Math 144 Sample Final 2
1. (a) Evaluate lim
x1
x1
x+32
(b) Find the horizontal asymptote of the graph of f (x) = 5x sin
1
.
x
2. (a) Dierentiate f (x) = (ln x)x
(b) Dierentiate h(x) = sin x + cos2 (x3 )
(c) Find y if ex/y = x y.
3. Evaluate the following

[7]
1. Newtons law states that F = dp , where p is the momentum of a body. In Newtonian physics,
dt
if the body has constant mass m, its momentum is mv, and Newtons law becomes the familiar
F = m dv = ma. However, in special relativity, the momentum of a

1. A particle moving back and forth along a straight line has position function given by x(t) =
sin (t 1) with t in sec.
[6]
(a) Estimate its instantaneous velocity at t = 1 sec using a table of values (up to two decimal
places).
[2]
(b) Using part (a), w

[8]
1. Use the denition of the derivative to prove that
d
cos x = sin x.
dx
[6]
2. Find the value of k such that the derivative of f (x) = xk has the same value as the function
itself at the point x = 7.
[8]
3. Assume that f and g are dierentiable functio

1. The gravitational force exerted by the planet Earth on a unit mass at a distance r from the
centre of the planet is
GM r
for r < R,
R3
F (r) = GM
for r R,
r2
where M is the mass of Earth, R is its radius, and G is the gravitational constant.
[4]
(a) Ev

[10]
1. Show that
cos 2 tan1 (x) =
1 x2
.
1 + x2
[10]
2. Find y if xy = y x .
[2]
Bonus question: For what value of k does e2x = k x have exactly one solution?

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 35
Back to Snells law
Let v1 be the velocity of light in air and v2 be the velocity of light in water. According
to Fermats principle, a ray of light will travel from a point A i

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 33
Curve sketching!
Its fun!
How would a corner in a curve (where the derivative cannot be taken) be
mathematically found?
Lightning question
Heres the graph of the derivative f

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 26
Feedback
Do you have to do the Riemann sum every single time?
If work is the integral of force, what is the integral of work?
Why cant you just go force times displacement?
Ho

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 34
Whats next
Wednesday: Calculus of baseball and more
Friday: Quiz 4
Monday: Review
Problem 1
The sum of two positive numbers is 16. What is the smallest possible value for the

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 31
Feedback on Quiz 3
Average is about 58%, not bad!
Comments:
Equal signs!
dx, du, etc. missing in integrals
Notation for denite integrals:
Mean Value Theorem: my race with Usai

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 28
Midterm feedback
Assignments are sometimes long
Videos are also sometimes too long
More Socrative
Intermediate Value Theorem
Intermediate Value Theorem
Let f be a continuous f

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 23
Feedback
How to undo the chain rule: substitution
To evaluate an indenite integral, do the substitution u = g (x), du = g 0 (x)dx, for
some function g (x).
You want to choose

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 26
Centre of mass for a 1D discrete system
Centre of mass for a 1D continuous system
Centre of mass for a 1D continuous system
Consider a rod between x = a and x = b with mass de

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 29
Mean Value Theorem
Mean Value Theorem
Let f be a function that is:
continuous over [a, b], and
dierentiable over (a, b).
Then there is a number c 2 (a, b) such that
f 0 (c) =

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 23
Feedback
All right, all right, this is getting tougher!
Part 1 is more obscure for most people (understandably)
Issues with x vs t
Do you need to know how to prove the FTC?
Fu

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 25
Question 1
Calculate the area bounded by the curves y = x 3 and y = x.
Question 2
Calculate the area bounded by x = y 2
4y and x = 2y
y 2.
Question 3
Use a substitution to com

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 20
Feedback
Question 2 was di cult!
Summation notation is quite abstract. but it is extremely useful, and trust me,
if you do maths or physics at a higher level, you will use it

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 19
Feedback
Why do we bother about C ?
Whats the dierence between denite and indenite integrals?
Do we need to memorize the antiderivative formulae?
Dierence between antiderivati

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 22
Feedback
Not too much feedback this week. everyones tired :-)
Notation is hard for some people. Lots of new stu!
Comparison properties?
Denite integrals
The denite integral
Z

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 15
Feedback
Assignment 3 was long! And di cult!
Working with logs is confusing
Change of base formula
Why isnt the derivative of ax just xax
1?
Question 1
If log2 x = y , then lo

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 14
Education is the process of turning mirrors into windows.
Feedback
To nd inverse functions, can you just switch x and y in the original function and
then solve for y ?
Is it a

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 12
Feedback
Chain rule and composite functions are a little tricky!
Need more examples!
Where does Leibniz notation come form?
Yippee for the piano! :-)
Chain rule
Consider the c

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 13
Feedback
Will we ever need to know how to graph implicit functions?
Are there cases where you cannot rearrange to solve for y 0 ?
If y is an implicit function of x, does that

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 16
Feedback
Simplications as in calculating the derivative of cos
1 (x)
are not so obvious
Review of unit circle, Pythogarean Theorem, etc. (see Video 0 R4 Trig
Function) in Supp

Math 144 Sample Final 1
x2 1
1. (a) Consider f (x) =
x3
i. Evaluate lim f (x)
x0
ii. Briey discuss the meaning of the result in terms of the graph of
y = f (x).
3x2 5x + 1
x2 1
i. Evaluate lim g(x)
(b) Consider g(x) =
x
ii. Briey discuss the meaning of th

University of Alberta
Department of Mathematical & Statistical Sciences
MATH 144, Lecture A1, Fall 2015
Calculus for the Physical Sciences I
Instructor:
Office:
E-mail:
Personal webpage:
Course webpage:
Vincent Bouchard
CAB 565
[email protected]

[10]
2
1. Determine the area bounded by the graphs of y = cos x and y = x + 1.
2. Consider a cylindrical water tank, with height 8m and radius 2m, standing upright (that is,
its footprint is a circle).
[7]
(a) If the tank is full of water, what is the min

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 32
Linear approximations
Example: pendulum
Other functions
Special relativity
The mass of an object is
m0
m= q
1
v2
c2
.
Its total energy and kinetic energy are:
E = mc 2 ,
K = m

Math 144 Fall 2015
Lecture A1
Vincent Bouchard
University of Alberta
Lecture 18
Feedback
These problems are really fun!
Seems ok for many of you! Although more examples would denitely help!
More often than not, the most di cult part of related rates probl