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
Iteration, Fixed points - MIT Mathematics
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
f(x+h) – f(x) 2 h NOTE
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 - MIT OpenCourseWare
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
Graph Transformations
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
More on functions
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
The Riemann Integral
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
58 Lagrange Multipliers
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
Exponential Distribution
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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