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5 Pages

### 1314L23

Course: MATH 1313, Fall 2010
School: U. Houston
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Word Count: 508

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23 Lesson Partial Derivatives When we are asked to find the derivative of a function of a single variable, f ( x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. With a function of two variables, we can find the slope of the tangent line at a point P from an infinite number of directions. We will only consider two directions, either parallel to the x axis or...

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23 Lesson Partial Derivatives When we are asked to find the derivative of a function of a single variable, f ( x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. With a function of two variables, we can find the slope of the tangent line at a point P from an infinite number of directions. We will only consider two directions, either parallel to the x axis or parallel to the y axis. When we do this, we fix one of the variables. Then we can find the derivative with respect to the other variable. So, if we fix y, we can find the derivative of the function with respect to the variable x. And if we fix x, we can find the derivative of the function with respect to the variable y. These derivatives are called partial derivatives. First Partial Derivatives We will use two different notations: f or f x consider y as a constant x f or f y consider x as a constant y We can also evaluate a partial derivative at a given point. Example 1: Find the first partial derivatives of the function f ( x, y ) = x 2 3xy 2 + 4 y 2 . Then evaluate the first partials at (-1, 2). Lesson 23 Partial Derivatives 1 Example 2: Evaluate the first partial derivatives of f ( x) = 2 x 2 y 2 5 ye x at the point (1, 2). Example 3: Find the first partial derivatives the of function f ( x, y ) = xy . x + y2 2 Example 4: Find the first partial derivatives of the function f ( x, y ) = y 1 x 2 . Lesson 23 Partial Derivatives 2 Example 5: Find the first partial derivatives of the function f ( x, y ) = (x 2 2 xy + 3 y 2 ) . 4 Example 6: Find the first partial derivatives of the function f ( x, y ) = ln 3 x 2 4 y 2 ( ) Lesson 23 Partial Derivatives 3 Second-Order Partial Derivatives Sometimes we will need to find the second-order partial derivatives. To find a second-order partial derivative, you will take respective partial derivatives of the first partial derivative. There are a total of 4 second-order partial derivatives. There are two notations, but we will only use one of them. f xx = 2 f f = 2 x x x 2 f f = 2 y y y f xy = 2 f f = xy y x 2 f f = yx x y f yy = f yx = Example 7: Find the second-order partial derivatives of the function f ( x, y ) = 3x 2 y 2 5 x 2 + 10 y . Lesson 23 Partial Derivatives 4 Example 8: Find the second-order partial derivatives of the function f ( x, y ) = 3x 2 x 3 y 3 + 5 xy + 6 y 3 . From this section, you should be able to Find first order partial derivatives Find second order partial derivatives Evaluate partial derivatives at a given point Lesson 23 Partial Derivatives 5
