Question

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))?

Answer 1

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