Problem 16 - Calculus Early Transcendentals

Question

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=(t+1)^{2}, y=t+2 ;-10 \leq t \leq 10$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter t from the given parametric equations, describe the curve and indicate the positive orientation. Parametric equations: $$x=(t+1)^{2}, y=t+2; -10 \leq t \leq 10$$ Answer: The parametric equations can be expressed as an equation in \(x\) and \(y\) as: $$x = (y - 1)^2$$ The curve described by the parametric equations is a parabola with its vertex at the point (0, 1), opening towards the positive x-axis. The positive orientation is in the top-right direction as x and y both increase.

Step 1: Eliminate the parameter

To eliminate the parameter t, we can solve one of the equations for t, and then substitute the expression into the other equation. Let's solve the second equation for t: $$y = t + 2$$ $$t = y - 2$$ Now, substitute this expression for t into the first equation: $$x = (t + 1)^2$$ $$x = (y - 2 + 1)^2$$

Step 2: Simplify the equation

Simplify the equation by combining the terms inside the parentheses and then squaring the result: $$x = (y - 1)^2$$ This is the equation in \(x\) and \(y\).

Step 3: Describe the curve and find its orientation

The simplified equation \(x = (y - 1)^2\) represents a parabola with its vertex at the point (0, 1) and opening towards the positive x-axis. To determine the orientation, we need to find the direction of the curve as t increases. Let's look at the behavior of x and y as t increases: For \(t=-10\): $$x=(-9)^2=81$$ $$y=-10+2=-8$$ For \(t=10\): $$x=11^2=121$$ $$y=10+2=12$$ As t increases from \(-10\) to \(10\), the value of x increases, and the value of y also increases. Thus, the positive orientation is in the direction of increasing x and increasing y (top-right direction). In conclusion, a. The parametric equations can be expressed as an equation in \(x\) and \(y\) as: $$x = (y - 1)^2$$ b. The curve described by the parametric equations is a parabola with its vertex at the point (0, 1), opening towards the positive x-axis. The positive orientation is in the top-right direction as x and y both increase.

Key Concepts

Eliminating Parameters

Understanding this conversion is crucial for graphing and analyzing curves within the realm of calculus and beyond. It not only aids in visualizing curves but also enables us to use standard methods of calculus, such as differentiation and integration, in relation to those curves.

Describing Curves in Calculus

By analyzing particular values of t (as illustrated by checking t = -10 and t = 10), we gain insight into how the curve behaves over the interval of t. This allows us to sketch the curve accurately and predict other characteristics, such as intercepts and bounds, which are essential in applications of calculus.

Parametric Curve Orientation

Grasping the concept of orientation enables students to tackle more complex problems involving motion, slopes, and velocities on parametric curves. It also enhances comprehension when dealing with integrals and derivatives of parametric equations, which are essential topics in further calculus studies.