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In the world of parametric equations, derivatives help us understand how the output values change concerning some parameter, often time. For parametric equations given as functions of a parameter, such as

• \( x = \cos t \)
• \( y = \sin(2t) \)

we need to take the derivatives with respect to this parameter—here, \( t \). This means we're looking at how \( x \) and \( y \) change as \( t \) changes. By finding \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \), we can compute the derivative of \( y \) with respect to \( x \), or \( \frac{dy}{dx} \), using the formula:\[\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.\]For the given equations:

• \( \frac{dx}{dt} = -\sin t \)
• \( \frac{dy}{dt} = 2\cos(2t) \)

This leads to \( \frac{dy}{dx} = -\frac{2\cos(2t)}{\sin t} \), which we then evaluate at a specific point, such as \( t = \pi/4 \), to find the slope of the tangent line.

The tangent line to a curve at a point is the straight line that just "touches" the curve at that point. It shows the direction in which the curve is headed. In parametric equations, we calculate this line by finding the slope, \( \frac{dy}{dx} \), at a specified parameter value.For the equations \( x = \cos t \) and \( y = \sin(2t) \), we've found the slope of the tangent line at \( t = \pi/4 \) to be zero \((\frac{dy}{dx} = 0)\), implying the tangent line is horizontal.To write the equation of the tangent line, we use the point \((\frac{\sqrt{2}}{2}, 1)\) determined by plugging \( t = \pi/4 \) into the parametric equations. The tangent line is thus:

• \( y = 1 \)

This represents a horizontal line at \( y = 1 \). The concept of a tangent line is essential in understanding the instantaneous direction of the curve at a given point.

While the tangent line indicates the path of the curve at a point, the normal line provides a perpendicular direction to this path. It's important because it offers a different perspective on how the curve behaves at that very point.The slope of the normal line is the negative reciprocal of the slope of the tangent line. If the tangent line's slope is zero, the normal is vertical, implying it doesn't have a defined slope.For our specific case, since the tangent line is horizontal \((slope = 0)\), the normal line through the point \((\frac{\sqrt{2}}{2}, 1)\) is vertical:

• \( x = \frac{\sqrt{2}}{2} \)

This equation tells us that as you move through the point where the tangent line intersects the curve, the normal line cuts through the x-axis without tilting forward or backward.

Sketching the graph of a parametric equation involves plotting points that are determined by the parametric functions for a range of parameter values. This creates a smooth trajectory or a path that represents the curve.For functions like \( x=\cos t \) and \( y=\sin(2t) \), plotting these for \( t \) over the interval \([0, 2\pi]\) helps visualize the shape that the curve takes.To accurately represent this curve, including any tangent and normal lines at specific points (like \( t = \pi/4 \)), ensures a complete understanding of how the curve sits in relation to these lines. From the example above:

• The tangent line is a horizontal line at \( y = 1 \).
• The normal line is a vertical line at \( x = \frac{\sqrt{2}}{2} \).
  Including these lines in the graph offers valuable insight into the nature of the curve and its motion.

Effective graph sketching combines understanding parametric form, calculating necessary derivatives, and applying these to draw the tangent and normal lines with precision.