Mathematics

Problem

Let

f

and

g

be differentiable functions on

[0,a]

and consider the parametric curve

C

defined by

x(t)=f(t)

y(t)=g(t)

. A new curve

C_c

is obtained by scaling each coordinate by a positive constant

c

x_c(t)=cf(t)

y_c(t)=cg(t)

. a. Explain how the arc length of

C_c

[0,a]

is related to the arc length of

C

on the same interval. Provide a clear justification based on properties of derivatives. b. Suppose instead that only the

x

–coordinate is scaled, giving the curve

D_d

defined by

x_d(t)=d\,f(t)

and

y_d(t)=g(t)

. Derive a general expression that compares the arc length of

D_d

[0,a]

to that of

C

. Express your answer in terms of

d

and derivatives of

f

and

g

. c. Using your expression from part (b), determine the value(s) of

d>0

(if any) for which the arc length of

D_d

[0,a]

is unchanged from that of

C

. Explain your reasoning.