MATH1010: Chapter 2 cont
1
LIMITS AND DERIVATIVES cont
Precise Definition of a Limit (Section 2.4)
Recall: Last day, we discussed the concept of finding the limit
determining how close youd need to be to
sufficiently close to f ( x ) L.
2 x 7,
Example: f
MATH1010: Chapter 2
1
LIMITS AND DERIVATIVES
Now that we have an understanding of functions, we will move on to study limits of
functions, which is the foundation of our future work with derivatives and integrals.
Tangent/Velocity Problems (Section 2.1 of
MATH1010: Chapter 3 cont
1
DIFFERENTIATION RULES cont
Implicit Differentiation (Section 3.5 of Stewart, pg.207) cont
Derivatives of Inverse Trigonometric Functions
Question: Earlier in the course, we studied inverse trigonometric functions. How in the wo
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Substitution Rule Contd (Section 5.5 of Stewart, pg. 400)
Last day we looked at the very important substitution rule. Lets get some more practice doing
more challenging questions.
x3
Example:
Example:
1 e
1
MATH1010: Chapter 5 cont
1
INTEGRATION cont
Indefinite Integrals and the Net Change Theorem (5.4, pg. 391)
cont
Recall: Last day, we took a look at The Fundamental Theorem of Calculus. Lets get
more practice at applying FTC to more indefinite integrals.
E
MATH1010: Chapter 4
1
APPLICATIONS OF DERIVATIVES cont
Max and Min Values (Section 4.1 of Stewart, pg. 271) cont
Recall: Last day we introduced the Extreme Value Theorem.
Question: What happens if the function isnt continuous? Can we still apply the
Extre
MATH1010: Chapter 4 cont
APPLICATIONS OF DERIVATIVES cont
Antiderivatives (Section 4.9 of Stewart, pg. 340)
So far, given a function, we know how to find a rate of change, but what if all we knew
was how a function was changing with time, and we wanted t
MATH1010U: Chapter 1
1
FUNCTIONS AND MODELS
Inverse Functions and Logarithms (Section 1.6, pg. 59)
Inverse trigonometric functions:
1
Recall: sin =
6 2
So, what if youre asked to solve sin ( x)
1
?
2
1
Notation: In the above example, we want x sin 1 .
2
MATH1010: Chapter 5 cont
1
INTEGRALS cont
The Definite Integral (Section 5.2 of Stewart, pg. 366) cont
Recall: Last day, we defined the approximation to the area under the curve as
n
A lim f ( xi *)x
n
i
1
and we worked on evaluating the limit of this inf
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Substitution Rule (Section 5.5 of Stewart, pg. 400) cont
Question: What about definite integrals? Can we still do usubstitution for these?
We have two options:
1. Deal with the indefinite integral for conve
MATH1010U: Chapter 1
1
FUNCTIONS AND MODELS
NOTE: This is a VERY fastpaced review of some of the precalculus knowledge
you should have for this course. To get more practice and make sure you
understand the concepts, check out the precalc package at
htt
MATH1010: Chapter 5
1
INTEGRALS
Areas and Distances (Section 5.1 of Stewart, pg. 355)
If we think of differentiation as being centered on finding slopes of tangent lines, then
well find that integration is focused on finding the area under a curve.
Questi
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MATH1020: Assignment Worksheet #6
What were working on today: Separable Differential Equations; Parametric Curves
Activity 1 (Max 5 min): Your TA will run a quick game to practice identifying whether a given
differential equation is separable or not.
Acti
MATH1020: Assignment Worksheet #10
What were working on today: Double Integrals over General Regions and Taylor Series
Activity 1 (~1520 min): Mary and Robert agree to meet at the coffee shop each day at noon
once their classes are over. They arrive ther
InTutorial Assignment Guide
Learning Objectives: Tutorials in MATH1010 run much like laboratories, where the assignments are
designed to allow you to practice new skills in an applied setting, and use the software relevant to the
course. The key learning
Calculus II Midterm 2
March, 2008
2
1. (3 marks each; total 9 marks) Answer each question in the space provided. You
MUST show your work.
a) Find the cardiac output over the time interval [0, 15] if the dye dilution method is
used with 10 mg of dye and th
Formula Sheet To Be Provided During MATH1020 Final Exam:
Derivatives:
d
sin x cos x
dx
d
sec x sec x tan x
dx
d
cos x sin x
dx
d
tan x sec 2 x
dx
d
csc x csc x cot x
dx
d
cot x csc 2 x
dx
1
d
(sin 1 x )
dx
1 x2
d
1
(sec 1 x )
dx
x x2 1
1
d
(cos 1 x)
dx
Calculus II Midterm 2
June, 2009
2
1. (3 marks each; total 9 marks) Answer each question in the space provided. You
MUST show your work.
1
a) Show that y x cos x is a solution of the differential equation y y sin x .
2
b) Given the polar equation
r 10 , f
MATH1010: Chapter 5
1
INTEGRALS
Areas and Distances (Section 5.1 of Stewart, pg. 355)
If we think of differentiation as being centered on finding slopes of tangent lines, then
well find that integration is focused on finding the area under a curve.
Questi
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Fundamental Theorem of Calculus (Section 5.3, pg. 379)
cont
Recall: Last day, we found definite integrals using Riemann sums, which was timeconsumingwe had discussed that the Fundamental Theorem of Calculus
Calculus I Midterm 1
2009
2
1. (2 marks each; total 8 marks)
(a) What is the domain of the function h(x) =
(b) If cos x f (x)
x
limit, if not, why not?
x
?
3x 1
3
(c) If f (x) = x2 + 1 and g(x) =
can you determine lim f (x)? If so, nd the
x
2x + 3 nd f (
MATH1010: Chapter 3 cont and Chapter 4
1
DIFFERENTIATION RULES cont
Linear Approximations and Differentials (Section 3.10, pg. 247)
Recall: We first started studying derivatives because we were interested in finding
tangent lines to curves.
The tangent li
MATH1010U: Chapter 3
1
DIFFERENTIATION RULES
Derivatives of Polynomials and Exponential Functions(3.1,p173)
Recall: Last day, we learned how to find the derivative from first principles.
Now, lets consider some useful rules to help us do differentiation.
MATH1010: Chapter 4
1
APPLICATIONS OF DERIVATIVES cont
Max and Min Values (Section 4.1 of Stewart, pg. 271) cont
Recall: Last day we introduced the Extreme Value Theorem.
Question: What happens if the function isnt continuous? Can we still apply the
Extre
MATH1010: Chapter 2 cont
1
LIMITS AND DERIVATIVES cont
Limits at Infinity; Horizontal Asymptotes (Section 2.6, pg. 130)
Recall: Previously, we talked about infinite limits and vertical asymptotes.
Horizontal asymptotes, on the contrary, are based on the b
MATH1010: Chapter 2 cont
1
LIMITS AND DERIVATIVES cont
Continuity (Section 2.5 of Stewart, pg. 119)
x 1 x 3
Question: How does f ( x)
x 3
2
compare with g ( x) x 1 ?
Definition: A function f is continuous at a number a if
lim f ( x) f (a )
x a
So, what
MATH1010: Chapter 4
1
APPLICATIONS OF DERIVATIVES cont
Max and Min Values (Section 4.1 of Stewart, pg. 271) cont
Recall: Last day we introduced the Extreme Value Theorem.
Question: What happens if the function isnt continuous? Can we still apply the
Extre