 5 l
INTEGRALS
Areas and Distances Section 5.1 0 Stewart . 355
If we think of differentiation as being centered on nding slopes of tangent lines, then
well nd that integration is focused on nding the area under a curve.
Question: How d
Students will know:
3.10: LINEAR APPROXIMATIONS AND DIFFERENTIALS
How to find a linearization (linear approximation) of a function, and use it to
estimate values of a function;
How to work with differentials, and use them to find errors in approximations.