This lecture.
In this lecture well think about some basic techniques for
understanding the shape of graph based on the
derivatives of a function.
The first derivative test.
What information is contained in the second
derivative?
Concavity.
Points of infle
MH1100/MTH112: Calculus I.
The Quiz: 1:30pm version.
SOLUTIONS
Problem 1:
(3 marks)
(i) (1 Mark.) Let f : R R be a function and let a and L be real numbers.
State the - denition of
lim f (x) = L.
x a
(ii) (2 marks.) Using the denition you gave in part (i)
MH1100/MTH112: Calculus I.
The Quiz: 10:30am version.
SOLUTIONS
Problem 1:
(3 marks)
(i) Let f : R R be a function and let a and L be real numbers. State
the - denition of
lim f (x) = L.
x a
(ii) Using the denition you gave in part (i), prove that
(
)
lim
MH1100/MTH112: Calculus I.
The Quiz: Thursday 10:30am version.
SOLUTIONS
Problem 1:
(3 marks)
(i) (1 Mark.) Let f be a function and let a be a real number. State the
M - denition of
lim f (x) = .
xa
(ii) (2 marks.) Using the denition you gave in part (i),
MH1100/MTH112: Calculus I.
The Quiz: 11:30am version.
SOLUTIONS
Problem 1:
(3 marks)
(i) Let f : R R be a function and let a and L be real numbers. State
the - denition of
lim f (x) = L.
x a
(ii) Using the denition you gave in part (i), prove that
(
)
lim
Calculus.
The Midterm Exam: SOLUTIONS
Problem 1:
(5 marks.)
Determine the following limits. Carefully justify your answer, using standard
properties of limits.
( )
x1
(a)
limx1 cos x2 1
)
(
1
(b)
limx0 sin x cos x
Solution to (a).
First well understand th
This lecture.
This lecture we are going to focus on the concept of the
derivative of a function.
Recall why we needed limits
Remember the main motivation for introducing limits?
We discovered we need limits because when we tried to
write down formulas for
MH1100/MTH112: Calculus I.
Problem set for Week #10.
Solutions.
Problem 1:
Consider the function f (x) = 2x3 3x2 12x. Use the information
contained in the rst and second derivatives of f (x) to draw the graph of
this function.
Solution
The rst thing well
MH1100/MTH112: Calculus I.
Problem list for Week #7.
Tutorial in the week beginning October the 8th.
Solutions
Problem 1: (#2.3.21 from [Stewart].)
Dierentiate the function u(t) =
5
t + 4 t5 .
Solution
d
[u(t)] =
dt
=
=
=
=
d
5
t + 4 t5
dt
1
d1
t 5 + 4 t5
MH1100/MTH112: Calculus I.
Problem list for Week #12.
Solutions.
Problem 1: (Various from Section 6.8 from [Stewart]).
In this problem you are asked to determine a number of limits. If you can
nd a more elementary method than LHospitals rule, please use i
MH1100/MTH112: Calculus I.
Problem list for Week #9.
SOLUTIONS
Problem 1: (#3.1.9 from [Stewart].)
Sketch the graph of a function which is continuous on [1, 5], and has all the
following properties:
It has an absolute maximum at 5.
It has an absolute mi
MH1100/MTH112: Calculus I.
Problem list for Week #6.
Solutions.
Problem 1:
Use the denition of the derivative to determine the derivative function of
f (x) = x3 3x + 5.
State the domain of the function and the domain of the derivative.
Solution
The deriva
Problem #1.
Suppose that and are continuous functions such that 2 = 6 and
lim 3 + () = 36.
2
Find (2).
Solution.
Because and are continuous functions we know that:
1.
The limit lim () exists and equals (2).
2.
The limit lim exists and equals (2).
2
2
Thus
MH1100/MTH112: Calculus I.
Problem list for Week #11.
Solutions
Problem 1: (#6.1.9 to #6.1.14 from [Stewart].)
Which of the following functions are 1-1. Briey justify.
(i) f (x) = x2 2x.
(ii) f (x) = 10 3x.
(iii) g (x) = 1/x.
(iv) g (x) = |x|.
(v) h(x) =
MH1100/MTH112: Calculus I.
Solutions to Problem list for Week #3.
Problem 1:
(i) For every a, b R, |ab| = |a|b|. Prove this identity by checking all the
possible cases for the signs of a and b.
(ii) For every a, b R, with b = 0,
a
b
=
|a|
|b | .
Prove thi
MH1100/MTH112: Calculus I.
Solutions to the Problem list for Week #4.
Problem 1: (Based on #1.7.30 from [Stewart])
In this problem well prove the limit limx2 (x2 + 2x 7) = 1.
(i) Find a number C such that if 0 < |x 2| < 1, then |x + 4| < C .
(ii) Use your
MH1100/MTH112: Calculus I.
Problem list for Week #8.
Solutions.
Problem 1: (#2.5.5 from [Stewart])
Identify the following function as a composition h(x) = f (g (x) then use the
chain rule to dierentiate it:
h(x) = sin x.
Solution
To present h(x) as a comp
MH1100/MTH112: Calculus I.
Solutions to the Week #2 problem set.
Solution to Problem 1.
In this problem we are asked to guess the slope of the tangent to the graph
x
of the function f (x) = 1+x at the point P (1, 1 ) by looking at the slopes of
2
various
MH1100/MTH112: Calculus I.
Solutions to the Week #1 problem set.
Problem 1: (Problem 1.1.35 from [St].)
What is the domain of
f (x) =
4
1
?
5x
x2
Solution:
This expression makes sense when x2 5x > 0. To understand when this is
true, we begin by writing