Question

Derivatives Consider the following parametric curves. a. Determine \(dy/dx\) in terms of t and evaluate it at the given value of \(t\). b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of \(t\). $$x=\cos t, y=8 \sin t ; t=\pi / 2$$

Short answer

Answer: The slope of the tangent line at t = π/2 is 0.

Step-by-step solution

Find \(dy/dx\) in terms of \(t\)

First, we need to find \(dx/dt\) and \(dy/dt\). From the given parametric curve, the derivatives are: $$\frac{dx}{dt} = -\sin t \quad \text{and} \quad \frac{dy}{dt} = 8\cos t$$ Now we can find \(dy/dx\) in terms of \(t\) by using the chain rule: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{8\cos t}{-\sin t}$$

Evaluate \(dy/dx\) at \(t = \pi / 2\)

Now we have to evaluate \(dy/dx\) at the given value of \(t\): $$\frac{dy}{dx}\Bigr|_{t = \pi/2} = \frac{8\cos (\pi / 2)}{-\sin (\pi / 2)} = \frac{8 \cdot 0}{-1} = 0$$ So, the slope of the tangent line at \(t = \pi / 2\) is \(0\).

Find the coordinates of the point corresponding to \(t = \pi / 2\)

In order to sketch the tangent line, we need the coordinates of the point corresponding to \(t = \pi / 2\): $$x(\pi / 2) = \cos (\pi / 2) = 0 \quad \text{and} \quad y(\pi / 2) = 8\sin (\pi / 2) = 8$$ So, the coordinates of the point are \((0, 8)\).

Sketch the curve and tangent line

Now, we can sketch the curve and the tangent line at the point corresponding to \(t = \pi / 2\). The curve is an ellipse given by the parametric equation \(x = \cos t\) and \(y = 8\sin t\). The tangent line at the point \((0, 8)\) has a slope of \(0\), which means it is a horizontal line passing through the point \((0, 8)\). You can use a graphing tool or plot this by hand to visualize the curve and the tangent line.

Key concepts

Parametric Equations

A parametric equation defines a group of quantities as functions of one or more independent variables called parameters. In this case, the equations for the curve use the parameter \(t\) to define both \(x\) and \(y\):

• \(x = \cos t\)
• \(y = 8 \sin t\)

These equations allow us to describe complex shapes like ellipses that might not be easily described using standard Cartesian equations. By varying the parameter \(t\) over a specific range, we can trace out the curve defined by these equations. For example, as \(t\) goes from 0 to \(2\pi\), the curve forms an ellipse in the \((x, y)\) plane.
The advantage of parametric equations is their ability to model motion and paths that are not easily represented by regular functions. This makes them incredibly useful in fields like physics, engineering, and computer graphics, where objects often follow non-linear and intricate paths.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When dealing with parametric equations, it helps us find the rate of change of one variable with respect to another, such as \(\frac{dy}{dx}\).
To derive \(\frac{dy}{dx}\) from parametric equations, we take the derivative of \(y\) with respect to \(t\) and the derivative of \(x\) with respect to \(t\):

• \(\frac{dx}{dt} = -\sin t\)
• \(\frac{dy}{dt} = 8\cos t\)

Then we find \(\frac{dy}{dx}\) by dividing these derivatives:\[\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{8\cos t}{-\sin t}\]
The chain rule allows us to relate these rates of change and analyze how changes in \(t\) affect \(x\) and \(y\), providing a clearer understanding of the curve's behavior at specific points.

Tangent Line

In calculus, a tangent line is a straight line that touches a curve at a single point, representing the curve's immediate direction at that point. By finding the derivative \(\frac{dy}{dx}\), we determine the slope of the tangent line for curves defined by parametric equations.
In our specific example, evaluating \(\frac{dy}{dx} = \frac{8\cos t}{-\sin t}\) at \(t = \pi/2\) gives us:

• Slope: \(0\) (as \(\cos(\pi/2) = 0\) and \(\sin(\pi/2) = 1\))

A slope of 0 means the tangent is a horizontal line. With parametric point coordinates \((0, 8)\), the tangent line can be drawn as a horizontal line through this point, which emphasizes the curve’s direction and behavior specifically at \(t = \pi/2\).
Tangent lines are crucial in understanding the local linear approximation of a curve and its instantaneous rate of change. They are extensively used in fields requiring precise measurements and estimates of a curve's behavior, such as physics, engineering, and computer simulations.