Question

Determine the value of h’(8)

Answer 1

1.5

Explanation:

At x = 8 :

• f(x) is linear and pass through (5, 7) & (9, 3)
• g(x) is linear and pass through (5, 1) & (9, -3)

Therefore:

`\boxed{(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1) }`

for f(x) :

`(y-7)/(3-7)=(x-5)/(9-5)`

`(y-7)/(-4)=(x-5)/(4)`

`y-7=-(x-5)`

`y=-x+12`

`f(x)=-x+12`

`f'(x)=-1`

`f(8) = -8+12`

`=4`

`f'(8)=-1`

for g(x):

`(y-1)/(-3-1)=(x-5)/(9-5)`

`(y-1)/(-4)=(x-5)/(4)`

`y-1=-(x-5)`

`y=-x+6`

`g(x)=-x+6`

`g'(x)=-1`

`g(8)=-8+6`

`=-2`

`g'(8)=-1`

`\boxed{d((u)/(v) )=(u'v-uv')/(v^2) }`

Therefore,

`h'(x)=(f'(x)\cdot g(x)-f(x)\cdot g'(x))/((g(x))^2)`

`h'(8)=(f'(8)\cdot g(8)-f(8)\cdot g'(8))/((g(8))^2)`

`=(-1(-2)-(4)(-1))/((-2)^2)`

`=1.5`