Parametric in calculus

Front

What is a parametric equation?

Back

A parametric equation expresses the coordinates of the points of a curve as functions of a variable, typically denoted as 't'. For example, x(t) and y(t) define a curve in the xy-plane.

Front

How do you find the speed of a particle given parametric equations x(t) and y(t)?

Back

The speed of a particle is found using the formula: \( v = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \). Calculate the derivatives of x and y with respect to t, then substitute into the formula.

Front

What is the relationship between parametric equations and rectangular equations?

Back

Parametric equations can be converted to rectangular form by eliminating the parameter 't'. This often involves solving one equation for 't' and substituting it into the other.

Front

What is the total distance traveled by a particle given its velocity function?

Back

The total distance traveled is found by integrating the speed (magnitude of velocity) over the given interval. For a velocity function \( v(t) \), the distance is \( \int_{a}^{b} |v(t)| dt \).

Front

What is the significance of the parameter 't' in parametric equations?

Back

The parameter 't' often represents time, allowing the equations to describe the motion of a particle over time in a two-dimensional space.

Front

How do you convert the parametric equations x = 4cos(θ) and y = 3sin(θ) to rectangular form?

Back

To convert, use the identity \( \cos^2(θ) + \sin^2(θ) = 1 \). From the equations, we have \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \).

Front

What is the formula for the distance between two points in parametric form?

Back

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Front

What is the derivative of a parametric equation?

Back

The derivative of a parametric equation \( y(t) \) with respect to \( x(t) \) is given by \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). This represents the slope of the curve at any point.

Front

How do you find the coordinates of a point given a parameter value in parametric equations?

Back

To find the coordinates, substitute the value of the parameter into the parametric equations. For example, if \( t = 1 \), substitute into \( x(t) \) and \( y(t) \) to find the point (x(1), y(1)).

Front

What is the equation of a circle in parametric form?

Back

The parametric equations for a circle of radius r centered at the origin are: \( x(t) = r \cos(t) \) and \( y(t) = r \sin(t) \).