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Short History of Calculus - Nipissing University Word
... powers) and implicit differentiation of polynomials in x and y Together with analytic geometry this made possible to find tangents, maxima and minima of all algebraic curves p (x,y) = 0 Newton’s calculus of infinite series (1660s) allowed for differentiation and integration of all functions expr ...
... powers) and implicit differentiation of polynomials in x and y Together with analytic geometry this made possible to find tangents, maxima and minima of all algebraic curves p (x,y) = 0 Newton’s calculus of infinite series (1660s) allowed for differentiation and integration of all functions expr ...
7.4 - The Fundamental Theorem of Calculus
... • But we have introduced a new symbol for our antiderivative the integral symbol (∫) • So now we can put our relationships together n ...
... • But we have introduced a new symbol for our antiderivative the integral symbol (∫) • So now we can put our relationships together n ...
Expected Values of Random Variables
... Already demonstrated in R for p = 1/2. Consider the random variables gt(X) = etX. By summing a geometric series we can show that mX(t) = E(etX) = pet/(1 – qet), for t in a neighborhood of t = 0. (Essentially, we need the denominator to be positive.) Using the expansion of ex: mX'(0) = [dmX(t) / dt]t ...
... Already demonstrated in R for p = 1/2. Consider the random variables gt(X) = etX. By summing a geometric series we can show that mX(t) = E(etX) = pet/(1 – qet), for t in a neighborhood of t = 0. (Essentially, we need the denominator to be positive.) Using the expansion of ex: mX'(0) = [dmX(t) / dt]t ...
Recurrence relations and generation functions
... sequence of squares 0, 1, 4, …, n2,….. Solution: by the above Newton’s binomial theorem with n =2 and r =1, (1-x)-2 = 1+2x+3x2+…+nxn-1+…. Hence x/(1-x)2=x+2x2 + 3x3+…+nxn +….. Differentiating, we obtain （1+x)/(1-x)3=1+22x+32x2+…+n2xn-1+….. Multiplying by x, we obtain the desired generating function ...
... sequence of squares 0, 1, 4, …, n2,….. Solution: by the above Newton’s binomial theorem with n =2 and r =1, (1-x)-2 = 1+2x+3x2+…+nxn-1+…. Hence x/(1-x)2=x+2x2 + 3x3+…+nxn +….. Differentiating, we obtain （1+x)/(1-x)3=1+22x+32x2+…+n2xn-1+….. Multiplying by x, we obtain the desired generating function ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.