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