Math021, week 1
Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by f (x). Remark 1.2 The objective of our course "Calculus" i
HKUST MATH021 Concise Calculus
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6 Apr 2011
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MATH021 Concise Calculus
Final Examination Solution, Spring 08
Part I: Multiple Choice Questions
Question
1
2
3
4
5
6
7
8
9
10
Answer
d
a
e
c
d
c
b
a
b
c
Question
11
12
13
14
15
Answer
e
a
d
b
c
1. Find the domain of the function f (x) =
Total
(x + 1)(x 3
HKUST
MATH021 Concise Calculus
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HKUST
MATH021 Concise Calculus
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13th Dec 2008
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HKUST
MATH021 Concise Calculus
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17th Dec 2009
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HKUST
MATH021 Concise Calculus
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Midterm Examination
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HKUST
MATH021 Concise Calculus
Final Examination
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21 Dec 2010
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HKUST MATH021 Concise Calculus
Final Examination
26 May 2011
16:3019:30
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Electronic calculators are NOT all
Math021, week 14
Definition 14.1 A series
n=1
an converges absolutely if
n=1
|an | converges.
Theorem 14.2 A series converges if it converges absolutely. proof: Let n=1 an be a series which converges absolutly. Define a+ = maxcfw_an , 0, and a- = maxcfw
Math021, week 2
Theorem 2.1 For every number x, y sin(x + y) = sin x cos y + cos x sin y sin(x - y) = sin x cos y - cos x sin y cos(x + y) = cos x cos y - sin x sin y cos(x - y) = cos x cos y + sin x sin y tan(x + y) = tan(x - y) =
tan x+tan y 1-tan x tan
Math021, week 3
Remark 3.1 loga x is NOT defined when x 0. In other words, the domain of loga is the collection of all positive numbers. Proposition 3.2 For any numbers x, y, a, b > 0 and for any number k, 1. loga xy = loga x + loga y. 2. loga
x y
= loga
Math021, week 4
One-Sided Limits Definition 4.1 Let f be a function and c be a number. If f (x) approaches a number L when x is bigger (smaller) than but close to the number c, we say that f (x) approaches (tends to) L as x approaches c from the right (le
Math021, week 5
Example 5.1 Evaluate the derivative of the function f (x) = solution: For h = 0 and x = 0,
f (x+h)-f (x) h 1 1 1 = h [ x+h - x ] x-(x+h) = hx(x+h) 1 = - x(x+h) .
1 for all numbers x = 0 x
So, f (x) = lim -
h0
1 1 =- 2 x(x + h) x
for all x
Math021, week 6
Chain Rule Theorem 6.1 (Chain Rule) Let f and g be functions. Then, (f g) (x) = f (g(x)g (x) proof (sloppy): For all numbers x and h = 0,
f g(x+h)-f g(x) h = f (g(x+h)-f (g(x) h = f (g(x)+(g(x+h)-g(x)-f (g(x) g(x+h)-g(x) g(x+h)-g(x) h
Thus
Math021, week 7
Definition 7.1 When there is is a quantity changing over time and f (t) is that quantity at time t. f (t) is called the rate of change of the given quantity at time t. Example 7.2 In a chemical reaction chamber, the concentration of a cert
Math021, week 8
Example 8.1 Find all the relative maximum and minimum of the function f (x) = xex - ex - x2 . solution: Since > 0 x > ln 2 or x < 0 = 0 x = 0 or ln 2 f (x) = x(ex - 2) < 0 0 < x < ln 2
f has a relative minimum at ln 2 and a relative maximu
Math021, week 9
Antiderivatives Definition 9.1 If f and F are functions such that F = f , F is called an antiderivative of f . Remark 9.2 If F is an antiderivative of f , so is F + C for any constant function C. Definition 9.3 If F is an antiderivative (o
Math021, week 10
Integration by Substitution Theorem 10.1 If f and g are functions, f (g(x) is an antiderivative of f (g(x)g (x). That is, f (g(x)g (x)dx = f (g(x) + C. proof: nothing different from the orinary chain rule. Example 10.2 Evaluate 2x(1 + x2
Math021, week 11
Example 11.1 Evaluate tan xdx. solution: Let u = cos x so that du = - sin xdx. Then, tan xdx sin = cos x dx x = - du u = - ln u + C = - ln cos x + C. Example 11.2 Evaluate sec xdx. solution: Let u = tan x + sec x so that du = (sec2 x + ta
Math021, week 12
Partial Fractions Example 12.1 Evaluate dx . x2 - 1 we will try finding two numbers A and B such that
solution: Since x21 = -1
1 (x-1)(x+1) ,
1 A B = + (x - 1)(x + 1) x-1 x+1 if possible. If such numbers A and B are there, they satisfy 1
Math021, week 13
Sequences Definition 13.1 An infinite sequence of numbers (or just a sequence) a1 , a2 , a3 , . is usually denoted by the symbol cfw_an . Example 13.2 Let a and d be numbers. A sequence cfw_an defined by an = a + (n - 1)d for all n is us
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MATH021 Concise Calculus
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