Find dx/dt in a similar fashion. Also find x & y, when t = 2.miglo said:Homework Statement
x^3+2t^2=9, 2y^3-3t^2=4, t=2
find the slope of the curve at the given value of t
Homework Equations
The Attempt at a Solution
i know dy/dx=(dy/dt)/(dx/dt) for parametrized curves
but how do i use implicit differentiation on parametrized curves?
i tried d/dt(2y^3-3t^2)=d/dt(4)
=6y^2*dy/dt-6t=0
=6y^2*dy/dt=6t
=dy/dt=t/y^2 ? but shouldn't dy/dt be defined in t's only? I am confused
An implicitly defined parametrization is a mathematical representation of a curve or surface that is defined by an equation that relates the coordinates of the curve or surface to a set of parameters. These parameters can be thought of as independent variables that determine the shape and position of the curve or surface.
The main difference is that an explicitly defined parametrization has a direct equation that relates the coordinates to the parameters, while an implicitly defined parametrization does not have a direct equation and instead relies on a set of implicit equations. This means that an explicitly defined parametrization is easier to manipulate and solve for specific values, while an implicitly defined parametrization is often more complex but can represent a wider range of curves and surfaces.
Implicitly defined parametrizations are used in various fields of science, including physics, engineering, and computer graphics. They are particularly useful in solving problems involving curves and surfaces that cannot be easily represented by explicit equations. For example, in physics, they are used in the study of motion and trajectories, while in engineering, they are used in designing and analyzing complex structures.
One of the main advantages is that they can represent a wider range of curves and surfaces compared to explicit parametrizations. They are also more flexible and can handle more complex shapes and structures. Additionally, they can be used to describe curves and surfaces that do not have a simple equation or formula, making them applicable in various scientific and engineering problems.
Some common methods for solving implicitly defined parametrizations include numerical methods, such as iteration and root-finding algorithms, as well as analytical techniques, such as substitution and differentiation. Additionally, computer software and programming languages can also be used to solve and graph implicitly defined parametrizations.