Given that f (3) = 9, f′ (3) = 3, g (9) = 3, and g′ (9) = 5, what is the approximate value of g (f (3.02)) ?

F (3) = 9

f' (3) = 3

L (x) = linear approximation to f (x) at x = a

L (x) = linear approximation to f (x) at x = 3

L (x) = f' (a) * (x-a) + f (a)

L (x) = f' (3) * (x-3) + f (3)

L (x) = 3 * (x-3) + 9

L (x) = 3x-9+9

L (x) = 3x

L (3.02) = 3*3.02

L (3.02) = 9.06

So f (3.02) is approximately 9.06 based on the L (x) linear approximation

g (9) = 3

g' (9) = 5

M (x) = linear approximation to g (x) at x = 9

M (x) = linear approximation to g (x) at x = a

M (x) = g' (a) * (x-a) + g (a)

M (x) = g' (9) * (x-9) + g (9)

M (x) = 5 * (x-9) + 3

M (x) = 5x-45+3

M (x) = 5x-42

M (9.06) = 5*9.06-42

M (9.06) = 3.3

So g (9.06) is approximately equal to 3.3 based on the linear approximation M (x)

In summary, this means

g (9.06) = g (f (3.02)) = 3.3

which are approximations

The final answer is 3.3