Is the approximation an under or overestimate under x 0 7e

Is the approximation an under- or overestimate? = under

x = 0 7e-5x f(x) x = a y = 5x - 5 a a = f(a) f(2.5) f(2.5) = y = sin x y = x -1 x 1 y = sin x y = x sin x 7e-5x -1 x 1 x > 0 8 + x x = 5

Solution

1.) f\'(0) = -35, f(0) = 7

tangent : y = -35x + 7

2.) y = 5x-5 => (y+5) = 5(x-0) => x = 0, y(0)=-5

f(2.5) = -5+ -5*2.5 = -17.5

above value is underestimate because the curve is lying above the tangent after x=0

3.) sinx can be approximated as y=x near x=0 as it is the tangent approximation.

max error will occur at +1 = 1-sin1 = 0.158,

which is overestimate after x=0

also

max error will occur at x=-1, = -0.158

which will be underestimate

4.) f(5) = 13^0.5 , f\'(5) = 1/2(13)^0.5

tangent : y = -1/2(13)^0.5x +13^0.5