Question

Write a formula for the distance between a point (x, y) on the graph of y = x^4 and the point (1, 10). Express your answer in terms of x.

Answer 1

Anonymous's answer is completely correct. I thought this problem was asking how to find the distance along the function from the point (2,2^8), and wrote the answer to that nice, tasty problem.

Simply integrate the line element with respect to some affine parameter!

L=∫10(∂x∂λ)2+(∂y∂λ)2−−−−−−−−−−−−−√dλL=∫01(∂x∂λ)2+(∂y∂λ)2dλ

In this case,
x(λ)=λ(X−2)+2,x(λ)=λ(X−2)+2,
y(λ)=(λ(X−2)+2)8.y(λ)=(λ(X−2)+2)8.

Note that this approach can also solve the original problem, with some simplification.