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Part (A): In your own words, discuss what is meant by parametric equations. When would we use...

Question:

Part (A): In your own words, discuss what is meant by parametric equations. When would we use parametric applications? (provide a real-world example).

Part (B): Given a set of parametric equations, how would you go about sketching the curve represented by the set of parametric equations?

Part (C): How can you determine concavity for the graph representing a set of parametric equations?

Parametric Equations:

{eq}\\ {/eq}

The Parametric Equation of a function {eq}F(x,y) {/eq}, with parameter as time {eq}t {/eq} are written as {eq}x=f(t), y=g(t) {/eq}, where {eq}x\**** \ y {/eq} are the functions of time {eq}t. {/eq}

Concavity - The curve is concave upwards where the double derivative of a function is greater than zero and concave downwards where the derivative of function is less than zero.

Answer and Explanation:

{eq}\\ {/eq}

(A) Parametric Equations are the way to find the position of an object w.r.t. time. They generally represent a curve or a surface.

A real-life application of Parametric Application is 'Throwing a ball in the sky and recording its position at a particular time with the help of parametric equations'.

(B) Suppose we have a set of parametric equations as {eq}x=f(t), \ y=g(t) {/eq}, so each value of {eq}t {/eq} defines a point {eq}(x,y) {/eq}, which can be plotted as {eq}(f(t),g(t)) {/eq}. The collection of all such points completes the Parametric Curve.

(C) Let {eq}x=f(t), \ y=g(t) {/eq}, so {eq}\dfrac{dy}{dx}=\dfrac{\frac{dy}{dt}}{\frac{dx}{dt}} {/eq}

And, {eq}\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right) {/eq}.

The curve will be concave upwards when {eq}\dfrac{d^2y}{dx^2}>0 {/eq} and concave downwards when {eq}\dfrac{d^2y}{dx^2}<0. {/eq}


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
3.8K

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