Question

TAYLOR POLYNOMIALS 1. LINEAR AND QUADRATIC APPROXIMATIONS Compute the linear approximation centered at a defined by L(x) = f(a) + f'(a)(x - a) and the quadratic approximation centered at a defined by Q(x) = f(a) + f'(a)(x - a) +- (x - a) 2 for the following functions when available: (a) f(1) = 23/2 with a = 1 (b) f(x) = V3 with a = 4 (c) f(x) = cos(x) with a = 7/4 (d) f(x) = x1/3 with a = 8 (e) f(0) = tan-(x) with a = 0

Answer 1

a) 6(2): 2312, a. 1. bla) = 6(1) 1972 , We Know d (xn)n.xn-1 bln)= dfx's): 274. = 31.23- b'C) = 3.04 B "cesta este cores) - x V) da xy 6"(); Li te u CSScanned with CamScanner

Substhede LCD - 6ca)t 6a) (x-a) = 1+ 31, (x-) - B - 9+) - Ź 6(x)=bCab + b'@).(x-a) + bylca) (x-a)2 . 1 § (2-4)+ & Cx-2x+1) + 34 - + porta b) f(x)= re , azu = x12 oca): 6(u) = (UJY() ' b'(): x 2 12 LC Scanned with CamScanner

x s'(x)= 2x2 b(W)a anche B!ca) 4.xv) - 6'(x) = 2 x 3/2 6"Ca)= (0)47 (2-3) B'(2) - web und zu -16%) = 6(a) + b'm) (x-4) 2+ Ź (x-4) - 7+ Scanned with CamScanner

G(A) = 24 ,62.4) + ) 4 (24)? 9 G61) = -(2-0) ? 4 642W +2 64 . = 62416-88) + X-1+2 1+2 6u 8(x) = argumentere & c) ble) = Copx with a = TI/ + (a). Có (Tu) - 6 b (3) 4 (18%). - Sing . B'(T/) = -sin t) = -412 Escared win "(20) e coral) = a (-sinx) ==cox

b"(W) = - cos (114) = . 5. l(a): 6 + 6) (2-The). 6(4)= + Costa (x-Toyd) + (6) 4 (x-7).)? 962) = (x-76) +- te (2.TVW) + da) f(x)= x 73 , as 6(8) = (23) 3.2 8(x)= 132487 4.x-23 t'C%) = (29)"? 14 27 and the b")= x 2 x ze" 3:4 3* zuze%87 1C Scanned with CamScanner

6'()= -2, b"Co): -2 (23) = 2(3): - Porto Tom b" (B) = ethernet luu |ач Q(x): (x-8) 2 2+ (x8) + 2 Gry) = (4.02 * t rote 2 12 268): 6.+2 e) 6(x)= tan-'62), 2:0 Scanned with CamScanner

blo): tan '(0) 20 b'mm): tan"x)] +72 1+02 b')=de $"cated in * 22(ligten , (+22 6"(*) - ethernet x2 8'60). O Lox): 0+ t. (2-0) = x AW) - 0+ 1x(x-@) + .5(x-o)" - 2 C Scanned with CamScanner