Chapter 11.1, Problem 48E

To determine

To determine: The correct option for the parametric curve to be,

(A) increasing and concave up.

(B) increasing and concave down.

(C) decreasing and concave up.

(D) decreasing and concave down.

(E) decreasing with a point of inflection.

Answer to Problem 48E

The correct option is (C).

Explanation of Solution

Given information:

The parametric equations: $x=\sin t$ , $y=\csc t$ for $0<t<\frac{\pi }{2}$

Calculation:

The parametric equations are $x=\sin t$ and $y=\csc t$ .

It is known that $\csc t=\frac{1}{\sin t}$ .

So, $y=\frac{1}{\sin t}$

Substitute $x$ for $\sin t$ in the equation $y=\frac{1}{\sin t}$ .

  $y=\frac{1}{x}$ for $0<x<1$

This means the curve is decreasing.

Now, differentiate the expression $y=\frac{1}{x}$ .

  $\frac{dy}{dx}=-\frac{1}{x^2}$

Again, differentiate the above derivative.

  $\begin{array}{c}\frac{d^{2}y}{d{x}^2}=-\frac{\left(-2\right)}{x^2}\\ =\frac{2}{x^3}\end{array}$

It is seen that the second derivative is positive. This means that the curve is concave up.

Thus, the curve is decreasing and concave up.

Hence, the correct option is (C).