PPT-AP Problems Involving the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus If then 1 2 One of the hardest calculus topics to teach in the old days was Riemann sums They were hard to draw hard to compute and many felt totally unnecessary. For example the graph of a di64256erentiable function has a horizontal tangent at a maximum or minimum point This is not quite accurate as we will see De64257nition Let an interval A point is a local maximum of if there is 948 0 such that wheneve 3 Theorem 1 Theorem Let be a discrete valuation ring with 64257eld of fractions and let be a smooth group scheme of 64257nite type over Let sh be a strict Henselisation of and let sh be its 64257eld of fractions Then admits a N57524eron model over By Jess Barak, Lindsay Mullen, Ashley Reynolds, and Abby . Yinger. The concept of unique factorization stretches right back to Greek arithmetic and yet it plays an important role in modern commutative ring theory. Basically, unique factorization consists of two properties: existence and uniqueness. Existence means that an element is representable as a finite product of . Learner Objective: Students will apply a Right Angle Theorem as a way of proving 
 that two angles are right angles and to solve problems involving right angles.. Advanced Geometry. Learner Objective: Students will apply a Right Angle Theorem as a way of proving 
 that two angles are right angles and to solve problems involving right angles.. 3: Indefinite and Definite . Integrals, . the Fundamental Theorem of . Calculus, Integration Via Substitution, Integration by Parts, Computing Areas, Computing Volumes by the Disk and Shell Methods. Part I: Indefinite and Definite Integrals and the Fundamental Theorem of Calculus. Chapter 7 Day 1. Basic Integration Rules. Fitting Integrands to Basic Rules. Fitting Integrands to Basic Rules. So far we have dealt with only basic integration rules. But what happens when our integral doesn’t fit into one of those categories? What then?. BARROW AND LEIBNIZ ON THE FUNDAMENTAL THEOREM . OF CALCULUS. In . 1693, . Leibniz . published . a geometrical . proof of the fundamental theorem of the calculus. During his notorious . dispute with . Fundamental theorem of calculus. Deriving the Theorem. Let. Apply the definition of the derivative:. Rule for Integrals!. Deriving the Theorem. This is average value of . f. from. x. to . x. + . h. Word Problems Involving One-Step. Equations and Inequalities. 25R. 25L. Word Problems Reflection. Observe,. Question,. Comment . 10/31/11. 10/31/11. Warm-Up:. Warm-Up: Write and solve inequalities. Thirteen plus a number . Vesta Coufal. Gonzaga University Philosophy Club. March 16, 2011. Fractal: Mandelbrot Set. http://. math.youngzones.org. /Fractal webpages/. Julia_set.html. Escher: Poincare Disk Model of Hyperbolic Geometry. As the number of rectangles increased, the approximation of the area under the curve approaches a value.. Copyright .  2010 Pearson Education, Inc.. Section 5.3 – The Definite Integral. Definition. Robert “Dr. Bob” Gardner. Based on Hungerford’s . Appendix to Section V.3 . in . Algebra. , Springer-. Verlag. (1974). The field of complex numbers, . , is algebraically closed..  . Lemma . V.3.17. Robert “Dr. Bob” Gardner. Based on Hungerford’s . Appendix to Section V.3 . in . Algebra. , Springer-. Verlag. (1974). The field of complex numbers, . , is algebraically closed..  . Lemma . V.3.17. Complex Numbers. Standard form of a complex number is: . a bi.. Every complex polynomial function of degree 1 or larger (no negative integers as exponents) has at least one complex zero.. a . and. b .