Question

A) Eliminate the parameter to find a Cartesian equation of the curve.

b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
1. x=3cost, y=3sint, 0 ⩽ t ⩽π
2. x=sin4θ, y=cos4θ, 0 ⩽ θ ⩽ π/2
3. x=cosθ, y=sec^2θ, 0 ⩽ θ < π/2
4. x=csct, y=cott, 0 < t < π
5. x=e^−t, y=e^t
6. x=t+2, y=1/t, t>0
7. x=lnt, y=√t, t ⩾ 1

Answer 1

The Cartesian equations for the given parametric curves are as follows:

1.
`\(x = 3\cos(t), \quad y = 3\sin(t)\), \(0 \leq t \leq \pi\)`

2.
`\(x = \sin(4\theta), \quad y = \cos(4\theta)\), \(0 \leq \theta \leq (\pi)/(2)\)`

3.
`\(x = \cos(\theta), \quad y = \sec^2(\theta)\), \(0 \leq \theta < (\pi)/(2)\)`

4.
`\(x = \csc(t), \quad y = \cot(t)\), \(0 < t < \pi\)`

5.
`\(x = e^(-t), \quad y = e^t\)`

6.
`\(x = t + 2, \quad y = (1)/(t)\), \(t > 0\)`

7.
`\(x = \ln(t), \quad y = √(t)\), \(t \geq 1\)`

Step-by-step explanation:

1. For the first parametric equations
`\(x = 3\cos(t), \quad y = 3\sin(t)\)`, eliminating the parameter yields
`\(x^2 + y^2 = 9\)`, which represents a circle of radius 3 centered at the origin.

2. In the second set of parametric equations
`\(x = \sin(4\theta), \quad y = \cos(4\theta)\)`, eliminating the parameter results in
`\(x^2 + y^2 = 1\)`, indicating a circle of radius 1 centered at the origin.

3. For
`\(x = \cos(\theta), \quad y = \sec^2(\theta)`, eliminating the parameter gives
`\(y = \sec^2(\cos^(-1)(x))`, representing a curve with vertical asymptotes where
`\(\cos^(-1)(x)\)`is undefined.

4. In
`\(x = \csc(t), \quad y = \cot(t)\)`, eliminating the parameter results in
`\(y = (1)/(x)\)`, representing a rectangular hyperbola.

5. The parametric equations
`\(x = e^(-t), \quad y = e^t\)` translate to
`\(y = e^(2x)\)` when eliminating the parameter, representing an upward exponential curve.

6. Eliminating the parameter in
`\(x = t + 2, \quad y = (1)/(t)\) gives \(y = (1)/(x-2)\)`, representing a hyperbola with a vertical asymptote at
`\(x = 2\)`.

7. For
`\(x = \ln(t), \quad y = √(t)\)`, eliminating the parameter results in
`\(y = √(e^x)`), representing a curve with exponential growth.

Parametric equations and their elimination to derive Cartesian equations for various geometric shapes and curves. Understanding the relationship between parameterization and the resulting algebraic representations provides insights into curve behavior and facilitates analysis in different coordinate systems.