Calculus with Parametric Equations Calculator

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Example

Created on 2024-06-20Asked by Michael Torres (Solvelet student)

Find the derivative

\frac{dy}{dx}

for the parametric equations

x = t^2

and

y = t^3

Solution

Step 1: Identify the parametric equations

x = t^2

and

y = t^3

. Step 2: Find the derivatives

\frac{dx}{dt}

and

\frac{dy}{dt}

\frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2.

Step 3: Use the formula

\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}

\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}.

Step 4: Conclusion. The derivative

\frac{dy}{dx}

for the parametric equations is

\frac{3t}{2}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Layla Young on Solvelet

1. Find the tangent line to the curve defined by the parametric equations, at the point. In order to find the equation of the tangent line to a curve defined parametrically through a system of two equations, one has to find the coordinates of the given point first and then evaluate the derivatives of both equations at this point. The system of the given equations can also be represented in the scalar form in the following way:,. Differentiating both of these equations, we can obtain the following system:,. At, = 0 and = 1, which means that the corresponding equation is. Thus, at the given point, this equation defines the tangent line, and its coefficients,, and are = 0, = −1, and = 1.2. Calculate the arc curve length defined by the parametric equations

x = t^2

y = t^3

from

t = 0

t = 2

DefinitionDerivative of Parametric Equations in Calculus with Parametric Equations, the applications of the functions are known directly with the help these equations, where the variables are expressed in terms of another parameter. This includes methods to differentiate, integrate, and plot parametrically defined curves and surfaces. In physics, engineering, and computer graphics, calculus with parametric equations are used to define motion, and also trajectories, and parametric surfaces. It is a powerful tool in analyzing curves and surfaces in two and three dimension. For instance, applications of 2D parametric equations are in describing the motion of objects in projectile motion problems, planetary orbits problem, and space curves.

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