coolingwater said:Homework Statement
f(x,y) = ln(3y-8x)
Derive the first and second order Taylor polynomial approx, L(x,t) and Q(x,t), for T(x,T) about the point (1,1)
Homework Equations
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The Attempt at a Solution
I do not understand what the question wants, nor do i want a solution. I would love to know how do i proceed to do the questions, probably a few steps and directions as i do want to learn how to tackle the question. Thank you!
A Taylor Polynomial Approximation is a mathematical method used to estimate the value of a function at a specific point by using a polynomial of a certain degree. It is based on Taylor's theorem, which states that any smooth function can be represented by an infinite series of polynomials.
To calculate a Taylor Polynomial Approximation, you need to determine the degree of the polynomial and the point at which you want to approximate the function. Then, you must take the derivatives of the function at that point and plug them into the Taylor Polynomial formula, which is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^n(a)(x-a)^n/n!
The main purpose of a Taylor Polynomial Approximation is to provide a good estimation of a function at a specific point, especially when the function is difficult to evaluate directly. It is also useful for simplifying complicated functions and making them easier to work with.
No, a Taylor Polynomial Approximation can only be used for smooth functions, which means that the function must have continuous derivatives of all orders. It is not suitable for functions with sharp corners or discontinuities.
The accuracy of a Taylor Polynomial Approximation depends on the degree of the polynomial used. Generally, the higher the degree, the more accurate the approximation will be. However, it is important to note that even with a high degree polynomial, the approximation may not be completely accurate, especially if the function is highly nonlinear or has a complex shape.