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### 13147

Course: MATH 1313, Fall 2010
School: U. Houston
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Word Count: 270

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7 Lesson 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 ' ' ( x). To find the second derivative, we will apply whatever rule is appropriate given the first derivative. Similarly, the third derivative is the derivative of...

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7 Lesson 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 ' ' ( x). To find the second derivative, we will apply whatever rule is appropriate given the first derivative. Similarly, the third derivative is the derivative of the second derivative, the fourth derivative is the derivative of the third derivative, the fifth derivative is the derivative of the fourth derivative, etc. The second, third, fourth, fifth, . . . derivatives of a function are collectively called higher order derivatives. In application, we will mostly use the first and second derivatives. If the derivative represents a rate of change, the second derivative can be used to how determine fast the rate of change is increasing or decreasing. For example, if costs are rising, the first derivative will give the rate of change of the costs, and the second derivative will give the rate of change of increase or decrease. Example 1: Find the second derivative: f ( x) = 4 x 5 0.3x 4 + 2 x 2 7 x + 5. Example 2: Find the second derivative: f ( x) = 3 . ( x 7 )2 Lesson 7 Higher Order Derivatives 1 Example 3: Find the second derivative: f ( x) = (x 3 + 8) . 4 Example 4: Find the third derivative: f ( x) = x(3 x + 1) 3 . Lesson 7 Higher Order Derivatives 2 Example 5: Find the second derivative: f ( x) = ln(2 x 3) Example 6: Find the second derivative: f ( x) = e 3 x + 2e x From this lesson you should be able to Find a higher order derivative of a function Lesson 7 Higher Order Derivatives 3
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Fudan University - ECON - 2965
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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
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Fudan University - ECON - 2965
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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
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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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
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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
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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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
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Fudan University - ECON - 2965
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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
Fudan University - ECON - 2965
0u*=A3801GT F29 , 37 , 52 hB3802G cfw_G 10,G 9,G 8, 1,1,2,3,8,9,10A R+ R+A 1,nE@ 3803cfw_G n,G (n-1),G (n-2),1,1,2,3,n- c ( b10 F @ 71 a e d hj iabcd efg3804 E * E (1) . (2) 3805AE@ . 2GHG G G DF ACG G GG ABC * c FE G CE EG = FD FGFBC G
Fudan University - ECON - 2965
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Fudan University - ECON - 2965
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Fudan University - ECON - 2965
+ (*, 4101C207B1 15ABCD A A AB =150 AD =240 BC =70 CD =200 ABD + BDC = 9000 0 ABCD 8 JD410224 2AA 20 + ! U 7 F @m(1m2005)w 20 8* J41030 x,y 8J 4104100 0,0 B0 cfw_(x,y)7 | x20 y20B0 a , b, c A111 1 += ,0 a b c a+b+c: 0 nA,1 a2 n +1+
Fudan University - ECON - 2965
4201 ABCD B ABD = 50o DBC = 20o DAC = 10o CAB = 40o ACD =DCA4202 H * * p M ; 1 1003 H AH B2005 B 1 B 2005 1H B B 4203 1. M C U H H N 1003 1B AA DY T GE H O B (B )S X DW O YE V O B (B )L E UB (B )B (B )X @+p)@AF * * p 1M ; F )@A9U \ A
University of Illinois, Urbana Champaign - CS - 105
FinalCS105Spring 2010May 7th, 2010DO NOT START UNTIL INSTRUCTED TO DO SO. YOU WILL LOSE POINTS IF YOU START WORKING ON THE TEST BEFORE WE TELL YOU. THIS IS A 150 MINUTE EXAM.Do not leave this blankfill it in now:Name: Discussion Section: TA:FORMA
University of Illinois, Urbana Champaign - CEE - 105
rF 1 ( 3)( O,O C ( t-,TFfrrfTF ( Z S O ( .1,'1' r i., , ,I II tr - r.r + g . . , -'r ; r . r'r rri s lDE2 f G 7 3 . !;.i '; c .i( i',rr'r r2 e eC@ .2 6 r '1-,r-.rr ' ' ' C :1i;7 4 t i.t'-z': 5 0 , A.)l.r?)'2-'t,;.'r'r'r t !)ig,-rf. r. ,'l 3 @ )ag)
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Midterm 1CS105Fall 2010September 28th, 2010DO NOT START UNTIL INSTRUCTED TO DO SO. YOU WILL LOSE POINTS IF YOU START WORKING ON THE TEST BEFORE WE TELL YOU. THIS IS A 60 MINUTE EXAM.Do not leave this blankfill it in now:Name: Discussion Section: TA:
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Midterm 2CS105Fall 2010November 2nd, 2010DO NOT START UNTIL INSTRUCTED TO DO SO. YOU WILL LOSE POINTS IF YOU START WORKING ON THE TEST BEFORE WE TELL YOU. THIS IS A 60 MINUTE EXAM.Do not leave this blankfill it in now:Name: Discussion Section: TA:F
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