Not sure I am working these problems correctly. I watched some youtube videos and think I figured it out but would like you to verify my work.

Find the differential of the function. Calculate dy for the given valuse of x and dx.

1) y=sqrt(1+x^2), x=0 dx=0.01

f'(x)=1/2(x^2+1)^(-1/2) (2x) = x/sqrt(x^2+1)

dy=f'(x)dx

dy=x/sqrt(x^2+1)dx

dy=0/sqrt(0^2+1)(0.01)=0

2) y=x^3-6x^2+5x-7, x=-2 dx=0.1

f'(x)=1/4 x^4 - 2x^3 + 5/2 x^2 - 7x

dy = f'(x)dx

dy = (1/4(-2)^2-2(-2)^3 +5/2(-2)^2-7(-2))(0.1)

dy = 4.4

3) cos^2(pit), t=1/3 dt=0.05

f'(t) = -2pi cos(pit)sin(pit)

dy=f'(t)dt

dy = -2pi cos(pi/3)sin(pi/3)(0.05)

dy = -2pi (1/2)(sqrt(3)/2)(0.05)

dy = 1/20(sqrt(3)/4)pi

dy = pi sqrt(3)/80 = 0.068

Thank you for looking,

Rod