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 functions is even, odd or none of them.
(a)
x 2 x 4 x 6 x
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)
x2 dx
=
1
3
Z
(b)
e
1
dx
x ln x
=
/2
Z
sin 2x dx
(c)
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 of Calculus.
2
Z
(a)
x2 dx =
1
3
Z
(b)
e
1
dx =
x ln x
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
= lim
= lim
= .
(i) lim
= lim
x 0 x ( 3 + x +
x 0
x 0
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 meaning of appearing in limit problems.
Understand the c
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 domain, which contains all the input values of the functi
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 value f (x), a curve is NOT the graph of any function if
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
whenever the limit exists.
Geometrically, f (x) is the
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 polynomials, natural
exponential or logarithmic functions can b
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 of change.
Related rate problems - the most important
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
nd
.
dx
Reversing the process:
Given a function
f
nd a
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 trending behaviour
of function values, instead of just the s
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 (largest function value) or
minimum (smallest function value)
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 of change.
Related rate problems - the most important
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
f (x + h) f (x) or
= lim
= lim
= lim
t x
x 0
x 0 x
h
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.
Solution By dierentiating y , we have dy/dx = 6 6x2 . Substi
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
(d) sin 2 cos 2 1
3.
3
(b) sec
1
cos
csc
(f)
cot 2
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. Give an explanation or a
counterexample.
(c)
2.
( x a) 5
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 ( x) f (a )
.
lim
x a
xa
xa
1.
4 x 2 if x 1
Is f ( x)
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 statements is true or not. Give an explanation or a
counter
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 curve) For what non-negative value(s) of b is the line y =
MATH1013-L4L5
Calculus I
Fall 2012
Practical Exercise 1 - Suggested Solution: Basic Algebra, Functions, Limits
Q1. (Solving inequality with absolute values) Solve |x2 10|
Solution
x2 10
This is equivalent to 6
2
x
4
6, 4
4
or
x2
6.
16, 2
x
1
2
x
4.
So, ei
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 following function is
f (x) = x5
54
x.
4
Q2. (First-derivat
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 intervals where the following function is
f (x ) = x 5
Solu
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 + x) f (x) or
f (t) f (x) or
y
= lim
= lim
= lim
t x
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, written notes) is allowed.
ANSWER AS MANY QUESTIONS AS