Linear approx questions

Question

Hello,

I would appreciate any help with this. I am just having trouble navigating these type of questions:

1.Suppose that we don't have a formula for g(x) but we know that g(2) = −1 and g'(x)=sqrt( (x^2)+5)

 for all x.

(a) Use a linear approximation to estimate g(1.99) and g(2.01).

2.find the linear approx of (1+x)^1/3 at a=0.

b)use it to approximate the numbers of (0.95)^1/3 and (1.1)^1/3

Answer 1

for this approximation we start at x=2 and then move forward or backward using the slope of the tangent line at 2. so g(1.99) = g(2) + (-0.01)(the derivative evaluated at 2).

Since the derivative at 2 is sqrt(2² +5) we calculate it as 3.

so , now g(1.99) = g(2) + (.01)*3) or -1 + -.03 = -1.03

similarly, g(2.01) = -1 + (3))(.01) or -0.97

For part 2. i am not sure what at a=0 refers to. lets assume we want to approximate from x=0. Then f(0) = 1.

Now, we must find the derivative odf f(x) which is 1/3 (1+x)^-2/3 . If we plug in 0 the slope is 1/3.

so, f(-0.05) = 1 + (-0.05)(1/3) = 0.983

f(0,1) = 1 + .1(0.3333) = 1.033