A Falling Ladder
http:/webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_app_rr_falling_ladder.html
A 16-ft ladder leans against a wall on the vertical axis (h-axis). The bottom of the ladder
slides along the x-axis. Let x be the distance from the b
Math1013 Calculus I, Fall 2012
Homework-3 : Due 10/12/2012 at 11:50pm HKT
Name: Tian TIAN
Answer(s) submitted:
The problems in this homework set cover the basic concept of limits of functions and their calculation. You need to know:
1. the idea of limits:
Derivatives of some basic functions:
(c) ' = 0 c is a constant
( )' = x
1
x
( )' = a
a
x
x
1
1
' = 2
x
x
( x )' = 2 1x
ln a ( e x ) ' = e x
( log a x ) ' =
1
1
( ln x ) ' =
x ln a
x
( sin x ) ' = cos x
( cos x ) ' = sin x
( tan x ) ' = sec
MATH1013 T05B / T05C / T06A / T06B / T06C
Tutorial Exercise (Week 9) Answers
Optimization
1. Of all boxes with a square base and a volume of 100 m3, which one has the minimum
surface area?
the box of dimensions 3 100 m 3 100 m 3 100 m
2. Suppose an airlin
MATH 1013 Tutorial 10
Discussion on selected questions in mid-term
1. Assume
x
f (x)
f (x)
g(x)
g (x)
(a)
d
dx
f and g are dierentiable everywhere, use the given table to evaluate the following limits
-2 -1 0 1 2
1 -4 -6 1 -2
2 -5 1 3 5
-1 1
3 6 10
-5 -2
MATH1013 Calculus I, 2012-13 Fall
Week 11 Worksheet: Newtons method/Anti-derivatives
Name:
ID No.:
(L13-L14)
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Soluti
MATH 1013 Tutorial 9
Review
Graphing y = f (x):
1. identify the domain of f and any symmetries
2. nd y and y
3. nd the critical points of f and identify the behaviour at each point
4. nd where the curve is increasing/decreasing
5. determine the concavity
MATH 1013 Tutorial 13
Review
Fundamental theorem of calculus:
if f is continuous on [a, b], then A(x) =
x
a
Substitutoin rule:
d
f (t) dt satises A (x) =
dx
let u = g(x), where g is continuous,
f (g(x) g (x) dx =
b
g(b)
f (g(x) g (x) dx =
for denite int
MATH 1013 Solutions to Exercises in Tutorial 9
1. from the table
x
(, 3) 3 ( 3, 1)
f (x)
f (x)
0
+
f (x)
inf.pt.
y-intercept: f (0) = 1
(x + 1)2
= x = 1
x-intercept: 0 =
1 + x2
local extreme points: f (1) = 0, f (1) = 2
4
1
0
+
min.
(1, 0)
+
+
0
+
0
inf.
MATH 1013 Solutions to Exercises in Tutorial 12
n
1
1. (a) R = lim
n
n
k=1
k
n
2
1
n
1
k2
3
n
n n
k=1
1 n(n + 1)(2n + 1)
= lim 1 3
n
n
6
2n2 + 3n + 1
= lim 1
n
6n2
2
=
3
n
2k
2
(b) R = lim
2 2+
n
n
n
= lim
n
k=1
8
8k
+
n n2
k=1
8 n(n + 1)
= lim 8 + 2
MATH 1013 Tutorial 12
Review
Definite integrals as riemann sums:
to approximate the net area under y = f (x) over a closed interval [a, b], the interval is partitioned into n subintervals
a = x0 < x1 < x2 < < xn1 < xn = b, then xk = xk xk1 for k = 1, 2,
MATH 1013 Tutorial 11
Review
Newtons method:
1. make an initial guess x0 near a root of f (x) = 0
2. use the rst approximation to get a second, the second to get a third, and so on, by the formula
f (xn )
, if f (xn ) = 0
xn+1 = xn
f (xn )
x(n+2)
x(n+1)
MATH1013 Calculus I, 2012-13 Fall
Week 06 Worksheet: Derivatives
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this worksheet
MATH1013 Calculus I, 2012-13 Fall
Week 07 Worksheet: Applications
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this workshee
MATH1013 Calculus I, 2012-13 Fall
Week 01 Worksheet: Functions (Part I)
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least THREE questions from the following questions! The worksheet must be
handed in at the end of the tutorial
(Solution of this
MATH1013 Calculus I, 2012-13 Fall
Week 10 Worksheet: Mean Value theorem, LHptals rule
o
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Sol
MATH1013 Calculus I, 2012-13 Fall
Week 05 Worksheet: Derivatives
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
( Solution of this workshee
MATH1013 Calculus I, 2012-13 Fall
Week 07 Worksheet: Applications
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this workshee
MATH1013 Calculus I, 2012-13 Fall
Week 09 Worksheet: Related rate problems
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this
MATH1013 Calculus I, 2012-13 Fall
Week 06 Worksheet: Derivatives
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this worksheet
MATH1013 Calculus I, 2012-13 Fall
Week 03 Worksheet: Trigonometry and Limits
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of th
MATH1013 Calculus I, 2012-13 Fall
Week 04 Worksheet: Limits and Continuity
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this
MATH1013 Calculus I, 2012-13 Fall
Week 03 Worksheet: Trigonometry and Limits
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of th
MATH1013 Calculus I, 2012-13 Fall
Week 02 Worksheet: Functions (Part II) and Trigonometry
Name:
ID No.:
(L13-L14)
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(S
Week 02 Worksheet: Functions (Part II) and Trigonometry
Name:
ID No.:
(L13-L14)
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this worksheet will be
MATH1013 Calculus I, 2012-13 Fall
Week 01 Worksheet: Functions (Part I)
Name:
(L13-L14)
ID No.:
Tutorial Section:
Complete at least THREE questions from the following questions! The worksheet must be
handed in at the end of the tutorial
(Solution of this
MATH1013 Calculus I, 2012-13 Fall Week 04 Worksheet: Limits and Continuity
(L13-L14)
Name:
ID No.:
Tutorial Section:
Complete at least TWO questions from the following questions! The worksheet must be handed
in at the end of the tutorial
(Solution of this
Integration
5.1
Antiderivatives
Definition 5.1 If f and F are functions such that F 0 = f , F is called an
antiderivative of f .
Remark 5.2 If F is an antiderivative of f , so is F + C for any constant
function C. Moreover, these are the only antiderivati
Limits
2.1
Limit of a Function
Definition 2.1 Let f be a function and c be a number. If there is number L
such that f (x) approaches L as x approaches c, we say that L is the limit of
f (x) as x tends to c. Symbolically we write
lim f (x) = L
xc
or
f (x)
Application of Derivatives
4.1
Linearization
Theorem 4.1 Let f be a function and a be a number. The function f (a + h)
f (a) is approximated by the linear function f 0 (a)h.
proof:
Since
f 0 (a) = lim
h0
We see that
f (a + h) f (a)
.
h
f (a + h) f (a)
f
Math1013 Calculus I, Fall 2013
Midterm Exam Short Answers
White Version
Question
1
2
3
4
5
6
7
8
9
10
Answer
c
e
a
a
d
a
e
d
c
c
Total
Part I: MC questions.
1. What is the color version of your midterm exam paper?
(a) Green
(b) Orange
(c) White
(d) Yellow
Functions
Definition (L1, Ch1.1)
y = f (x)
MATH 1013 Calculus I Derivatives (Review)
Yichao Zhu
Department of Mathematics, HKUST
I
x - independent variable; y - dependent variable
I
x Domain of definition; y Range.
General Properties
mathematical definiti