Calculus I Integration: A Very Short Summary
For example, the function f(x) = 2x has antiderivatives such as x 2, x + 3, x −π, and x2 + 002, just to name a few Definition: General Antiderivative The function F(x) + C is the General Antiderivative of the function f(x) on an interval I if F0(x) = f(x) for all x in I and C is an arbitrary constant
2) = 0 So this is very very attracting f(x) = 5 2 x 3 2 x 2 has a xed point at x = 1, and f0(1) = 5=2 1 = 1=2, so it is attracting (the negative sign means that we reach the xed point by bouncing around from left to right) By the way, there’s another xed point at x = 0, but there f0(0) = 5=2, so that one is repelling f(x) = 13 4 x 3 2 x
10 5 Sketch the graph of the function f(x) = x 2 (x–2)(x + 1) Label all intercepts with their coordinates, and describe the “end behavior” of f That f(x) is a 4th-degree POLYNOMIAL* function is
Example 2 f(x) = x n where n = 1, 2, 3 d In this example we answer the question “What is x n ?” Once we know the dx answer we can use it to, for example, find the derivative of f(x) = x4 by replacing n by 4 At this point in our studies, we only know one tool for finding derivatives – the difference quotient
We can use this graph that we know and the chart above to draw f(x)+2, f(x) 2, 2f(x), 1 2f(x), and f(x) Or to write the previous five functions without the name of the function f, these are the five functions x2+2,x22, 2x2, x2 2,andx2 These graphs are drawn on the next page 68
16 ) f(x)= 1 x3 17 ) g(x)= 2x4 x+3 18 ) h(x)= 3x2 4x+5 3x+4 19 ) Use the formula n k = n k(nk) to write the number 6 2 as a natural number in standard form (For example, 7, 13, 25, etc are standard forms
1 2 Examples of the Riemann integral 5 Next, we consider some examples of bounded functions on compact intervals Example 1 5 The constant function f(x) = 1 on [0,1] is Riemann integrable, and
The region D is a circle of radius 2 p 2 • fx(x,y)=y • fy(x,y)=x We therefore have a critical point at (0 ,0) and f(0,0) = 0 Now let us consider the boundary
2 If X i, i = 1,2, ,n, are iid exponential RVs with mean 1/λ, the pdf of P n i=1 X i is: f X1+X2+···+Xn (t) = λe −λt (λt) n−1 (n−1), gamma distribution with parameters n and λ 3 If X1 and X2 are independent exponential RVs with mean 1/λ1, 1/λ2, P(X1 < X2) = λ1 λ1 +λ2 4 If X i, i = 1,2, ,n, are independent
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COVID-19 ICI, LE MASQUE EST OBLIGATOIRE Ensemble, 17 juillet 2020 faisons bloc contre le coronavirus
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Le portail de la validation des acquis de l'expérience
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15 Demande de crédits d’heures Déclaration du salarié Réservé à la CCQ ID M058 DL11 CT 24-07 (1908) SECTION 1 : Renseignements sur l’identité
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Considere uma função f que associa x = (x1x2
g x dx para f e g integráveis no intervalo fechado a
x2 e fyy y f y. 2f y2. 2.2.1 DERIVADAS PARCIAS SEGUNDAS MISTAS. TEOREMA 1: Seja f uma função de duas variáveis x e y. Se ffx
(x2 + y2)2. fy = x5 - xy4 - 4x3y2. (x2 + y2)2. (1). (b) fx(0
2e?2x . Resta mostrar que sua integral é 1. Mas sabemos a (c) Determine P(X ? 1/2) P(X > 1/2) e P(1/4 ? X ? 3/4). ... P(X < m)=0
x2+y2+z4. (b) (4 pontos). lim. (xy)?(0
1) Determinar a área limitada pela curva. 2 xx5y. -. = e pelo eixo x. 0xx5. 2= -. 0)x5(x 0 1. 2. ?. ?. = - a a a dxxf dxxf. 0. )(2. )( y f(x)=x2. X.
1. Encontre uma equação da reta tangente à parábola y = x2 no ponto P(1 1). 2. Determinar a equação da reta tangente à curva f(x)= x no ponto P da ...
Deter- mine a função de probabilidade de X. Solução: O espaço amostral S é formado por 36 pares. S = {(1
of change of f as x varies between x 1 and x 2 is the quotient average rate of change = y x = y 2 y 1 x 2 x 1 = f(x 2) f(x 1) x 2 x 1 (6 1) It’s a linear approximation of the behavior of f between the points x 1 and x 2 7 Quadratic Functions The quadratic function (aka the parabola function or the square function) f(x) = ax2 + bx+ c (7 1
Functions)Worksheet) Domain)Range)and)Function)Notation) 1 #Find#the#domain# ####a € f(x)= x?4 x?2 #####b € g(x)= x2+5 x+1 # #####c € h(x)= x x2?9
What is f(x) = 1x?
Key Point. A function of the form f(x) = ax (where a > 0) is called an exponential function. The function f(x) = 1x is just the constant function f(x) = 1. The function f(x) = ax for a > 1 has a graph which is close to the x-axis for negative x and increases rapidly for positive x.
What is the formula for calculating the PDF of X and Y?
Y. S. Han Multiple Random Variables 111 • Marginal pdfs of X and Y are fX(x) = e?(x?m1)2/2?2 1 ? 2??1 , fY(y) = e?(y?m2)2/2?2 2
Which function is defined by f (x) = 3x + 2?
Prove the function f: R ? R defined by f ( x) = 3 x + 2 is one-to-one. Assume f ( x 1) = f ( x 2), which means 3 x 1 + 2 = 3 x 2 + 2. so x 1 = x 2.
What is PDF/X?
PDF/X was the first ISO standard based on PDF technology. A subset of the PDF specification, PDF/X was designed to enable PDF files to meet specific user needs. For example, the relevant files must be complete, i.e., self-sufficient.
Aula 3 Derivadas Parciais .5cm MA211 - Cálculo II