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U. Houston - MATH - 1313
Lesson 24 Maxima and Minima of Functions of Several VariablesWe learned to find the maxima and minima of a function of a single variable earlier in the course. Although we did not use it much, we had a second derivative test to determine whether a critic
U. Houston - MATH - 1313
1. Given the following graph of a function, f(x).y6 5 4 3 2 1x9 8 7 6 5 4 3 2 1 1 2 3 4 1 2 3 4 5 6 7 8 9 10Find: a. lim f ( x)x 2b.lim f ( x)x 1xc.xlim f ( x)5d.limf ( x) 2+e.lim f ( x)x 1f. f(-2) h. f(2)g. f(1)2. Given the follow
U. Houston - MATH - 1313
Math 1314 Test 1 Review Review Properties of Exponents 1. a 0 = 12. a n =1 n1 an3. a = n am4. a n = n a m 5. a m a n = a m + n am 6. n = a m n , a 0 a 7 . ( a m ) n = a mn 8. (ab) n = a n b nan a 9. = n , b 0 b b 10. For b 1 , b x = b y if and only
U. Houston - MATH - 1313
Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on the curve?y 1
U. Houston - MATH - 1313
Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specific point.We need a way to find the
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives which will make this easier.d [ f
U. Houston - MATH - 1313
Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x ) f ' ( x) dx Example 1: Use the produ
U. Houston - MATH - 1313
Lesson 6 The Chain Rule In this lesson, you will learn the last of the basic rules for finding derivatives, the chain rule. Example 1: Decompose h( x) = (3 x 2 5 x + 6 ) into functions f ( x) and g ( x ) such that h( x) = ( f o g )( x).4Rule 10: The Cha
U. Houston - MATH - 1313
Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ' ' (
U. Houston - MATH - 1313
Math 1314 Test 2 ReviewIn numbers 1 7, find the limit. 1.xlim x3 + 2x 2 1 2 x + 32. limxx3 + x 1 x 13. limxx3 x 1 x 14. Given the following graph of a function f, find lim f ( x) if it exists.x 4Math 1314 Test 2 Review1Limits at Infinity: I
Missouri (Mizzou) - IST - 5
Case 2 a. Most should be close-ended, but should allow customers to provide additional comments. Questions: Visit Frequency Food Quality The order taking process Delivery Speed Potential New Services Overall satisfactionb. Asking open-ended question in a
CUNY Queens - ECON - 247
Bus 247 Homework Set 1 Summer 2009 Queens College Professor Bradbury This assignment is due at the beginning of class on Monday June 15. The assignment must be typed and stapled in order to receive credit. (though graphical solutions may be handwritten in
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
Home Work II Exam II Preparation Assignment This assignment is due in class Tuesday April 29. The assignment will not be accepted if it is turned in late. The assignment will not be accepted unless it is typed. 1) Consider the following Game: Compete Comp
