Let f(x)=x^4. If a=1 and dx=Δx=1/2, what are Δy and dy?

Answer 1

y = x^4
dy = 4x^3 dx
= 4*1(1/2)
= 2

Δy = f(x+Δx)-f(x)
= (1+1/2)^4 - 1^4
= 81/16 - 1
= 65/16
= 4.01625

They are nowhere near the same. This is because dx uses a linear approximation (the tangent line at x=1), but the actual graph curves steeply upward.

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y%3Dx^4%2C+y%3D4%28x-1%29%2B1