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As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity of the graph on that interval. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 4, a point of inflection is the point on a curve where the concavity changes from concave up to concave down or concave down to concave up. The second derivative equals zero at a point of inflection. This is formalized in Theorem 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Theorem 3 gives us a second means of justifying that a minimum or maximum occurs at a critical point. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Optimization – the process of finding the “best” solution to a problem, e.g., • the largest area that a given length of fence can enclose • the path that minimizes travel time • the dimensions that maximize the volume of a box made from a rectangle of cardboard The process of optimization usually involves three steps: 1. Choose variables. 2. Determine the function. 3. Optimize the function. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In some cases, we may know the derivative and wish to find the function itself. A function F (x) whose derivative is f (x) is called an antiderivative of f (x). Theorem 1 gives us a method of finding the general antiderivative of a function, namely: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The process of antidifferentiation is denoted by the Leibniz symbol ∫ which specifies finding the indefinite integral of a function. Just as we have rules for differentiation, we also have rules for integration, of which the first is: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figure 2, we see that the slope of the graph of y = ln |x| is x–1 for all x. This leads to Theorem 3, which covers the exception to Theorem 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The linearity rules for differentiation have their counterparts for integration as noted in Theorem 4. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling derivatives of the basic trigonometric functions, we can recognize the basic trigonometric integrals. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling the Chain Rule, we have the following trigonometric integrals. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Remembering that the exponential function is its own derivative, we see how the indefinite integral of ex is ex + C. Applying the Chain rule in reverse to the exponential function, we obtain: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Since we used rectangles of common width, the approximate area under the curve becomes: Using the second rule of linearity of summations, we obtain Some texts refer to this method as the Rectangular Approximation Method or RAM. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The summations we use for the left-hand and midpoint approximations are respectively: The Rectangular Approximation Methods may further be abbreviated as RRAM, LRAM, and MRAM to indicate using the y-values on the right, left, and midpoint respectively. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The largest subinterval in a Riemann sum is known as the norm of the partition. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A function is integrable over [a, b] if all of the Riemann sums (not just the endpoint and midpoint approximations) approach one and the same limit L as the norm of the partition tends to zero, which we may write as:. N L lim R f , P , C lim P 0 P 0 f c x i i i 1 Assuming |R(f, P, C) – L| gets arbitrarily small as the norm||P|| tends to zero, the limit is called the definite integral of f (x) over [a, b]. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company R eferring to b a f x dx The definite integral is usually referred to as the integral of f over [a, b]. The function f (x) inside the integral symbol is called the integrand. The numbers a and b are called the limits of integration. The independent variable in the function is used as the variable of integration. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As illustrated in Figure 3, when f (x) is not positive for all x on [a, b], the definite integral yields the signed area between the graph and the x-axis. Signed area is defined as: If the graph in Figure 3 represented the velocity of a particle, then the integral from a to b would tell us the net displacement or movement of the particle from its starting point. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 8 illustrates the definite integral of f (x) = C over [a, b] for some C > 0. The integral may be evaluated by using Theorem 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Similar to limits, definite integrals have linearity properties as noted in Theorem 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we reverse the limits of integration, we change the sign of the signed area yielded by the definite integral as noted in the following definition: If the upper and lower limits of integration equal one another, the width of the interval is zero and Rogawski Calculus Copyright © 2008 W. H. Freeman and Company This theorem is illustrated in Figure 10. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 11 illustrates Theorem 5, the Comparison Theorem. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If a function has a lower bound m and an upper bound M on [a, b], then the Comparison Theorem may be written algebraically as: This is illustrated in Figure 12. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The Fundamental Theorem of Calculus ties together the operations of differentiation and integration in a relationship. Specifically, the integral of f (x) evaluated from a to b is equal to the antiderivative of f (x) evaluated at x = b minus the antiderivative of f (x) evaluated at x = a. W hen evaluating a definite integral, w e frequently have an interm ediate step as follow s: b a f x dx w here F x b a F x b a F b F a indicates w e are evaluating F x betw een a and b . Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling that the antiderivative of f (x) = x–1 is F(x) = ln x, the FTC tells us: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Basic Antiderivative Formulae x dx n x n 1 n 1 C dx ln x C x e dx e C x x sin xdx cos x C cos xdx sin x C sec xdx tan x C csc xdx cot x C sec x tan xdx sec x C csc x cot xdx csc x C 2 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company 2 If we take the derivative of the area function, we obtain the equation of the function under whose graph we are finding the area, or the derivative of the area function is the original function as stated in Theorem 1. S im ilarly, applying the C hain R ule: d dx gx a f t dt f g x g x Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If s′(t) is both positive and negative on [t1, t2], the integral will total the signed areas and give us the net change in s. This is the basis of Theorem 1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Theorem 2 formalizes what we observed in Figure 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company R em em bering that the derivative of ln x is 1 and that ln 1 0, x and that ln is not defined for x 0, w e can state: Figure 1 shows a graphical interpretation of this statement. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The following integral formulas apply to the inverse trigonometric function derivatives we learned in Section 3.9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company R em em bering that to find the derivative of y b , w e use x y bx e ln b x =e x ln b , thence y ln b e x ln b w e evaluate the follow ing integral: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company ln b e ln b x ln b b , x Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A frequently asked question is: “When will this process produce twice the original amount?” This question can be answered by solving 2 = 1 ekt for t. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In the case of exponential decay, the half-life of the substance is calculated by using: Archeologists use the half-life of carbon-14 to date organic materials. The process was developed following World War II. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Chapter 5, we learned that the definite integral gives us the signed area between the graph of the function and the x-axis. If we want to find the area between two graphs and f (x) > g (x) on [a, b], then we can use the definite integral to find this area. b f a x dx a g x dx a f x g x dx b b Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Sometimes it is more convenient to integrate with respect to y. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company More formally, Rogawski Calculus Copyright © 2008 W. H. Freeman and Company For some urban planning functions, it might be appropriate to consider the population density P as a function of distance r from the urban center P (r). To calculate the population living between r1 and r2 miles of the urban area, we would use the integral: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Using the fact that the volume of a solid is equal to the integral of the cross sectional area over the interval, we obtain: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company When calculating the volume of the solid obtained by rotating the region between curves f (x) and g(x), where f (x) > g(x) > 0 on [a, b], about the x-axis, use the following formula: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If the region is rotated about an axis other than the x- or y-axis, use the washer method, calculating the inner and outer radii from the stated axis. In Figure 7, the outer radius is R = g(x) – (–3) = g(x) + 3 and the inner radius is r = f (x) – (–3) = f (x) + 3 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Notice how the sketch in Figure 8 helps us determine the inner and outer radii and what to subtract from what to obtain them. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As shown in Figure 9, the same process works for axes parallel to the y-axis. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we divide the interval [a, b] into N even intervals, the area is found using the Trapezoidal Rule Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rogawski Calculus Copyright © 2008 W. H. Freeman and Company • Differential Equations • Slope Fields