differentiable function has value y(3)=4 the derivative value y′(3)=-2 Approximate value of y(3.1)

Question

it is a tangent line approximation problem

Answer 1

The derivative is the tangent to the curve at the point where y=3, so the slope of the tangent is -2. As x increase to 3.1, y will decrease because of the negative slope. If we assume the slope doesn't change significantly over the small change in x we can write dy/dx=-2 at 3.1, so the small change in y corresponding to dx=0.1, dy=0.1*(-2)=-0.2, and y(3.1) is approximately 4-0.2=3.8.