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";s:4:"text";s:7879:"The derivative of the integral of a function is usually the original function. If we compare differentiation and integration based on their properties: Both differentiation and integration satisfy the property of linearity, i.e.,k1 and k2 are constants in the above equations. In the figure, h is positive and Every differentiation formula, when read in reverse, gives us an example of a primitive of some functionfand this, in turn, leads to an integration formula for this function.From the differentiation formulas worked out thus far we cari derive the following integration formulas as consequences of the second fundamental theorem. As you go to the right on the cumulative graph, the height of each successive column simply grows by the amount of profits earned in the corresponding single year shown in the upper graph. Similarly, if we operate on a continuous function f by integration, we get a new function (an indefinite integral off) which, when differentiated, leads back to the original function f. For example, if f(x) = x2, then an indefinite integral A off may be defined by the equation$A(x)=\int_c^x f(t) \ dt = \int_c^x t^2 \ dt = \frac{x^3}{3} - \frac{c^3}{3},$where c is a constant. Let x be a point of continuity off, keep x fixed, and form the quotient[A(x + h) - A(x)]/hTo prove the theorem we must show that this quotient approaches the limit f(x) as h → 0. At that point on C, you run across 1 year and rise up $1,250,000, the ’08 profit you see on the frequency distribution graph (F for short). Integration of rational powers. Using the definition of limit, we must show that for every ε > 0 there is a δ > 0 such that (5.4) |G(h)| < ε whenever 0 < |h| < δ.Continuity of f at x tells us that, if ε is given, there is a positive δ such that(5.5) |f(t) -f(x)| < ε/2whenever(5.6) x - δ < t < x + δ.If we choose h so that 0 < h < δ, then every t in the interval [x, x + h] satisfies (5.6) and hence (5.5) holds for every such t. Using the property$|\int _x^{x+h} g(t) \ dt| \ \leq \ \int _x^{x+h} |g(t)| \ dt$with g(t) =f(t) -f(x), we see that the inequality in (5.5) leads to the relation$|\int _x^{x+h} [f(t) - f(x)] \ dt | \leq \int _x^{x+h} |f(t) - f(x)| \ dt \leq \int _x^{x+h} \frac{1}{2} \epsilon \ dt = \frac{1}{2} h \epsilon < h \epsilon$If we divide by h, we see that (5.4) holds for 0 < h < δ. This Will enable us to extend (5.9) to all real exponents except - 1.Note that we cannot get P'(x) = 1/x by differentiation of any function of the form P(x) = xn. This formula, proved for all integers n ≥ 0, also holds for all negative integers except n = -1, which is excluded because n + 1 appears in the denominator. The numerator is$A(x+h) - A(x) = \int_c^{x+h} f(t) \ dt - \int_c^x f(t) \ dt = \int_x^{x+h} f(t) \ dt.$If we write f(t) =f(x) + [f(t) -f(x)] in the last integral, we obtainfrom which we find(5.3)$\frac{A(x+h) - A(x)}{h} = f(x) + \frac{1}{h} \int_x^{x+h} [f(t) - f(x)] \ dt $Therefore, to complete the proof of (5.1), all we need to do is show that$\lim_{h\rightarrow 0} \ \frac{1}{h} \int_x^{x+h} [f(t) - f(x)] \ dt = 0$It is this part of the proof that makes use of the continuity off at x.Let us denote the second term on the right of (5.3) by G(h). A frequency distribution histogram (above) and a cumulative frequency distribution histogram (below) for the annual profits of Widgets-R-Us show the connection between differentiation and integration. If x > c, then$\int _c^x f'(t) \ dt \geq 0$, and hence f(x) ≥ f(c). You can see that the ’02 column shows the ’02 rectangle sitting on top of the ’01 rectangle which gives that ’02 column a height equal to the total of the profits from ’01 and ’02. Let's see how this works by differentiating 4 x to the power of 7 and then integrating 4 x to the power of 7 and seeing how it is different. Geometric motivation. I struggled with the exact same thing for a while. To exhibit such a function all we need to do is write a suitable indefinite integral; for example,$P(x) = \int _x^c \frac{1}{t} \ dt \qquad \qquad if \ x > 0$This integral exists because the integrand is monotonie. Thus, the slope on C (at ’08 or any other year) can be read as a height on F for the corresponding year. Now let’s go through how these two graphs explain the relationship between differentiation and integration. We proved this fact earlier as an immediate consequence of the definition of derivative. This formula, which expresses f in terms of its derivative f â, enables us to deduce prolperties of a function from properties of its derivative. Integration of the sine and cosine. No significant relationships between integration and differentiation were associated with mortality with the exception of the validation sample in 1997 in which a higher likelihood of mortality was found with an increase of Acute Service Differentiation, and a lower likelihood associated with Long Term Differentiation. The upper graph in the figure shows a frequency distribution histogram of the annual profits of Widgets-R-Us from January 1, 2001 through December 31, 2013. In: Integration and Modern Analysis. Look at the ’02 column in the lower graph and the ’01 and ’02 rectangles in the upper graph, for example. Integration and differentiation effectively un-do each other. The difference is simply that in the cumulative graph, the height of each column shows the total profits earned since 1/1/2001. Still trying to grasp how differentiation and integration work? Therefore, if f' is continuous and increasing on 1, Equation (5.11) shows that f is convex on Z. Therefore, we use a different method to prove the theorem under this weaker hypothesis. How Statistics Shows the Connection Between Differentiation and Integration, The Integration by Parts Method and Going in Circles, Trig Integrals Containing Sines and Cosines, Secants and Tangents, or…, The Partial Fractions Technique: Denominator Contains Repeated Linear or Quadratic…. The first fundamental theorem of calculusWe corne now to the remarkable connection that exists between integration and differentiation. Then consider a function and its derivative. The lower graph in the figure is a cumulative frequency distribution histogram for the same data used for the upper graph. Therefore, A(x) - P(x) = -P(c), from which we obtain (5.7). For example, if 0 < a < b and n = -1/2, we find The reverse is also true, to a point. And you’ve seen how the same ’01 through ’08 rectangles that lie along the stair-step top of C can also be seen in a vertical stack at year ’08 on C. The cumulative graph is drawn this way so it’s even more obvious how the heights of the rectangles add up. Take a look at the figure. the sine, the second fundamental theorem also gives us the following formulas:$\int _a^b \cos x \ dx = \sin x |_a^b = \sin b - \sin a$We also proved, as part (c) of Theorem 4.7, the converse of this statement which we restate here as a separate theorem. (Note: Most cumulative histograms are not drawn this way. (Make sure you see how this works.)$\int_x^{x+h} f(t) \ dt = \int_c^{x+h} f(t) \ dt - \int_c^x f(t) \ dt = A(x+h) - A(x)\$The example shown is continuous throughout the interval [x, x + h]. Geometric motivation. In a nutshell (keep looking at those rectangles with the bold border in both graphs), the slopes of the rectangles on C appear as heights on F. That’s differentiation. So here’s the calculus connection. No significant relationships between integration and differentiation were associated with mortality with the exception of the validation sample in 1997 in which a higher likelihood of mortality was found with an increase of Acute Service Differentiation, and a lower likelihood associated with Long Term Differentiation. As to the more broad integral/derivative question. 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