University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Substitution Rule (Section 5.5) cont
Last day, we introduced the concept of substitution for tougher integralslets continue.
Example:
2x3
x 4 1
dx
Often, the hard part is figuring out what substitution to ma
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Assignment Worksheet #8
What were working on today: Optimization, Antiderivatives, and Numerical Integration
Activity 1 (~15 min): There is a traditional problem that goes like this We want to make an
open topped box from an
8.5 11
inch sheet of
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Assignment Worksheet #3
What were working on today: IVT, Infinite Limits, Derivatives, and Differentiation Rules
Activity 1 (max 10 min): Get warmed up by playing a quick game to practice your basic skills
regarding limits at infinity and the de
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Assignment Worksheet #4
What were working on today: Derivatives (and getting lots of practice!)
Activity 1 (max 10 min): Come up with a question asking for the derivative of a function (the
function has to be complicated enough so that the deriv
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Assignment Worksheet #5
What were working on today: Applications of Derivatives (incl. Related Rates)
Activity 1: A plane is cruising at an altitude of 2 miles at a distance of 10 miles from an airport.
Choosing the airport to be at the point (0
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont; 6
INTEGRATION cont
The Substitution Rule (Section 5.5) cont
Last day, we further explored the very important substitution rulelets try some more
challenging questions involving usub.
x3
Example:
Example:
1 e
1 x
dx
x
2
dx
1
MAT
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
Indefinite Integrals and the Net Change Theorem (Section 5.4)
cont
Applications
Recall: Last day, we stated the following relationship between velocity and position of
an object:
b
v(t )dt s(b) s(a)
a
The Net C
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Fundamental Theorem of Calculus (Section 5.3)
Recall: So far, we have been interested in finding derivatives (slope) and integrals
(area)how are these two things related?
x
Consider the function
g ( x ) f (t
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Substitution Rule (Section 5.5) cont
Last day, we introduced the concept of substitution for tougher integralslets continue.
Example:
2 x3
x4 1
dx
Often, the hard part is figuring out what substitution to ma
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRALS cont
The Definite Integral (Section 5.2)
Recall: Last class, we were interested in computing the limit of an infinite sum. Just like
f ( x h) f ( x )
earlier in the course when lim
came up frequently and was given the
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Fundamental Theorem of Calculus (Section 5.3)
Recall: So far, we have been interested in finding derivatives (slope) and integrals (area)
how are these two things related?
x
f (t )dt with a x b
Consider the
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont; 6
INTEGRATION cont
The Substitution Rule (Section 5.5) cont
Last day, we further explored the very important substitution rulelets try some more
challenging questions involving usub.
Example:
y g (x)
Example:
1 e
dx
x
And finall
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Chapter 5 cont
1
INTEGRATION cont
Indefinite Integrals and the Net Change Theorem (Section 5.4)
cont
Applications
Recall: Last day, we stated the following relationship between velocity and position of
an object:
b
v(t )dt s(b)
s (a)
a
The Net
University of Ontario Institute of Technology (UOIT)
Calculus
ENGINEERIN 1010u

Fall 2012
MATH1010: Assignment Worksheet #6
What were working on today: Applications of Derivatives (Linear Approximations and MVT)
Activity 1: Instead of being satisfied with a linear approximation to a function, it is possible to
find a quadratic or higherorder