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- #Partial derivatives using calculus made easy ti89 manual#
- #Partial derivatives using calculus made easy ti89 series#
Then the m + n-th order partial derivative of g ( x, y), Suppose the domain of the two-variables function If the assumptions are the same as Theorem 1.
#Partial derivatives using calculus made easy ti89 series#
Next, we find the infinite series forms of any order partial derivatives of the two-variables function (2). Using differentiation term by term theorem, differentiating n-times with respect to x, and m-times with respect to y on both sides of (9), we have Then the m + n-th order partial derivative of f ( x, y), If the domain of the two-variables function
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Suppose a, b are real numbers, a ≠ 0, p is an integer, and m, n are non-negative integers. In the following, we determine the infinite series forms of any order partial derivatives of the two-variables function (1). Suppose α, β are real numbers, α > 0, and p is an integer. īefore deriving the first major result in this study, we need a lemma. Then is uniformly convergent and differentiable on. Differentiation Term by Term Theorem ()įor all non-negative integers k, if the functions satisfy the following three conditions: (i) there exists a point such that is convergent, (ii) all functions are differentiable on open interval, (iii) is uniformly convergent on. Next, we introduce an important theorem used in this study. Taylor Series Expression of Complex Cosine Function Taylor Series Expression of Complex Sine Function , where p is any integer, and is any real number. Suppose r is any real number, m is any positive integer. For the two-variables function, its n-times partial derivative with respect to x, and m-times partial derivative with respect to y, forms a -th order partial derivative, and denoted by. We denote a the real part of z by, and b the imaginary part of z by. Let be a complex number, where, are real numbers. Main Resultsįirstly, we introduce some notations and formulas used in this paper. Therefore, Maple provides insights and guidance regarding problem-solving methods.
#Partial derivatives using calculus made easy ti89 manual#
This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. The research methods adopted in this paper involved finding solutions through manual calculations and verifying these solutions by using Maple. In addition, we provide some examples to do calculation practically. The study of related partial differential problems can refer to. We can obtain the infinite series forms of any order partial derivatives of these four types of two-variables functions using differentiation term by term theorem these are the major results of this study (i.e., Theorems 1-4), and hence greatly reduce the difficulty of calculating their higher order partial derivative values. Where a, b are real numbers, a ≠ 0, and p is an integer. In this paper, we study the partial differential problem of the following four types of two-variables functions q is large), and hence to obtain the answers by manual calculations is not easy. These two procedures will make us be faced with increasingly complex calculations when calculating higher order partial derivative values ( i.e. On the other hand, calculating the q-th order partial derivative value of a multivariable function at some point, in general, needs to go through two procedures: firstly determining the q-th order partial derivative of this function, and then taking the point into the q-th order partial derivative. For example, Laplace equation, wave equation, as well as other important physical equations are involved the partial derivatives. In calculus and engineering mathematics curricula, the evaluation and numerical calculation of the partial derivatives of multivariable functions are important. The computation results of Maple can be used to modify our previous thinking directions, thereby forming direct and constructive feedback that can aid in improving understanding of problems and cultivating research interests. In addition, through the numerical and symbolic computations performed by Maple, the logic of thinking can be converted into a series of instructions. The superiority of Maple lies in its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. Maple possesses significance among mathematical calculation systems and can be considered a leading tool in the CAS field. The rapid computations and the visually appealing graphical interface of the program render creative research possible. The computer algebra system (CAS) has been widely employed in mathematical and scientific studies.