Math 1013 Fall 2013-14
Tutorial Exercise 1 (Week 2)
1.
Use interval notation to indicate the domain of
ln x
(a)
(b)
4
1
ln x
(c)
2.
( x 1)( x 2)( x 3)( x 4)
Determine whether each of the following fu
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 4 : Ch. 5 Integration
Q1. (Fundamental theorem of calculus) Evaluate the following denite integrals by the Fundamental
Theorem of Calculus.
2
Z
(a
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 4 - Suggested Solution: Ch. 5 Integration
Q1. (Fundamental theorem of calculus) Evaluate the following denite integrals by the Fundamental
Theorem
1
Math1013 Calculus I
The basics about limits, continuity, and derivatives
1. Find the limits:
3+x 3
(i) lim
x0
x
x3 + x2 sin
(ii) lim
x0
x
(iii)
lim
x
sin x
x
Solution
3+x3
1
1
3+x 3
3+x 3 3+x+ 3
=
1
Math013 Calculus I
The basics about limits, continuity, and derivatives
Basic algebraic tricks (e.g., factor cancelling, limit laws) and the Squeeze Theorem in limit calculation.
Understand the me
Math1013-L2/L3 Calculus I
Week 1 Brief Summary Slides
Functions and Graphs
p. 1/36
Functions
Functions are useful for showing relationships between quantities.
Basic ingredients of a function:
the do
Math1013-L2/L3 Calculus I
Week 2 Brief Summary Slides
Inverse Functions and Graphs
p. 1/?
Vertical Line Test
As every number x in the domain of a function f can give rise only to a
unique function va
Math1013 L2/L3 Calculus I
Week 5-6 Brief Summary Slides
Derivatives - Basic Computation
p. 1/26
Derivative
The derivative f (x) of a function y = f (x) is dened by
f (x + h) f (x)
h0
h
f (x) = lim
w
Math1013 L2/L3 Calculus I
Week 6-7 Brief Summary Slides
More on Differentiation Techniques
p. 1/30
Derivative Formulas & Differentiation Rules
Differentiation of functions built by +, , , of polynomi
1
Math1013 Calculus I
Implicit Dierentiation and Rates of Change
Working with implicit dierentiation and logarithmic dierentiation - just another usage of the chain rule.
Using derivatives as rates
Math1013-L2-L3 Calculus I
Week 11-13 Brief Summary Slides
Antiderivatives/Indenite Integrals, and
Denite Integrals
p. 1/?
Anitderivatives/Indenite Integrals
Differentiation :
Given a function
f
df
n
Math1013 L2/L3 Calculus I
Week 3-4 Brief Summary Slides
Limits of Function Values and Continuity
p. 1/45
Trending Behaviour of Function Values
The fundamental idea of Calculus is to look at the trend
Math1013 -L2-L3 Calculus I
Week 9-11 Summary Slides
Extrema, Graph Sketching, and Other
Applications of Derivatives
p. 1/34
Extreme Values of Functions
Recall that we could locate the maximum (larges
1
Math1013 Calculus I
Implicit Dierentiation and Rates of Change
Working with implicit dierentiation and logarithmic dierentiation - just another usage of the chain rule.
Using derivatives as rates
1
Math1013 Calculus I
Basic Problems on Derivatives
Get use to the limit denition of derivative: (i.e., by looking at slopes of nearby secant lines)
f (x) = lim
h0
f (x + x) f (x) or
f (t) f (x) or
y
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 2 - Suggested Solution: Dierentiation
Q1. (Tangent line of a curve) Find an equation of the tangent line to the curve y = 5 6x 2x3 at
x = 2.
Solut
Math 1013 Fall 2013-14
Tutorial Exercise 2 (Week 3)
1.
Fill in the blanks.
0
3
2
2
2
2
2
2
cos
4
2
sin
6
2
1
2
2
2
tan
2.
Fill in the blanks.
1
(a)
(d)
3.
cot
_ cos 2 1
(b) sec
(e)
1
_ 1 sec 2
1
(c
Math 1013 Fall 2013-14
Tutorial Exercise 2 (Week 3) Answers
1.
Fill in the blanks.
0
4
3
2
sin
0
2
1
2
2
2
3
2
4
2
cos
4
2
1
2
0
2
0
3
2
1
2
2
tan
2.
6
1
3
undefined
Fill in the blanks.
cot
(a)
1
tan
Math 1013 Fall 2013-14
Tutorial Exercise 3 (Week 4)
1.
Evaluate the following limits.
(a)
(b)
(2 x 1) 2 9
x 1
x 1
lim
3t 1 3a 1
t a
Determine whether each of the following statements is true or not. G
Math 1013 Fall 2013-14
Tutorial Exercise 5 (Week 6)
Differentiability
A function f (x) is differentiable at x = a if and only if f ' (a) lim
x a
lim
x a
f ( x) f ( a )
exists, i.e.
xa
f ( x) f ( a )
f
Math 1013 Fall 2013-14
Tutorial Exercise 3 (Week 4) Answers
1.
Evaluate the following limits.
(a)
(b)
(2 x 1) 2 9
12
x 1
x 1
lim
3t 1 3a 1
3
t a
ta
2 3a 1
Determine whether each of the following stat
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 2 : Dierentiation
Q1. (Tangent line of a curve) Find an equation of the tangent line to the curve y = 5 6x 2x3 at
x = 2.
Q2. (Normal line of a cur
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 3 : Ch. 4 Applications of Derivatives
Q1. (Increasing and decreasing functions)
increasing and decreasing.
Determine the intervals where the follo
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 3 - Suggested Solution: Ch. 4 Applications of Derivatives
Q1. (Increasing and decreasing functions)
increasing and decreasing.
Determine the inter
1
Math1013 Calculus I
Some More Basic Problems on Derivatives
Get use to the limit denition of derivative: (i.e., by looking at slopes of nearby secant lines)
f (x) = lim
h0
f (x + h) f (x) or
f (x +
MATH 1013
APPLIED CALCULUS I, FALL 2009
SECTIONS
NAME:
STUDENT #:
A: Professor Szeptycki
B: Professor Toms
C: Professor Szeto
SECTION:
Final Exam: Sat 12 Dec 2009, 09:00-12:00
No aid (e.g. calculator,