Improper Integral AP X Version 4.0 x E +∞ x dx x2 + 1 ∫-∞ 1 + x2 u = 1 + x2 +∞ x du = 2x dx CALCULUS II = 1 2 dx 0.4 2 ∫-∞ 1 +x2 +∞ 1 2 0.2 = 2 ln(1 + x ) ∞ -10 -5 - X = {∞ - ∞}, 5 10 indeterminate form -0.2 -0.4 Does Not Exist. Gregory Hartman, Ph.D. Department of Applied Mathematics Virginia Military Institute Arc Length What is the arc length, the length of the curve, of y = f(x), a ≤ x ≤ b? Y Contributing Authors Troy Siemers, Ph.D. y = f(x) ds Department of Applied dy dx Mathematics Virginia Military Institute Brian Heinold, Ph.D. Department of Mathematics and Computer Science X a x b Mount Saint Mary’s University Theorem Let y = f(x) be differentiable for a ≤ x ≤ b. Then its arc length on that interval is b Dimplekumar Chalishajar, Ph.D. _dy 2 s = ∫ 1 + dx. a dx Department of Applied Mathematics Derivation Virginia Military Institute f is differentiable on the interval. So f is differentiable at x and therefore locally linear or equivalently 'asymptotically straight' there. Therefore Editor 2 2 2 ds ≈ dx + dy Theorem of Pythagoras Jennifer Bowen, Ph.D. _dy 2 2 = [1 + ( dx dx Department of Mathematics and Computing Science, + dy 2 ds ≈ 1 dx dx b The College of Wooster dy -2 ⟹ s = 1 + dx. ∫a dx Apex Infinitesimal Calculus, Volume II Second Preliminary Version William Freed Concordia University of Edmonton CLICK on desired section Volume II Review 1- Infinitesimals. Review 2 - Derivatives and Integrals 6 Other Elementary Functions. Inverse Functions. 6.1 Theory of Inverse Functions 6.2 Exponential Functions Review 6.3 The Natural Logarithmic Function. Other Bases 6.4 Inverse Trig Functions Algebra 6.5 Calculus of Inverse Trig Functions 6.6 Hyperbolic Functions 6.7 Inverse Hyperbolic Functions 6.8 L'Hôpital's Rule Exam #1 7 Techniques of Integration 7.1a Method of Substitution Review 7.1b Method of Substitution Advanced 7.2 Integration by Parts 7.3 Trig Integrals 7.4 Trig Substitutions 7.5 Partial Fractions 7.6 Improper Integrals 7.7 Numerical Integration Exam #2 8 Applications of Integration 8.1 Area between Curves 8.2 Volumes: Method of Slicing. The Disk Method 8.3 Volumes: the Shell Method 8.4 Arc Length: Surface Area 8.5 Work Problems 8.6 Present and Future Value Exam #3 9 Generalized Functions 9.1 Generalized Functions 9.2 Geometric Calculus of Generalized Functions 9.3 Analytic Calculus of Generalized Functions 10 Differential Equations 10.1 First order Differential Equations. Variables Separable 10.2 Homogeneous First Order Linear Equations 10.3 Nonhomogeneous First Order Linear Equations Exam #4 Preface Infinitesimal Calculus, Volume II In Volume I we set up, in detail, the basics of hyperreal calculus. The payoff was being able to do a thorough, elementary and intuitive development of calculus. Proofs of the all basic operational formulas and theorems were accomplished as well as more difficult foundational theorems such as the Extreme Value Theorem and the Riemann Integrability of Continuous Functions over a Closed Interval. In this Volume II, early/late transcendentals, we continue the theoretical development - especially applications of integration - using hyperreal analysis. Historically, the fundamental starting point of calculus was the differential . This is because the basic laws of science and geometric are often simpler to discover over an infinitely short interval of space or time. Now, with the basics of elementary applications better understood, it makes sense to start where students have better mathematical strengths. To find the rate of change of Q with respect to t, the derivative , you divide the differential by dt. To find the change in Q from time t1 to t2 , thea integral of f(t) over the interval, you sum the infinitesimal f(t) dt’s from t1 to t2 and find the closest real number to this sum. dQ = f (t ) dt ↑ dQ = f (t) dt the differential ↓ Δ = t 2 ( ) Q ∫t 1 f t dt The greatest benefit of infinitesimal methods is providing a reliable, intuitive and foolproof guide to setting up applications of integration. The differential in modern textbooks is a real number and is relegated to a method of approximating a function near a point. It is also