Calculus II - Tangents with Parametric Equations. In this section we want to find the tangent lines to the parametric equations given by, To do this let’s first recall how to find the tangent line to at . Here the tangent line is given by, Now, notice that if we could figure out how to get the derivative from the parametric equations we could simply reuse this formula since we will be able to use the parametric equations to find the x and y coordinates of the point. So, just for a second let’s suppose that we were able to eliminate the parameter from the parametric form and write the parametric equations in the form . Now, differentiate with respect to t and notice that we’ll need to use the Chain Rule on the right hand side.
Let’s do another change in notation. At this point we should remind ourselves just what we are after. Or that is in terms of the parametric formulas. Notice as well that this will be a function of t and not x. As an aside, notice that we could also get the following formula with a similar derivation if we needed to, ECCS'11 European Conference on Complex Systems. Safety verification via barrier certificates « Rod Carvalho.
Laplace transform. History[edit] The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called z transform) in his work on probability theory. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich. From 1744, Leonhard Euler investigated integrals of the form as solutions of differential equations but did not pursue the matter very far.[2] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form which some modern historians have interpreted within modern Laplace transform theory.[3][4][clarification needed] akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation.
Formal definition[edit] The parameter s is the complex number frequency: with real numbers and ω. instead of F.