Mathematics
Accumulation Function
An accumulation function is a mathematical function that represents the accumulation of a quantity over time. It is used to calculate the total amount of a quantity that has accumulated over a given period of time, based on the rate at which it is accumulating. The accumulation function is often used in calculus to model real-world phenomena such as population growth or the accumulation of wealth.
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eBook - ePubThe Learning and Teaching of Calculus
Ideas, Insights and Activities
- John Monaghan, Robert Ely, Márcia M.F. Pinto, Mike Thomas(Authors)
- 2023(Publication Date)
- Routledge(Publisher)
t and sometimes not?The Accumulation Function entails a generalisation on the part of the students to imagine a function that provides a distinct integral for each different stopping-valuet ^. This has similarities to the reasoning used to imagine a gradient function that provides a curve’s gradient at each point x, as discussed in Chapter 4 .The context of area under a curve really helps for picturing an Accumulation Function, so we revisit the quadrature problem. If we picture the area under the curve y = xk not up to the fixed stopping-point x = 1 but up to the variable stopping-pointx =, we can imaginex ^A (as an area that accumulates asx ^) =∫ 0x ^x kd xx ^moves to the right (Figure 5.13a ). We can also call this an area-so-far function because it outputs an area accumulated ‘under’ the graph up to the input pointx ^.Two area-so-far functionsFIGURE5.13AFigure 5.13b shows this forf (, where the graph of y = r(t) is a little more exciting than y = xk .t ^) =∫ 4r ( t ) d tt ^FIGURE5.13BOnce again, we can help students to understand this accumulation in a dynamic way by employing some digital technology. The GeoGebra program seen in Figure 5.14 has been set up to demonstrate the area accumulated under the graph of the functionf ( x ) =from x = 1 to x = k. The use of a slider for the value1 xenables students not only to visualise the accumulation but also the fact that, as the area is plotted, the graph of the area-so-far function emerges. Appreciating that this is a function is an important step, and they may also speculate on its closed algebraic form. The values of this function when k∫ 1 k1 xd x
eBook - PDF- Lorenzo Sadun(Author)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
Most of all, we had to learn how to estimate bulk quantities with Riemann sums, and how to get them exactly as limits of Riemann sums. The upshot of Chapter 7 is that lots and lots of real-world quantities can be represented as definite integrals. Remember that a definite integral is a number, not a function. Between 1:00 p.m. and 3:00 p.m. we might travel 105 miles (by car), or we might travel 5 miles (on foot), but we don’t travel () miles. Also, there isn’t any single value of (or ) to associate with the integral. The integral ∫ () adds up the contributions of all values of between and . Accumulation Functions. The Accumulation Function ∫ 0 () is a running total. It is a function of the endpoint . For instance, if the speed of your car is (), then ∫ 0 () is the reading of your odometer at time . It gives the total distance that the car drove between time 0 (the day the car was made) and time . The dummy variable in the integral runs from 0 to , as the odometer reading today is the result of all of the driving that you did between time 0 and time . Remember that the Accumulation Function is a function, not a number. It takes or as the input and outputs the number (8.2) () = ∫ 0 () or () = ∫ 0 (). Accumulation Functions are closely related to definite integrals. Partly this is be- cause the output of the Accumulation Function is a definite integral. Partly it is because we can use Accumulation Functions to compute definite integrals. As we saw in Chap- ter 7, (8.3) ∫ () = ∫ 0 () − ∫ 0 () = () − (). So far, we have defined our Accumulation Functions starting at 0, but we could just as well have used a different starting point or starting time. For instance, a trip odometer is an Accumulation Function that starts each time you press the reset button.
eBook - PDFApplied Calculus
for Business, Life, and Social Sciences
- Denny Burzynski(Author)
- 2014(Publication Date)
- XYZ Textbooks(Publisher)
In writing the function F(x), we omit the constant of integration, C. 3. The total accumulation was the difference between F(b) and F(a); that is, F(b) − F(a). It is common to use F(x) a b to indicate the evaluation F(b) − F(a). The Fundamental Theorem of Calculus We now present the Fundamental Theorem of Calculus, a theorem that summa- rizes this evaluation process. The Difference Between Indefinite and Definite Integrals It is worthwhile noting the difference between the indefinite integral f (x) dx and the definite integral a b f (x) dx 1. The indefinite integral f (x) dx is a function of x and represents a family of anti- derivatives of f (x). For example, x 3 dx = x 4 __ 4 + C, which is a function of x The Fundamental Theorem of Calculus If f (x) is a continuous function on an interval [a, b], where a < b, and if F(x) is an antiderivative of f (x), then the definite integral of f (x) from a to b is a b f (x) dx = F(x) a b = F(b) − F(a) and represents the total amount accumulated by F(x) as x increases from a to b. The number a is called the lower limit of integration and the number b the upper limit of integration. 5.3 The Definite Integral 411 2. The definite integral a b f (x) dx is an accumulator and represents a real number, a constant that records the total amount accumulated by a particular member of a family of antiderivatives. For example, 1 3 x 3 dx = x 4 __ 4 1 3 = 3 4 __ 4 − 1 4 __ 4 = 81 ___ 4 − 1 __ 4 = 80 ___ 4 = 20, which is a real number Properties of the Definite Integral The definite integral has the following properties. 1. a b k f (x) dx = k a b f (x) dx, where k is a constant. 2. a a f (x) dx = 0 No accumulation without change. 3. a b f (x) dx = − b a f (x) dx 4. a b [f (x) ± g (x)] dx = a b f (x) dx ± a b g (x) dx 5. a b f (x) dx = a c f (x) dx + c b f (x) dx, where a ≤ c ≤ b.
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