Question

If $x=x\left(t\right)$and $y=y\left(t\right)$are differentiable functions of $t$determining the direction of motion along the curve when

(a) $x^{\text{'}}\left(t\right)>0$and$y^{\text{'}}\left(f\right)>0$.

(b) $x^{\text{'}}\left(t\right)>0$and $y^{\text{'}}\left(t\right)<0$.

(c) $x^{\text{'}}\left(t\right)<0$and $y^{\text{'}}\left(t\right)>0$

(d) $x^{\text{'}}\left(t\right)<0$and $y^{\text{'}}\left(t\right)<0$

Short answer

(a). When the functions' derivatives $x^1\left(t\right)>0$and $y^{\text{'}}\left(t\right)>0$then the curve moves to the right and moves up.

(b) the derivative of the functions $x^1\left(r\right)>0$and $y^1\left(r\right)<0$then the curve moves to the right and moves down.

(c) The derivative of the function$x^{\text{'}}\left(t\right)<0$and $y^{\text{'}}\left(t\right)>0$,then the curve moves to the left and moves up.

(d) The derivative of the function $x^1\left(l\right)<0$and $y^{\text{'}}\left(l\right)<0$,then the curve moves to the left and moves down.

Step-by-step solution

Part (a) Step 1 : Given information.

Consider the parametric equations $x=x\left(t\right),y=y\left(t\right)$which are differentiable at t.

The objective is to find the direction of motion along the curve for the given conditions.

The derivative values of $x^{\text{'}}\left(t\right)$ denotes the function moves right or left,the derivative values of the function $y^1\left(t\right)$ denotes whether the function moves up or down.

Part (a) Step 2 : Calculation 

$x^1\left(t\right)>0$and $y^1\left(t\right)>0$

When the derivative of the functions $x^1\left(t\right)>0$and $y^1\left(t\right)>0$then the curve moves to the right and moves up.

Part (b) Step 1: Given information

Consider the parametric equations $x=x\left(t\right),y=y\left(t\right)$which are differentiable at t.

The objective is to find the direction of motion along the curve for the given conditions.

The derivative values of $x^{\text{'}}\left(t\right)$ denotes the function moves right or left,the derivative values of the function $y^1\left(t\right)$ denotes whether the function moves up or down.

Part (b) Step 2: Calculation

$x^{\text{'}}\left(t\right)>0$ and $y^{\text{'}}\left(t\right)<0$

When the derivative of the functions $x^{\text{'}}\left(t\right)>0$ and $y^{\text{'}}\left(t\right)<0$ then the curve moves to the right and moves down.

Part (c) Step 1: Given information

Consider the parametric equations $x=x\left(t\right),y=y\left(t\right)$which are differentiable at t.

The objective is to find the direction of motion along the curve for the given conditions.

The derivative values of $x^{\text{'}}\left(t\right)$ denotes the function moves right or left,the derivative values of the function $y^1\left(t\right)$ denotes whether the function moves up or down.

Part (c) Step 2: Calculation

$x^1\left(t\right)<0$and $y^1\left(t\right)>0$

When the derivative of the function$x^{\text{'}}\left(t\right)<0$ and $y^{\text{'}}\left(t\right)>0$, then the curve moves to the left and moves up.

Part (d) Step 1: Given information

Consider the parametric equations $x=x\left(t\right),y=y\left(t\right)$which are differentiable at t.

The objective is to find the direction of motion along the curve for the given conditions.

The derivative values of $x^{\text{'}}\left(t\right)$ denotes the function moves right or left,the derivative values of the function $y^1\left(t\right)$ denotes whether the function moves up or down.

Part (d) Step 2: Calculation

$x^{\text{'}}\left(t\right)<0$ and $y^{\text{'}}\left(t\right)<0$

When the derivative of the function $x^{\text{'}}\left(t\right)<0$ and $y^{\text{'}}\left(t\right)<0$, then the curve moves to the left and moves down.