Q 8. Curve plot - Calculus

Question

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve. $x=\sinh t,y={\cosh }^{2}t,t\in \mathrm{ℝ}$

Step-by-Step Solution

Answer

The Curve plot  

Step 1: Given information

The parametric curves, $x=\sinh t,y={\cosh }^{2}t,t\in \mathrm{ℝ}$

Step 2: Calculation

The goal is to draw the parametric curve.

Assume $t=-2,-1,0,1,2$ when drawing the graph for the parametric equations.

Find the values $x,y$ by substituting different $t$ values in the parametric equations.

The point $(x, y)$ When $t=-2$ is,

$(x,y)=\left(\sinh t,{\cosh }^{2}t\right)$

$(x,y)=\left(\sinh (-2),{\cosh }^2(-2)\right)[$ since by substituting $t=-2]$

$(x, y)=(-3.62,14.13)$ simplify

The point $(x, y)$ When $t=-1$ is,

$$\begin{gathered} (x,y)=\left(\sinh t,{\cosh }^{2}t\right) \\ (x,y)=\left(\sinh (-1),{\cosh }^2(-1)\right)[\text{ since by substituting }t=-1] \\ (x,y)=\left(-\sinh (-1),{\cosh }^2\left(1\right)\right) \\ (x,y)=(-1.17,2.37) \end{gathered}$$

The point $(x, y)$ When $t=0$ is,

$$\begin{gathered} (x,y)=\left(\sinh t,{\cosh }^{2}t\right) \\ (x,y)=\left(\sinh \left(0\right),{\cosh }^2\left(0\right)\right) [\text{ since by substituting }t=0] \\ (x,y)=(0,1)\text{ simplify } \end{gathered}$$

The point $(x, y)$ When $t=1$ is, 

$$\begin{gathered} (x,y)=\left(\sinh t,{\cosh }^{2}t\right) \\ (x,y)=\left(\sinh \left(1\right),{\cosh }^2\left(1\right)\right)\text{ [since by substituting }t=1\left] \\ \right(x,y)=(1.17,2.37) \end{gathered}$$

The point $(x, y)$ When $t=2$ is,

$$\begin{gathered} (x,y)=\left(\sinh t,{\cosh }^{2}t\right) \\ (x,y)=\left(\sinh \left(2\right),{\cosh }^2\left(2\right)\right)\text{ [since by substituting }t=2\left] \\ \right(x,y)=(3.62,14.13)\text{ simplify } \end{gathered}$$

Step 3: Calculation

The points are represented in tabular form as follows: 

Below is a graphical illustration of the situation. 

Therefore, the solution is the required graph.