Need help solving this problem

Question

Suppose that we don't have a formula for g(x) but we know that g(1)=-4 and g'(x)= sqroot (x^2+8) for all x

a) Use a linear approximation to estimate g(0.9) and g(1.1).

(b) Are your estimates in part (a) too large or too small? Choose one.

1)The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie above the curve. Thus, the estimates are too large.

2)The slopes of the tangent lines are positive and the tangents are getting steeper, so the tangent lines lie below the curve. Thus, the estimates are too small.   

3) The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie above the curve. Thus, the estimates are too large.

4) The slopes of the tangent lines are positive but the tangents are becoming less steep, so the tangent lines lie below the curve. Thus, the estimates are too small.

Answer 1

Straightforward using differentials to approximate value of function near a given point. However it is possible to find the solution for this problem as in this video.

Answer 2

g(a+h) ~ g(a) + h*g '(a)

g(0.9). Let a=1 (since we know what g(1) equals) and h=-0.1 so that a+h = 0.9

You know what g(a)=g(1) equals.

You know what h equals.

you know what g'(a) equals.

Now just use the top equation (or should I say the top approximation?) to approximate g(.9)