111 Principles of Calculus, Fall 1996 Kok-Wee Phan This is a terminal calculus course designed for students who need a basic understanding of the principles of the calculus. Course material includes basic properties of functions, followed by the differentiation and integration of funtions. Some manipulative skills will be required, but this course is not intended to prepare the student for more advanced work in calculus. 3 credits: 2 lecturs and 2 recitation sections per week. Textbook: Calculus and Its Applications, Sixth Edition Goldstein, Lay and Schneider Sample Syllabus: Sections 0.1 0.6 Sections 1.1 1.8 Sections 2.1 2.3 & 2.7 Sections 3.1 & 3.2 Sections 4.1 4.5 Sections 5.2 & 5.3 Sections 6.1 6.3 Rationale: If Notre Dame is to have a mathematics requirement for everyone and if it intends to admit a certain number of very weak students, there is a need for a course which takes these students and tries to accomplish two aims. The first is that they should see something of the beauty and power of mathematics in everyday use, and secondly they should be encouraged to develop their skills at an appropriate rate. The proposed course does this by requiring four hours a week with the instructor from these students, and by utilizing examples whose non-mathematical principles are easy to understand. The extra hour per week will enable the instructor to cover a standard one semester calculus curriculum. This class will be listed in DART as for "First Year of Students Only" and the Department intends to teach but one section with limited enrollment. 0 FUNCTIONS, 3 0.1 Functions and Their Graphs, 3 0.2 Some Important Functions, 22 0.3 The Algebra of Functions, 31 0.4 Zeros of Functions---The Quadratic Formula and Factoring, 37 0.5 Exponents and Power Functions, 46 0.6 Functions and Graphs in Applications, 55 1 THE DERIVATIVE, 71 1.1 The Slope of a Straight Line, 72 1.2 The Slope of a Curve at a Point, 84 1.3 The Derivative, 92 1.4 Limits and the Derivative, 103 1.5 Differentiability and Continuity, 114 1.6 Some Rules for Differentiation, 121 1.7 More About Derivatives, 128 1.8 The Derivative as a Rate of Change, 134 2 APPLICATIONS OF THE DERIVATIVE, 157 2.1 Describing Graphs of Functions, 158 2.2 The first and Second Derivative Rules, 173 2.3 Curve Sketching (Introduction), 189 2.4 Curve Sketching (Conclusion), 199 Optimization Problems, 207 Further Optimization Problems, 218 Applications of Calculus to Business and Economics, 230 3 TECHNIQUES OF DIFFERENTIATION, 249 The Product and Quotient Rules, 249 The Chain Rule and the General Power Rule, 261 Implicit Differentiation and Related Rates, 268 4 THE EXPONENTIAL AND NATURAL LOGARITHM FUNCTIONS, 287 4.1 Exponential Functions, 288 4.2 The Exponential Function ex^ , 292 4.3 Differentiation of Exponential Functions, 299 4.4 The Natural Logarithm Function, 305 4.5 The Derivative of ln x , 311 4.6 Properties of the Natural Logarithm Function, 316 5 APPLICATIONS OF THE EXPONENTIAL AND NATURAL LOGARITHM FUNCTIONS, 325 5.1 Exponential Growth and Decay, 326 5.2 Compound Interest, 340 5.3 Applications of the Natural Logarithm Function to Economics, 347 5.4 Further Exponential Models, 355 6 THE DEFINITE INTEGRAL, 373 6.1 Antidifferentiation, 374 6.2 Areas and Riemann Sums, 386 6.3 Definite Integrals and the Fundamental Theorem, 398 6.4 Areas in the xy Plane, 413 6.5 Applications of the Definite Integral, 424