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Suppose a curve is given by the parametric equations x=f(t), y=g(t), where the range of f is...

Question:

Suppose a curve is given by the parametric equations x=f(t), y=g(t), where the range of f is -1,7 and the range of g is 1,3. What can you say about the curve? You must select all correct choices to get full credit on this problem.

A. The curve must lie inside a circle with center (-1, -1) and radius 0.5.

B. Nothing can be said about the curve.

C. The curve is completely contained in the rectangle -1, 1 by 1, 3.

D. The curve is the line with endpoints (-1, 1) and (7, 3).

E. The curve must lie outside the rectangle -1, 7 by 1, 3.

D. The curve is a circle with center (-1, -1) and radius 3.

Parametric Equations:

{eq}\\ {/eq}

A function of the form {eq}F(x,y) {/eq} can be represented in the Parametric form as {eq}x=f(t), \ y=g(t) {/eq}, where {eq}t {/eq} is the parameter of the functions {eq}f\&g {/eq}. Here {eq}f(t) \& g(t) {/eq} represent the position of particle w.r.t. time {eq}t. {/eq} A curve is said to lie inside a region if it lies inside the domain of that region. In case of a parametric function with some parameter {eq}t {/eq}, the curve should lie within the domain of both functions, then only it is said to lie within that region. A rectangle is a is a region with four vertices in which the opposite vertices are of same length. A curve is said to lie with a rectangle if it is enclosed within the vertices of that rectangle.

Answer and Explanation:

{eq}\\ {/eq}

All possible paths joining {eq}(1,1) \ \& (7,3) {/eq} within the rectangle formed by {eq}(-1,1),(-1,3),(7,1) \ \& \ (7,3) {/eq} may be the possible path traced by the curve given by the parametric equations : {eq}x=f(t),y=g(t) {/eq}.

So, option {eq}(B) \ \& \ (D) {/eq} are correct.


Learn more about this topic:

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Parametric Equations in Applied Contexts

from Precalculus: High School

Chapter 24 / Lesson 6
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