suppose f(2)=-3, f'(2)=10, f'(4)=6 and g'(2)=1. Evaluate the following: a. (f^2-g)'(2) b.(f(g))'(2)

Answer 1

Given the following information...

f(2) = -3

f'(2) = 10

f'(4) = 6

g'(2) = 1

(f²-g)'(2) = (f²(2))' - g'(2) = 2f(2)*f'(2) - g'(2) = 2 * (-3) * (10) - 1 = -61

(f(g))'(2) = f'(g'(2)) = f'(1)

Since we don't have f'(1) we will attempt a linear interpolation from f'(2) and f'(4)

[ f'(4) - f'(2) ] / [ 4 - 2 ] = (6 - 10)/2 = -4/2 = -2 (slope of line connecting derivative points...

y = mx + b --> 6 = (-2) * (4) + b --> b = 14

So y = -2x + 14 for the linear interpolation of slope values...

When x = 1 --> y = 12 which we would use to estimate f'(1)

f'(g'(2)) = f'(1) = 12