CUNY Queens - ECON - 247
HW III Bus 247 Spring 2008 Professor Bradbury Please answer each of the following questions. This assignment must be turned in typed together with your final paper by May 20th. The assignment will be graded for accuracy.1) When firms are able to achieve
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 1-Fall 2008 Prof. Bradbury 1. The follow equation represents the Total Cost Function for a representative F i rm: TC= C (q) = 480 + 4q^2 a. Fixed Cost = 480 Variable Cost=4q^2b. Expression for Average To
CUNY Queens - ECON - 247
Bus 247 Business Economics Homework I Fall 2008 Prof. Bradbury To the best of your abilities, please answer each question in its entirety. This assignment will be due at the beginning of class on Thursday October, 2. All assignments must be typed and stab
CUNY Queens - ECON - 247
S teven Kordvani Business Economics (BUS 247) HW # 2-Fall 2008 Prof. Bradbury 1. The behavior of average cost which we associate with natural monopoly is falling cost 2. I t makes more sense for one fi rm to satisfy all market demand rather than m any fi
CUNY Queens - ECON - 247
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
CUNY Queens - ECON - 247
1.Profit q2=320 Maximization Reaction Function, 2 firms competing by quantity:q1=R1 q2= 480- q22| q2=R2 q1= 480- q12A. WAY # 1: B. WAY # 2 :q2R1 q2=R2 q1 480- q22 = 480- q12 2q2 32)q2=48023 =12q2=960-480 480 (480 - q22 ) 0 = 480 240-q24 240 = q2 24
CUNY Queens - ECON - 247
Homework III Bus 247/ Fall 2008 Prof. Bradbury This assignment is due at the beginning of class, 12/02/2008. Assignments must be typed and stabled in order to receive credit. 1) The following reaction functions describe the profit maximization problem for
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
CUNY Queens - ECON - 247
Steven Kordvani Business Economics (BUS 247) HW # 4-Fall 2008 Prof. Bradbury 1. When fi rms are able to achieve perfect price discrimination: P1= 3 5 10 a. The value of consumer surplus is zero because all the consumer surplus is t ransferredto the fi rm
CUNY Queens - ECON - 247
Bus 247 Homework Set IV Fall 2008 This assignment is due, together with your final paper, December 16th in class. Assignments must be typed and stabled in order to receive credit. 1) When firms are able to achieve perfect price discrimination, a. What is
Fudan University - ECON - 2965
8940113614111039177d HF yE F 7F 7 d HyEF F 78A-1 830 z 18 138 68 148 33 98 17+78 33 13+11+98 33 6+10+178 33 14+10+178 33 F 33 30 8 F 9 U 30 8 8 F 8 8 8 8 8F=yZ E F 90 8 y 8 33 9 8 x 13 11cfw_ 6 10 17 14 3 7| 8 33888 888 888 888 888 F X= 7i
Fudan University - ECON - 2965
89501 na a 1, a *4 2 n * 1P p&lt;0~aa 2n 1 = a 2 a 15 *9 E 2 n 1 x aa a 2O 8 * ka 8 a = 2k + 1 a 2 2 a = 4k + 4k + 1 2 n 1 = 4k 2 + 4k + 1 8 2 n = 4k 2 + 4k + 2 2 n 1 = 2k 2 + 2k + 1 8 = xO * 8 , 2n 1 n 8 * 2 1O x n aa 2n 1 = a3 a 25 2 * 1 `#E aa a 3O X a2
