BIT Basic Mathematics

Definition, domain range, Graphs of functions, Representing a function numerically, the ver- tical line test for a function, Piecewise defined functions, Increasing and decreasing functions, Even and odd function, Common functions: linear, power, polynomial, rational functions All worked out examples of 1.1. Exercises 1.1: 1-8, 15, 18, 23, 25, 26. 1.2: Combining functions:Shifting and Scaling graphs Sums, differences, products and quotients, Composite functions, Shifting a graph of a func- tion. Worked out examples: 1-5 Exercises 1.2: 1-8. 1.4: Graphing with calculator and computers (desmos may be easy) to plot the graph of the functions (some of the functions): y = x, y = x^2 , y =1/1 − x, y = sin x, y = cos x, y = sin 100x 1.5: Exponential functions: Definition, Exponential behavior, Exponential growth and decay. Worked out examples: 1-4. Exercises 1.5: 29-33 1.6: Inverse Functions and Logarithms Worked out examples: 1 - 4, 6, 7. Exercises 1.5: 79 - 81 2.1: Rate of change and tangent to curves. Worked out examples: 1-5. Exercises 2.1: 1, 3, 6, 7, 9, 15, 17.

2.2 Limit of a Function and Limit Laws Limits of function values, The limit laws, Eliminating zero denominators algebraically, The Sandwich theorem(no proof). Worked out examples: 1-11 2.3 The Precise Definition of a Limit Definition of limit Worked out examples: 1-5 One sided limit: Worked out Examples 1-4 2.5 Continuity Worked out examples: 2, 3 Intermediate Value Theorem for Continuous Functions Worked out examples: 11, 12 2.6 Limits Involving Infinity; Asymptotes of Graphs Worked out examples 1, 2, 3 Horizontal Asymptotes Worked out examples: 4-9 Oblique asymptotes Worked out examples: 10-14 Vertical asymptotes Worked out examples: 15-19. Some related problems

3.1 Tangents and the Derivative at a Point Finding a Tangent to the Graph of a Function Rates of Change: Derivative at a point Worked out Examples: 1, 2 Exercises 3.1: 5-8, 11, 12, 13, 23, 24, 25 3.2 The Derivative as a function Worked out Examples: 4, 5 Differentiable Functions are continuous 3.4 The Derivative as a rate of change Worked out Examples: 1-7 Ideas of derivatives of trigonometric, inverse trigonometric, log- arithm, exponential functions and ideas of chain rules. 3.10 Related rates Worked out Examples: 1-6.

4.1 Extreme values of functions: Introduction Worked out examples: 1-4 Exercise 4.1: 21, 22, 23, 31, 32 4.2 The mean value theorem Rolle’s Theorem(no proof), Lagrange mean value theorem(no proof) Worked out examples: 1-4 4.3 Monotonic functions and the first derivative test Increasing functions and decreasing Functions Worked out examples: 1, 2, 3 4.4 Concavity and curve sketching Worked out examples: 1-9 4.5 Indeterminate Forms and LHpitals Rule Indeterminate form, LHpitals rule All worked out examples Exercises 4.5: 1-7, 13, 15. 4.6 Applied optimization Worked out examples: 1-5 4.7 Newton’s method. Worked out examples: 1, 2 Examples 4.7: 1-4

4.8 Antiderivatives Worked out examples: 1, 2, 3 5.1 Area and Estimating with Finite Sums Area worked out examples: 1-4 Exercises 5.1: 1-4 5.2 Sigma notation and limits of finite sums Worked out examples: 1-5 5.3 The definite integral Worked out example: 4, 5 5.4 The fundamental theorem of calculus Mean value theorem for definite integrals, Fundamental theorem of calculus Part 1 and 2 (no proof), The net change theorem Worked out examples: 2-7 5.5 Indefinite integral and substitution method: All worked out examples 5.6 Area between the curves Worked out examples: 4, 5, 6, 7 Exercises 5.6 : 63-66

6.1 Volumes using cylindrical shells Worked out examples: 1-10 6.2 Volumes using cross-sections Worked out examples: 2, 3 6.3 Arc length Worked out examples: 1, 2 3, 4, 5 6.4 Areas of surfaces of revolution Worked out examples: 1, 2

Review of integration by parts, trigonometric substitutions, integration of rational functions by partial fractions. Computer algebra system (Maple) 8.6 Numerical Integration Numerical Integration Simpsons Rule: Approximations Using Parabolas Error Analysis Worked out examples:1-6 Exercises 8.6: 1, 2, 3, 4, 7, 8, 9, 10. 11, 12, 13, 17, 19, 21. 8.7 Improper integrals Worked out examples: 1-9

9.1 Solutions, Slope Fields, and Eulers Method General first order differential equations and solutions Worked out examples: 1, 2. Slope Fields: Viewing Solution Curves Eulers Method Worked out examples: 3, 4 Exercises 9.1: 11, 12, 13 9.2 First order linear equation Worked out examples 1, 2, 3 Exercises 9.2: 1-10, 15-21 9.3 Applications Motion with resistance proportional to velocity 7.2 Exponential change. Worked out Examples: 1, 2, 3, 4, 5. 9.4 Graphical solutions of autonomous equations Example worked out: 1

10.1. Sequences Worked out Examples: 1-11 Exercises 10.1: 1,2,3,7,8.13,16,27-32 Infinite series Worked out examples: 1-10 Related problems from exercise 10.2 Ideas of Integral test, comparison test: worked out examples, Alternating series, absolute and conditional convergence, with at least one worked out examples. !0.7. Power series Worked out examples 1-6. 10.8. Taylor and Maclaurin series Exercises 10. 8: 1, 2 ,3, 4, 7, 9, 11, 12

14.1. Functions of several variables Worked out examples: 1, 2, 3, 4 14.2 Limits and continuity in higher dimensions Worked out Examples: 1-6 Exercises: 1, 2, 3, 4, 5, 6, 13, 14 14.3. Partial derivatives Worked out examples: 1-10 Examples 14.3: 1-18 14.5. Chain rule Worked out examples: 1-6 14.5. Directional derivative Worked out examples: 1-5 14.6. Tangent planes and differentials Worked out examples: 1-4 14.7. Extreme values and saddle points Worked out examples: 1-5 Exercises 14.7: 1-7