Problem 6 - Active Calculus Multivariable

Question

Find a parametrization of the curve \(x=-5 z^{2}\) in the xz-plane. Use \(t\) as the parameter for all of your answers. \(x(t)=\)________ \(y(t)=\)________ \(z(t)=\)________

Step-by-Step Solution

Answer

The parametrization of the given curve \(x = -5z^2\) in the xz-plane using \(t\) as the parameter is: \(x(t) = -5t^2\) \(y(t) = t\) \(z(t) = t\)

Step 1: Rewrite the equation using t as the parameter

We have the curve given by the equation x = -5z^2 in the xz-plane. Since it's in xz-plane, there is no constraint on y. We can use the parameter 't' to rewrite this equation by considering z as a function of t, say z = g(t). Now, we need to find an appropriate function g(t) and rewrite x(t) accordingly.

Step 2: Express z as a function of t

We can consider z as a linear function of t, i.e., z = t. This is a simple function of t and will allow us to rewrite the given equation using t as the parameter.

Step 3: Rewrite the equation in terms of x(t) and z(t)

Now that we have z = t, we can substitute this back into the x = -5z^2 equation: x = -5(t^2)

Step 4: Write the parametrization using x(t), y(t), and z(t)

Now we have x(t) and z(t) expressions, and since there is no constraint on y in the xz-plane, we can write y(t) as a free variable. So, the parametrization of the curve is: x(t) = -5t^2 y(t) = t (or any other function of t if needed, since it's a free variable) z(t) = t

Step 5: Final parametrization

The parametrization of the given curve x = -5z^2 in the xz-plane using t as the parameter is: x(t) = -5t^2 y(t) = t z(t) = t

Key Concepts

Multivariable Calculus

Multivariable calculus extends the principles and techniques of calculus to functions of more than one variable. Unlike single-variable calculus where curves are defined explicitly by equations like y=f(x), in multivariable calculus, we often encounter curves and surfaces in three dimensions. Parametrization is a powerful tool in multivariable calculus that allows us to describe these curves and surfaces with several equations, each dependent on a common parameter.

Understanding parametric equations is crucial for mastering the concepts of multivariable calculus. These equations help to simplify complex problems, such as calculating lengths of curves, areas of surfaces, and volumes of solids. Moreover, multivariable calculus is fundamental in fields like physics, engineering, and economics where multiple variables interact in complex ways.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a single parameter, typically denoted as t. This approach contrasts with the more familiar Cartesian equations, which give a direct relationship between x and y. By introducing a parameter t, we can describe not only the position of a point at any given time but also the path and direction of motion along the curve.

For example, the parametric equations x(t) = -5t^2, y(t) = t, and z(t)= t from the exercise specify the coordinates of a point on the curve in the xz-plane. Each value of t yields a specific point, and as t varies, we trace out the entire curve. This representation is especially useful when dealing with motion or whenever the independent variable is time.

The xz-Plane

In three-dimensional space, the xz-plane is one of the three primary coordinate planes, formed by the x-axis and the z-axis intersecting at right angles. Curves in the xz-plane have equations that do not involve the y-coordinate, so their projection onto the xz-plane is essential for analysis.

When parametrizing a curve in the xz-plane, as shown in the original exercise, we focus solely on expressing x and z in terms of the parameter t. The y-coordinate is typically treated as a free variable, represented by y(t) = t (or any function of t), as it doesn't affect the curve's shape in the xz-plane. This simplification allows us to understand the behavior of the curve within the plane without the distraction of the third dimension.