Fudan University - ECON - 2965
a 89601 i *aE 20 Z3 \$+paE` i *aEA ,B,C*aE i 8 45 X* 55 840 8896028 x1 , x2 , x3 ,., x 7 8 * x1&lt;x2&lt;x3&lt;.&lt;x78 8 x1 + x2 + x3+.+x7=20008 x1+x2 + x 3 8 89603 ABC 8 AM . 8 8 BAC 8 * . ( AE A )AD A A BAC 848A8BMEDC89604 8 E Pe*, * * E Pe*, 88
Fudan University - ECON - 2965
89701 h D@8* 2000 x n 8 (n&gt;2)PC* * +89702 8 O*-ABC D h AO CBC = a, OA = a , AC = b, OB = b , AB = c, OC = c , a + a , b + b , c + c * +C*B89703 A 8 260 8 150 4 8 A 8 * D 45 h 8 45 n + u 0 e D8BCEF89704 8 nn h 8 n=3 n 3x * C 7@ 4 4=2 3 n 2=2
Fudan University - ECON - 2965
j 1 x0 8 89801 8 a , b , c * , d h a + b + c + d a+b = c ,b+c = d ,c+d = a8 * b h 889802 8 * a h x x + [ y ] + ( z ) = 1.5 y + [ z ] + ( x) = 7.7 z + [ x] + ( y ) = 2.6 [a ] H * a h (a ) ca [a ]c 89803 * H 0 @ + B n 3 H c 89804 0 * 0 h E I 8 C H
Fudan University - ECON - 2965
0 89901 8 89902 * 1| 89903 8 @ 7 E B@ C E @ 9 )7 x 2 ux 89904 ABCD cx 8 * ;O&amp; C * AOCD 1| * 1 ( | * 1| 8 89905 8 @ 9 )7 m E @ 9 )7 E @ 9 )7 * 1 | 4* uX 2E 8 (1)9 ) E @ 8 + 3 Y E H E52 8111111111111 1000000000005 + 1n + YE1 | *+=42 u * 2 E 1O |
Fudan University - ECON - 2965
0 891001(8 )8 *E 4 (1)@E9 7, * 2 (2) : *E E@8 8 8 8 8(8 )bt*f2 ( : *E )8 8 88 8 8 8 888 8 8889@8910028 E@ K*E 812 * 3 K*E 40 2 V+ u 4 K*E 5 K*E 60 8 12 * 891003 * n D E 7@ x , k 1 8 2 . 8 k . 8 * t n b2 k 891004 , * t b 2 (8 H * 1~n b2
Fudan University - ECON - 2965
111 8* 0 90110112 @ 7 E k901102*0@&lt; C,D x* (1)x (2)x / C,DE,F x* EFPx xx ODC x xxxCD P x &lt;C,D x* EFP EFP 0*iE 6CPDP CDAEOFBAEOFBBB901103 * 0@*&lt; kE @5 7O 9 kEf @5 7O 9 * 29 PiE 901104 G + + +* .+ G x1, x2, x3, ,x2001 x1= xk+1=
Fudan University - ECON - 2965
12 2K 6 * 0 901201(1)2 3 3 h* S B 2 1~9(h S B 2 K 2*yE (2)2 (1)h* S B 2 1~n (B Sh 2 6 E* 2 E y 9 )S B h* n n hS * B )B Sh* ( 6 *yE E K 6 (*yE 6H 2 *yE n 9 )2)2901202 B 2ABCD 2 2 AEF 22 ABD 2 2 ACD B S ABCD x L B42 12 BE B B ABD 2D E F A B C90
Fudan University - ECON - 2965
*E* 0 901301k Q *1000110000* k ,k = n, k Qn t,t ? Q110000901302 c . 7@E&quot;0 c . 7@E0 99 901303 * 9 17 Q EG @c 7. 0 Ej @c 7. 0 Q 9 901304 0 m * Q200 2 R+* 15m Q 52 R+* m E 4 9 901305 Q * 10 Q 9 * mE , * y ,k 3*2,99 (1)0 E) @c 7. 0 * ( 2)XE m
Fudan University - ECON - 2965
*)F* 0 901401*)F 9 *)F 20~1500 *pR3 B B1 *pR3 901402*F@+9014036 H|*)F @| 9 (1)X* Qa B*pR3 (2)X* Qa r p*)F Q 6 X* a 3 *F@+ B(*pR3 1~6 *pR3 B Q )X a5 *F@+901404 * XFF @ F (9 1234 94 321)H * 2088H 9 901405 C * _ ` F* ABCpR B *ABC 9AC B9 (
Fudan University - ECON - 2965
0 90150119 E H*i ! d ! *iEd H 8.99901502a=1+2+.+109 b=12+22+.+1029 c=13+23+.+103 9 d=14+24+.+104 9 1~10 9 * 2 S D a 1~10 9 * T + 9 S=1 2+1 3+1 4+.+8 10+9 109 T=1 2 3 4+1 2 3 5+1 2 3 6+.+6 8 9 10+7 8 9 109 9 a,b,c,d S9 T 949014034 4 1 4 4 9 4 4 2 4
Fudan University - ECON - 2965
*D* 0 901601 9 32000 9* 32001 ^ 9 901602 x4 0 9 ,9 * 0.5 4 D 9 1.5 9 , * ^37 9,x4 709 100 9 * =,H D , ^ 4 9 ,x4 1 9 ,99 901603 * ^ * 2001 @ 5 * 9 * 2001 ^A* 1 ^9 901604 * ^ ABCDE 9 1, 2, 3, 4, 59 9 9 12+ 34+ 5=_912 B E 53D C49 901605 @D&quot; 7
