f(x)=x^3-3x^2+3x+1 near 2 (at a=2)

Question

A. Use Linear approximation to approximate.
b.is this underestimate or overestimate? Justify answer.
C.Approximate f(2.25) using the linear approximation and compare the answer with the exact value.

Thanks.

Answer 1

f'(x) = 3x^2 - 6x + 3
f''(x) = 6x - 6

f''(2) > 0, so the graph is concave up at x=2. So, a linear approximation will underestimate f(x)

f'(2) = 3
f(2) = 3
the line through (2,3) with slope 3 is

y = 3(x-2)+3= 3x-3

y(2.25) = 3.75
f(2.25) = 3.953
the line underestimates