Parametric Equations
Definition
Parametric equations are a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. They allow us to represent complex shapes and motions by breaking them down into simpler components.
Analogy
Think of parametric equations as instructions for drawing a picture. Instead of giving you the exact x and y coordinates, they give you step-by-step directions on how to move your pencil to create the desired shape.
Related terms
Parametric Functions: These are functions defined by parametric equations, where each coordinate is expressed as a function of the parameter(s). For example, x = f(t) and y = g(t) would define a parametric function.
x = f(t): This is one part of a parametric equation that represents the x-coordinate in terms of the parameter t. It tells us how the x-value changes as t varies.
y = g(t): This is another part of a parametric equation that represents the y-coordinate in terms of the parameter t. It tells us how the y-value changes as t varies.
"Parametric Equations" appears in:
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Additional resources (3)
Practice Questions (19)
- Henry draws a heart defined by the parametric equations x(t) = sin(t) and y(t) = 1 + cos(t) - cos^2(t). What is the slope dy/dx of the curve at the point (sqrt(3)/2, 5/4)?
- Which of the following are a pair of parametric equations with the parameter t?
- The motion of a rolling ball on the coordinate plane is given by the set of parametric equations x(t) = 12sin(t) and y(t) = 6t^2. Which of the following derivatives is incorrect?
- Liam's height in inches, h(t), and weight in pounds, w(t), as a child at t years old can be modeled by the parametric equations h(t) = 3t+25 and w(t) = t^2/2+20. When Liam weighed 70 pounds, how tall was he in inches?
- Henry draws a heart defined by the parametric equations x(t) = sin(t) and y(t) = 1 + cos(t) - cos^2(t). What is the concavity of the curve at the point (sqrt(3)/2, 5/4)?
- Which of the following pairs of parametric equations are concave down at t = 1?
- The motion of a rolling ball on the coordinate plane is given by the set of parametric equations y(t) = 12cos(t) and x(t) = 6e^t. Which of the following derivatives is incorrect?
- The path of a lost cow on the xy-plane is given in parametric equations by x(t) = t^2 + t and y(t) = t^2-t. Is the path of the cow concave up or down at t=0?
- A video game character's attack and defense increase over time (t > 0) is given by the parametric equations A(t) = 3t^2+5t and D(t) = 2t^2+8t. James the gamer plots these values on a plot with attack as the x-axis and defense as the y-axis. Is the resulting curve concave up or concave down?
- Consider the parametric equations: x(t) = t^2 y(t) = 3t What is the arc length of the curve between t = 0 and t = 2?
- The parametric equations of a curve are: x(t) = 2cos(t) y(t) = 2sin(t) What is the arc length of the curve between t = 0 and t = π/2?
- Consider the parametric equations: x(t) = e^t + e^(-t) y(t) = e^t - e^(-t) What is the arc length of the curve between t = 0 and t = ln(2)?
- Consider the parametric equations: x(t) = t^2 - 1 y(t) = t^3 + 2t What is the arc length of the curve between t = -1 and t = 1?
- When dealing with parametric equations, what do we need to take the integral of in order to find the distance traveled?
- A particle moves along a curve defined by the parametric equations x = 2cos(t) and y = 3sin(t). What is the magnitude of the acceleration vector of the particle at time t = π/6?
- A particle moves along a curve defined by the parametric equations x = 3t^2 and y = 2t^3. What is the approximate distance traveled by the particle from time t = 0 to t = 2?
- A particle moves along a curve defined by the parametric equations x = t^3 and y = t^2. What is the acceleration vector of the particle at time t = 2?
- Consider a particle moving along a curve in the plane defined by the parametric equations x = 2t and y = t^2. What is the velocity vector of the particle at t = 3?
- A particle moves along a curve defined by the parametric equations x = t^2 and y = t^3. What is the velocity vector of the particle at time t = 2?
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