Fudan University - ECON - 2965
E Gv*/Gv*A0 3 p5 2 42 DBC901701* 9 cC `ABCD ` C c =39 =49 =59 ABCD 9P901702a1,a2,.,a2001 9 2001,2002,2003,.4000,4001 = * ^ E (1)(2001 1) (2002 2) (2003 3) . (4001 2001) a a a a Xb (2)(1 1) (2 2) (3 3) . (2001a2001) a a a cb 9017039C + c ` h
Fudan University - ECON - 2965
0 901801 X * (1) u (2) X] aX xx 12 3 + = 0x x a x+a ax xax 901802 9 n+1 X] 9 A,B,C C L 1, 5 , 5 2 , 5 3 , 5u n * * 2L 6 F A 4 n+1 99 901803 L C BH x DH u u a E9 F9 G9 H9 I9 J 9 DFJ * L 16 4 F A bx AB x AD x EF x BD x(1) a = b (2) a = b + 1 (3) a = b
Fudan University - ECON - 2965
*E* 0 901901100 ) q * 7 b @ G E 50 E * c E * q * ) q * ) x a b c 5 c x c a1 2 3 x 345* abc * x901902p cfw_ y x z x w cfw_* dq = 0.123 xyzw = 0.123xyzw23 xyzw23 xyzw 10 p pxq cfw_ dxx9019034k * + 1 y (1) 9 n d 1,5,9, *y ny an x a1 + a2 + a3 +
Fudan University - ECON - 2965
*@* 0 912001 * I Yx ^ p (k 0 2 9 * 6 2 1 * 48 k3 2@9 912002 ZI * 8 k 3 8 q * 8 q p q p a&lt;b * a,k b3 8 p, q 9 * a,k b3 8 , p* 0 0 k a,k b0 9 912003 CK @e &quot; 6 @GM*p@ @je * h |*4 (1) @U * I (2) 4 * * @I U* 83 k* 83 k912004 &quot; @e 1 1 @ @ 1 8 @ @ U I@
Fudan University - ECON - 2965
0 912101 u (1)9 n (2)9 n nff ( n) f * n n * i `- *9E f ( n) n * (N *9E f ( n)f ( 2307) = 2 + 3 + 0 + 7 = 12 ff 9121029 * 1 : * * N (9E 9 912103 *7 9 912104 )91 N 6(917 + B 24 9f 912105 9 * n : * nf180* n : * n 6 j( 8 i*9E )i*9E 9 (1)9 n = 7
Fudan University - ECON - 2965
0 912201 zC &lt; * M&lt; C z 2 2 2 (1) AH + BK +CP = H B 2+ KC 2 +PA2 (2) AH+BK+CP=HB+KC+PA ABC 9 MH,MK,MP9 9 H ,K,P &lt; z9 912202 * . E z = * . 1E 2 2912203 W1 9 W 2 9 W 3z C 9 &lt; W1 9 W 2 9 W 3 9 *z= N @ * A&lt; C z 829 359 21 9 zC &lt; W1 9 W 2 9 W 3 9 A9 B 9 C C
Fudan University - ECON - 2965
0 912301 9 9 ABCD AB + BM AD + DN AM = AN M9 N9 MAN=45099 912302 ABC E9 F 9 P9 / P: B c) =m9 =n9 / F: B = r9 m9 n9 a9 b9 c 9 EB : r 9 (9 = a 9 = b9 =D99 912303 (1)a9 b9 c9 d9 eR9 a+b9 c+d 9 b+c9 d+e 9 c+d9 e+a 9 d+e9 a+b 9 a9 b9 c9 d9 e(2)a9 b9 c9 d9
Fudan University - ECON - 2965
0 912401 (20021 )2 1 20022002 1 * (1 n1 =n (n1) (n2) 2 1 , n )1 912402 ( HH HH H H H * A B @+ F* ACB t 2 @ B A *C C 912403 1 2x2 11[x]+12=0 1 ([x] * x x=3.8[x]=31 x=-0.4[x]=-11 x=7[x]=7)1 912404A BCD= =10 ABC=100 0 1 CDA=130 0 1=9124058 7 6 5
Fudan University - ECON - 2965
E a * p * x 0 912501 h n n 4 * p8 K n n * p8 K1 912502 * 1 a @a E @ 7 E @a 1.a 2. * 3. &gt; * bd * a a E ac * 0h K 1912503 (1) A BCD A BCD &gt; * * x (2) A BCD A BCD &gt; * * x 2 U + ! x( 3 U + ! x(D F Cx 1) K hx 1) K hAE BB912504 *OH 24 * &gt;*4 *1H O (1)
Fudan University - ECON - 2965
*@* 0 912601 (1) (2) * x 2003200420052006+1 @ * ( 4567+1 , 9899100101+1)x * 912602 * x 1 1x * * l p * a ( @ M @ @ * @ 5 p @ * * l p * * l p * * l p ( ) an 1 bk 1 (1-bk)an 2003 a11 2 x a26 ) b11 b26 b20031 0 1 * 1 @ 2003 / a 2003( / *2003 / 0&lt; b1&lt;b2&lt;b20
Fudan University - ECON - 2965
