Math 120 Homework 7 (due on Wednesday, Oct 30)
1. In a mining operation, the cost C (in dollars) of extracting each tonne of ore is given by
x
20
+
,
C(x) = 10 +
x
1000
where x is the number of tonnes extracted each day.
(a) Write the total cost function
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Math 120 Midterm 2
Nov 13, 2013
Duration: 50 minutes
Name: . Stu dent Number:
This exam should have 9 pages. No textbooks, calculators, or other aids are
allowed. There are 5 problems in this exam: problem .1 is 18 points and
Full Name: .- Signature:
Student Number:
Math 121] Midterm Test 2 Oct. 26, 2007 50 min.
,r \ 1. Emii of tlieee shortrenswer questions is worth 3 marks. If e eorreet answer is 1written in the
i' J, r R 1 box provided! you get 3. otherwise you eon earn at m
Full Name:
Student Number:
Signature:
Math 120 Midterm Test 2
Oct. 27, 2006
50 min.
1. For each of these short-answer questions, write your nal answer in the box provided. Only
your nal answer will be graded, preliminary work will not be graded.
(a) Simpl
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Math 120 Midterm 1
Oct 9, 2013
Duration: 50 minutes
Name: m . Student Number:
This exam should have 8 pages. No textbooks, calculators, or other a
The University of British Columbia
Final Examinations - December 2010
MATHEMATICS 120
CLOSED BOOK EXAMINATION
Notes, calculators are not permitted.
All eight questions are of equal value.
Time: 3 hours
1. Differentiate each of the following functions and
Be sure that this examination has 12 pages including this cover
The University of British Columbia
Sessional Examinations - December 2012
Mathematics 120
Honours Dierential Calculus
Time: 2 1 hours
2
Closed book examination
Name
Signature
Student Number
I
Dierentiation Rules
Statement
Suppose that f (x) and g(x) are both dierentiable at x0 .
a) Let S(x) = af (x) + bg(x), with a and b constants. Then S(x) is dierentiable at x0
and S (x0 ) = af (x0 ) + bg (x0 ). That is,
d
[af (x)
dx
+ bg(x)]
x=x0
= af (x0 )
c _.
Full Name: M Signature:
Student Numbm:
Math 12!] Midterm Test 2 Oct. 27, E 5|] min.
1. For each of these shnrbanswer questiom, write jmur nal mm in the hx prm'ided. (My
your nal smurf will be gmdcd, preliminary work will not be graded.
(a) Simplify E
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Math 120 Homework 10 (due on Wed, Nov 27)
1. Use the Newton Method to nd the smallest and the second smallest positive roots of the
equation tan x = 4x, with error smaller than 105 .
2. Use Newtons method to calculate 10 with error smaller than 103 .
3. L
Math 120 Homework 8 (due on Wednesday, Nov 6)
1. Find the values of
d2 y
dy
and 2 (a) at the point (7,0), and (b) at the point (32,5), on the curve
dx
dx
x = y 3 4y 2 + 7
if they exist.
2. A tropical storm is approaching a straight coastline at a speed of
Math 120 Homework 1 (due on Tuesday, Sep 10)
1. Recall that we dene the domain of a function f as the set of all real numbers x for
which f (x) is a real number (unless otherwise specied). Find the domains of the
following functions.
(a) f (x) = x2 + 1.
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Math 120 Homework 6 (due on Wednesday, Oct 23
1. Consider the function
f (t) =
at + b
if t < 0
2 sin(t) + 3 cos(t) if t 0
(a) Find all values of a and b that make f continuous at x = 0.
(b) Find all values of a and b that make f continuous at x = 0. Expla
Math 120 Homework 3 (due on Wednesday, Sep 25)
1. Calculate the following limits if they exist. Justify your answer.
x3 + 1
x1 x 1
x4 4
(b) lim 2
x2 x 5x + 6
x17 + x + 3
(c) lim
x 3x17 + 5x12 + 1
(a) lim
(d)
lim x +
x
x2 4x + 1
x3
x3
x+52
|1 2x| |x|
(f) l
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Basic Trig Identities
(1) tan =
sin
cos
1
sin
csc =
(2) sin() = sin
sec =
1
cos
cot =
1
tan
=
cos
sin
cos() = cos
(3) sin( + 2) = sin
cos( + 2) = cos
sin( + ) = sin
cos( + ) = cos
sin( ) = cos
2
cos( ) = sin
2
(4) sin2 + cos2 = 1
(5) sin(2)
The Binomial Theorem
In these notes we prove the binomial theorem, which says that for any integer n 1,
n
n
n
(x + y) =
+m
x y n =
=0
x y m
n
where
=
n!
!(n)!
(Bn )
,m0
+m=n
Here n! (read n factorial) means 1 2 3 n so that, for example,
n
1
=
n
n1
=
n
2
=
Approximating Functions Near a Specied Point
Suppose that you are interested in the values of some function f (x) for x near some xed point x0 . The
function is too complicated to work with directly. So you wish to work instead with some other function F
The Fuel Tank
Question
Consider a cylindrical fuel tank of radius r and length L, that is
lying on its side. Suppose that fuel is being pumped into the tank at
a rate q . At what rate is the fuel level rising?
r
L
Solution
Here is an end view of the tank.
Derivatives of Exponentials
Fix any a > 0. The denition of the derivative of ax is
ah 1
ax+h ax
ax ah ax
ah 1
dx
a = lim
= lim
= lim ax
= ax lim
= C (a) ax
h0
h0
h0
h0
dx
h
h
h
h
where we are using C (a) to denote the coecient lim
h0
ah 1
h
that appears i
Dierentiation Rules
Statement
Suppose that f (x) and g (x) are both dierentiable at x0 .
a) Let S (x) = af (x) + bg (x), with a and b constants. Then S (x) is dierentiable at x0
and S (x0 ) = af (x0 ) + bg (x0 ). That is,
d
[af (x)
dx
+ bg (x)]
x=x0
= af