Question

Tentukan fungsi F(x) jika, F"(x) = √x dan F'(9) = 25, F(1) = 22

Answer 1

F''(x) = √x = x^1/2
int F''(x) = F'(x) = 2/3 x^3/2 + C
F''(9) = 2/3 (9)^3/2 + C = 25
2/3 * (3^2)^3/2 + C = 25
2/3 * 27 + C = 25
18 + c = 25
C = 7
so,
F'(x) = 2/3 x^3/2 + 7
int F'(x) = F(x) = 2/3 * 2/5 * x^5/2 + K
F(x) = 4/15 x^5/2 + K
F(1) = 4/15 * (1)^5/2 + K = 22
4/15 * + K = 22
K = 22 - 4/15 = 326/15
F(x) = 4/15  x^5/2 + 326/15

Answer 2

F'(x) = integral ( F''(x))
       = interal (√x )dx
       = integral (x pangkat 1/2)
       = 2/3 X pangkat 3/2 + C atau 2/3 (√X pangkat 3)

F'(9) = 25
    25 = 2/3 (√9 pangkat 3) + C
    25 = 2/3 x √729 +C
    25 = 2/3 x 9 + C
    25 = 6 + C  
      C = 19
jadi, F'(x) = 2/3 (X pangkat 3/2 ) +19 atau 2/3 (√X pangkat 3)+19

F(x) = intergral (F(x))
         = interal (2/3 (X pangkat 3/2) + 19 )dx
         =  4/15 X pangkat 5/2 + 19X + C atau 4/5 (√X pangkat 5) + 19X + C
       
F'(1) = 22
    22 = 4/15 (√1 pangkat 5) + 19(1) + C
    22 = 4/15 x 1 + 19 + C
    22 = 99/15 + C  
      C = 231/15

jadi, f(x) = 4/15 (√X pangkat 5) +19x + 231/15