0912701 1 1 t1, t2, t3, t4, t2003 1 t1=2 , tn+1 = , n=1,2,3,1 912702 1 1A 1 S1 M1,M2 1 300 1 (1 0 = = e1 H d D e )H D ( d e S H D S R9Ed 0 ) S S 1 912703 a,b He D a,b He D 7H de D 7H d e D M21d M2H eDM1K M1A V(1) (2)10a+b 5a+4ba-2b 4a-b7 79127
Fudan University - ECON - 2965
X2 920901 x C + Y E 1 2 382 * J Y * 2 920902 * x C F A *M o F A @ 7 F A @ 7 F A @ 7 * 7 M92 2 9 2(2 28 2 ) 212 22 38 50 2 26)1 + Y E * 9209 xCA F ( x C 121 2 3 4 * 5 6 x CA F 100100 * * J Y100100 1 + Y E 2 920903* 2M 1 7 F A @ * x F A 2M 2 7 F A
Fudan University - ECON - 2965
T 2 921001 + R 8 @ E ) + R 892 2 10 2(2 29 2 ) 21002 22DCAB2 9210021*d t * d t 22 38 * E 100 P ) 62 12 52 921003 @)E @ e @%V+p)E ) @ * d t 2 2 2 2 2 20 10 5 O * 3 t d 2 2 E *) 4 8 2 2 2 2 95 2 1 2 3 4 5 E) 02521021522022921004 1331 = 113
Fudan University - ECON - 2965
2 921101 A 2 + e 2 * p1x 92 2 11 2(2 30 2 ) 2100 &gt; 100 r @ E2 921102 * / &gt; E r *5 h8 gH * L20h8 * gH * L2 921103 2 X d2 p n * /8 ` E r d + e 72 2 * pn 22 9211042 x y y ( * / E rx + x 2 + x8 = y + y 2 + y 8x= y921105 2 ABC 6 / / 6 ABC y APBy B
Fudan University - ECON - 2965
921201 2 Q, N 2 H AM M ABC 2 BC H L 2 AB M AM M AC M P,AB AM AC , , u C AP AN AQAM 1 AB AC =( + )2 AN 2 AP AQ P B M NA2QCL2 921202 Ew 7@, 1 71 2 / , @ 7@E1 71 H * uC * 2 1X 7 * 2 2X 7 @ @ EF @,@ 5C @ E @,2 921203 uC AB 2 Em @, 7@ 1 * 7X 1P 56 2
Fudan University - ECON - 2965
&gt; 3 930201 * (1) * * * 393 3 2 3(3 32 3 ) 31,2,3,4 8 4,3,2,18 1 2 3 43 * ( 2) * * E *1 1r X 1 2 3 1,2,3,4,5,6 6,54,3,2,1X * 1r 4 5 63 * 1r X * E1. *1 2. F@993 930202 Fe 9 @D 2 * E *1 *1r X 7@13471897639233 13 121 3 2 &gt; * 122 X 1r 13 * 1 2X r930
Fudan University - ECON - 2965
393 3 3 3(3 33 3 )3933301 (1) 3 ` F p * Z d ^ T * W ( (2) 3 L F @ 7 ( G13 34 @ 7 F 23 G23 3n @ F n) G13G2933302m w* (1) (2) (3) (4) F@33 3 3m3 n *F n m3 n m3 nn w* (3)2x( 933303 3w p ABCD Phz0* *M PAB3 PBC3 PCD3 PDA F * (3P3933304 * M
Fudan University - ECON - 2965
y3 933401 + V u 1 3* 13K F 7 3* 23K F 7 93 3 4 3(3 34 3 ) 33 933402 3 3 2( 1( 1+2+-+108 * w 1 + 2 + L + 20 8 * 12 + 2 2 + + 10 2 13 + 2 3 + + 20 3933403 3 BD H w 3 C3 AB 3 P 3 CD H w D3 * AB H w PA A CE A E3 F H w Q3 PB A CF A AD3 R3QR / ABCP Q R
Fudan University - ECON - 2965
93 3 5 33 933501 3* N p * @ *j * p * @ * 10 fXI E 2(3 35 3 )3* 10 j * 10 j 10 j 10 3103 933502 3 M3 O3P QQQA BA3 B * Q M P Q 3 C3 D 3 3 A3 B3 * C3 D 3 O P C D MQ3 933503 * 3 3 3* abcde * 13 f W 23 33 3 bcJ K @ E * * f W E IaJ K @ E de33 93
Fudan University - ECON - 2965
3601(1)* C (2)1 3602 @ 8@cfw_E'X*p 1 5* S 3603 1 1 360411111 1PQRS S *yE EFGH 1 PQRS 1ABCD1 E1 F1 G1 H 1AP, BQ, CR, DS 11111 1 1 1 1 1 1 a = 1 + + + + + + + 1 b= S 2345 2003 2004 2005 1003 1004 2005 c = 1+ 3605 S EI P*yE 100 100 @cfw_E) 7H 31 201 11
Fudan University - ECON - 2965
*8=9 &quot;H3701 1A = cfw_ 2, 1,0,1,2,3 , B = cfw_ 1,2,3, 4,5 f :A B x + f ( x) + xf ( x) = 2k + 1, k f ( x) = 1, 2,3, 4,5 H E N *y x B 2, 1,0,1, 2,3 * 8 C* f 8C 3702 1 B 187 3703 1 x+ y =4 2 2 3 3 ( x + y )( x + y ) = 280 3704 5 K 5 AC B P 5 AK C * BP B