used somewhat dishonestly and without further explanation in order to be able to employ powerful infinitesimal techniques such as the method of change of variable in integration and separation of variables in differential equations. The symbol dy now is a fraction dx and needs no awkward discussion and the chain rule is obvious and proves itself (check it out in Volume I. The great tools of infinitesimal calculus are infinitesimals, which facilitates the ultra-precise calculations used to do calculus and the equivalence relation ≈ which enables the rapid comparison and simplification of hyperreal expressions. Misuse of the key comparison symbols =, ≈ and ≈> is not a big problem because their outputs tend to differ only by an infinitesimal. ≐ is the approximate equality for real numbers. 4 Chapter 6 Advanced Transcendental Functions 6.1 Inverse Functions Theory - a Review The main purpose of the inverse function is to solve for the independent variable in y = f(x): y = f(x) ⟺ x = f-1(y) I. When does f have an inverse? Answer: if for each y in the range of f, there is exactly one x. Y Y y=g(x) y=f(x) y ⟶ ↓ y ⟶ ↓ ↓ X X x x1 x2 Each y, one x. Some y's, more than one x. We say that f is one-to-one or invertible if for each y, there is exactly one x (or equivalently f passes the Horizontal Line Test.) Note that increasing functions are invertible; so are decreasing functions. II. How to find the formula for f-1? y = f(x) Solve for x: -1 x = f (y) -1 -1 III. The inverse function y = f (x) We prefer as usual, in working with or studying f to call the dependent variable y. So interchanging x and y: -1 y = f (x) -1 -1 (In applications, f is usually of interest only when it cannot be found exactly! f is then found numerically by -1 computer.) Note that y = f(x) and y = f (x) have different graphs and therefore are different functions. -1 -1 IV. How to graph y = f (x)? Since we interchanged x and y to obtain the graph of y = f (x) from that of y = f(x): Reflect the graph of y = f(x) across the line y = x. Y y = x 4 (3, 4) Y 3 (4, 3) x = f -1(y) y = x y = f(x) 2 y = f -1(x) (x, y) 1 x = f(y) ↘ X (y, x) 1 2 3 4 How it works The point (a, b) reflected across ithe line y = x is the point (b, a). X 5 V. Inverse Function Identities These identities are verified by following the f -1(f (x)) = x arrows in the graphs below. Start at x in the f f -1(y) = y left graph and at y in the right graph. Y Y y = f(x) y = f(x) ⟶ y ⟶ f(x) ⟵ ⟵ f( f -1(y)) ↑ ↓ ↑ ↓ X X x f -1(y) f -1(f(x)) VI. Application: Universal Equation Solver Solve: f(x) = c f -1(f (x)) = f -1(c) Taking f -1 of both sides The solution: x = f -1(c) . Inverse function identity VII. Calculus of a inverse function y = f -1(x) ⟺ x = f (y) 1 = f '(y) dy Differentiating implicitly; Chain Rule dx dy = 1 dx f '(y) or dy = 1 dx dx . Live math. dy Easy Example y = f(x) = 2x + 1 Y 5 4 I. 3 2 1 X 1 2 Passes the Horizontal Line Test. So f -1(x) exists. II. Find the formula for f-1 ? y = f(x) = 2x + 1 Solve for x : x = f -1 (y) 1 1 = f -1 ( ) = . x y 2y - 2 III. The inverse function. Interchange x and y. 1 1 f -1 ( ) y = x = 2 x - 2. 6 IV. Its graph. Y 5 y = f(x) 4 3 2 y = x 1 y = f -1(x) X -2 -1 1 2 3 4 5 V. Inverse function Identities. Let us check. -1 1 1 1 1 f (f(x)) = f(x) - = (2 x + 1)- = x. 2 2 2 2 1 1 f(f -1(x)) = 2f -1(x) + 1 = 2( x - ) + 1 = x. 2 2 Properties VI and VII are left as exercises. They are important for more difficult examples. Harder Example y = g(x) = x - 2 ? If you solve for x and then interchange x and y, you get y = g -1(x) = x2 + 2. Lets see. Y 4 ? y = g -1(x) 3 2 y = g(x) 1 X -2 -1 1 2 3 4 5 -1 Recall that squaring an equation can give spurious solutions. Clearly, reflecting y = g(x) about the line y = x, the correct inverse is -1 2 y = g (x) = x + 2, x ≥ 0. 2 Another Example with a problem y = h(x) = x + 1 -1 ?? If you solve for x and then interchange x and y, you get y = h (x) = ± x - 1 . Let's see. Y 4 y = h(x) 3 2 1 X -2 -1 1 2 3 4 -1 The problem here is that the function y = h(x) is not one-to-one. So it does not have an inverse. Nevertheless, it will be useful at times to do the best we can in finding a related inverse.