Math 155 - D100 Midterm 2 - Version 3
Simon Fraser University, Department of Mathematics
March 16, 2016
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Functions of Several Variables
Functions of Several Variables
So far we have dealt with the calculus of functions of a single
variable. But, in the real world, physical quantities often depend
on two or more variables, so we now turn our attention to
func
Midterm 2 Review
Integration by Parts
Understand how integration by parts can be used to evaluate
indenite and denite integrals
Integration by Parts
Example
1
Calculate
0
tan1 x dx.
Integration of Rational Functions by Partial Fractions
Understand how to
Maximum and Minimum Values
Maximum and Minimum Values
As we saw for functions of a single variable, one of the main uses
of ordinary derivatives is in nding maximum and minimum values
(extreme values). In this lecture we see how to use partial
derivatives
Limits and Continuity
Limits
Let us compare the behaviour of the functions
f (x, y ) =
sin(x 2 + y 2 )
x2 + y2
and g (x, y ) =
x2 y2
x2 + y2
as x and y both approach 0 (and therefore the point (x, y )
approaches the origin).
Limits
f (x, y ) =
sin(x 2 +y
Partial Derivatives
Partial Derivatives
If f is a function of two variables, x and y , suppose we let only x
vary while keeping y xed, say y = b, where b is a constant. Then
we are really considering a function of a single variable x, namely,
g (x) = f (x
The Leslie Matrix
The Leslie Matrix
In this lecture we discuss an age-structured model for population
growth for a population with discrete breeding seasons.
The Leslie Matrix
Problem
Assume that a population is divided into four age classes and that
40%
Midterm 1 Review
The Area Problem
Understand how the area under a curve or the distance
travelled by an object moving in one dimension can be
described as a limit of a Riemann sum
Understand how to use sigma notation to express Riemann
sums more compactly
Computing Inverse Matrices
Computing Inverse Matrices
There is an important connection between invertible matrices and
row operations that leads to a method for computing inverses.
Computing Inverse Matrices
Example
Find the inverse of A =
2 5
.
1 3
Compu
Systems of Autonomous Equations
Systems of Autonomous Equations
The dierential equations we have considered so far have described
the dynamics of a single quantity. Most biological systems cannot
be fully described without the interaction of multiple quan
The Inverse of a Matrix
The Inverse of a Matrix
In this lecture we investigate the matrix analogue of the reciprocal,
or multiplicative inverse, of a nonzero number.
The Inverse of a Matrix
Recall that the multiplicative inverse of a number such as 5 is 1
Basic Matrix Operations
Basic Matrix Operations
The denitions and theorems in this lecture provide some basic
tools for handling the many applications of linear algebra.
Basic Matrix Operations
If A is an m n matrix - that is, a matrix with m rows and n
c
Linear Systems
Systems of Linear Equations
In this lecture we present a systematic method for solving systems
of linear equations.
Systems of Linear Equations
A linear equation in the variables x1 , x2 , . . . , xn is an equation
that can be written in th
Equilibria and Their Stability
Equilibria and Their Stability
In this lecture we discuss techniques for understanding the
qualitative behaviour of solutions to autonomous dierential
equations without explicitly solving them.
Equilibria and Their Stability
Quiz 6 Solutions
Math 155 - D100 Spring 2016
1. [Now zero points, as it was not written!)] Find the equilibria of the following dierential
equation. For each equilibrium y, find its associated eigenvalue y , and determine whether
y locally stable or local
Quiz 4 Solutions
Math 155 - D100 Spring 2016
1. [5 points] Use partial-fraction decomposition to evaluate the following integral.
Z 3
x
x2 + x 4
I=
dx
(x2 + 1)(x2 + 4)
Solution: The denominator of the integrand is the product of two irreducible quadratic
Quiz 5 Solutions
Math 155 - D100 Spring 2016
1. [4 points] Compute the Taylor polynomial P3 (x) of degree 3 about a = 1 of the function
p
f (x) = x.
p
Then use P3 (x) to estimate the value of 2 (you do not have to simplify your answer).
Solution:
We have
Homework Summary and some Additional Practice Questions
for MT2
Calculus II for the Biological Sciences
Math 155 - D100 (Spring 2016)
Questions are from Textbook (Tan-Menz-Ashlock, Applied Calculus):
1. Here is a summary of past homework questions that I
Quiz 7 Solutions
Math 155 - D100 Spring 2016
1. [4 points] Find the augmented matrix, and use it to solve the following system of linear
equations.
y+x=3
z
y=
1
x+z =2
Solution:
The reduced matrix is the first one below. By replacing (R3 ) by (R6 ) = (R3
6.2.1 The Fundamental Theorem of Calculus (Part 1) and
6.2.2 Antiderivatives and Indefinite Integrals
1. Quote. If you cant run, then walk. And if you cant walk,
then crawl. Do what you have to do. Just keep moving forward
and never, ever give up
(Dean Ka
7.1.1 The Substitution Rule for Indefinite Integrals
1. Quote. Persuasion is often more effectual than force.
(Aesop, Greek fabulist, 6th century BC)
2. Problem. Find
Z
2
2xex dx
3. Hint. What if we think of the dx above as a differential? If
2
u = ex , w
6.1.3 Properties of the Riemann Integral
1. Quote. No human investigation can be called real science if it cannot
be demonstrated mathematically
(Leonardo da Vinci (1452-1519)
2. Two Special Properties of the Integral.
(a) If a > b then
Z
b
Z
f (x)dx =
a
6.2.3 The Fundamental Theorem of Calculus (Part II)
1. Quote.Each problem that I solved became a rule, which served
afterwards to solve other problems
(Rene Decartes, 1596-1650)
2. The Fundamental Theorem of Calculus, Part 2.
If f is continuous on [a, b],
6.3.2 Cummulative Change and 6.3.3 Average Values
1. Quote. The real danger is not that computers will begin to
think like men, but that men will begin to think like computers
(Sydney J. Harris, American Journalist, 1917-1986)
2. The Cumulative or Net Cha
7.1.2 The Substitution Rule for Definite Integrals
1. Quote. In the depth of winter I finally learned that there was
in me an invincible summer.
(Albert Camus, 1913-1960, French philosopher)
2. Substitution Rule for Definite Integrals.
If g 0 is continuou
6.3.1 Areas
1. Quote. It is not worth an intelligent mans time to be in the majority.
By definition, there are already enough people to do that.
(G.H. Hardy)
2. Goal: To find the area between two curves.
3. Theorem: The area A of the region bounded by the