01544

Tarrant County College

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Algebra Integrals
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juni 2023

2022/2023
Integrals 1.1. Areas and Distances. The Definite Integral 1.1.1. The Area Problem. The Definite Integral. Here we try to find the area of the region S under the curve y = f (x) from a to b, where f is some continuous function. Y y=f(x) S O a b X In order to estimate that area we begin by dividing the interval [a, b] into n subintervals [x0 , x1 ], [x1 , x2 ], [x2 , x3 ], . . . , [xn−1 , xn ], each of length ∆x = (b − a)/n (so xi = a + i∆x). Y O a b 6 X 1.1. AREAS AND DISTANCES. THE DEFIN...
kizokira
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Algebra applications of integration
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24

juni 2023

2022/2023
Applications of Integration 2.1. More about Areas 2.1.1. Area Between Two Curves. The area between the curves y = f (x) and y = g(x) and the lines x = a and x = b (f , g continuous and f (x) ≥ g(x) for x in [a, b]) is Z b Z b Z b f (x) dx − g(x) dx = [f (x) − g(x)] dx . A= a a a Calling yT = f (x), yB = g(x), we have: Z b A= (yT − yB ) dx a Example: Find the area between y = ex and y = x bounded on the sides by x = 0 and x = 1. Answer : First note that ex ≥ x for 0 ≤ x ≤ 1. So: ¸1...
kizokira
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Algebra differential equations
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9

juni 2023

2022/2023
Differential Equations 3.1. Differential Equations and Separable Equations 3.1.1. Population Growth. The growth of a population is usually modeled with an equation of the form dP = kP , dt where P represents the number of individuals an a given time t. This model assumes that the rate of growth of population is proportional to the population size. A solution to this equation is the exponential function: P (t) = Cekt . Check: P 0 (t) = kCekt = kP (t). A more realistic model takes into account tha...
kizokira
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Overig

Algebra Infinite sequences and series. plus Appendix
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38

juni 2023

2022/2023
Infinite Sequences and Series 4.1. Sequences A sequence is an infinite ordered list of numbers, for example the sequence of odd positive integers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 . . . Symbolically the terms of a sequence are represented with indexed letters: a1 , a2 , a3 , a4 , a5 , a6 , a7 , . . . , an , . . . Sometimes we start a sequence with a0 (index zero) instead of a1 . Notation: the sequence a1 , a2 , a3 , . . . is also denoted by {an } or {an }∞ n=1 . Some seque...
kizokira
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