   Chapter 10.2, Problem 67E

Chapter
Section
Textbook Problem

If f′ is continuous and f′(t) ≠ 0 for a ≤ t ≤ b, show that the parametric curve x = f(t), y = g(t), a ≤ t ≤ b, can be put in the form y = F(x). [Hint: Show that f −1 exists.]

To determine

To show: The parametric curve defined by equations x=f(t) and y=f(t) , atb can be put in the form y=F(x),axb .

Explanation

Given data:

The parametric equation for the variable x is as follows.

x=f(t) (1)

The parametric equation for the variable y is as follows.

y=g(t) (2)

The range of parameter t is a to b .

Calculation:

For the value of parameter t from a to b , f' is continuous and f'(t)0 when f'(t)>0 for all parameter value t in (a,b) or f'(t)<0 for all parameter value t in (a,b) .

Since f either increases or decreases, it is monotonic on (a,b) and has an inverse.

Therefore, the parametric equation for the variable x can be rearranged as,

x=f(t)f1(x)=t

Substitute f1</

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