Applied Mathematics

1. First Notes on Real Functions. 1.1. Introduction. 1.2. A Function of Real Numbers. 1.3. The Cost Function. 1.4. Function Representation in Table and Graphic. 1.5. Proofs and Mathematical Reasoning. 1.6. The Inverse Rationale. 1.7. Discussion of Results. 1.8. Concluding Remarks. 2. Sequences of Real Numbers. 2.1. Introduction. 2.2. Preliminary Notions. 2.3. Limit and Convergence of a Sequence. 2.4. Theorems About Sequences. 2.5. Study of Important Sequences. 2.6. Notes on Numbers Computation. 2.7. Concluding Remarks. 3. Limit of a Function. 3.1. Introduction. 3.2. Notions About Function Limits. 3.3. Lateral Limits at a Point — the Extended Cost Function. 3.4. Properties of Function Limits. 3.5. Remarkable Limits. 3.6. Concluding Remarks. 4. Continuity. 4.1. Introduction. 4.2. Continuity at a Point. 4.3. Continuity on a Range. 4.4. Properties of Continuous Functions. 4.5. Theorems about Continuous Functions. 4.6. Roots of Non-linear Equations. 4.7. Concluding Remarks. 5. Derivative of a Function. 5.1. Introduction. 5.2. Derivatives and Geometric Interpretation. 5.3. Derivation Rules. 5.4. Derivation of Important Functions. 5.5. Derivative of Inverse Function. 5.6. Derivatives of Different Orders. 5.7. Concluding Remarks. 6. Sketching Functions and Important Theorems. 6.1. Introduction. 6.2. Important Theorems on Differentiable Functions. 6.3. Maxima and Minima. 6.4. Asymptotes. 6.5. Sketching the Extended Cost Function. 6.6. Other Important Applications. 6.7. Concluding Remarks. 7. First Steps on Integral Sums. 7.1. Introduction. 7.2. Integral Sum and Geometric Interpretation. 7.3. Calculation of Areas. 7.4. Integral Sums. 7.5. Concluding Remarks. 8. Indefinite Integral. 8.1. Introduction. 8.2. Indefinite Integral. 8.3. Properties of Indefinite Integral. 8.4. General Methods of Integration. 8.5. Specific Methods of Integration. 8.6. Concluding Remarks. 9. Definite Integral. 9.1. Introduction. 9.2. Properties and Theorems. 9.3. The Fundamental Theorems of Calculus. 9.4. Applications to the Cost Function. 9.5. Area Calculations. 9.6. Improper Integrals. 9.7. Concluding Remarks. 10. Series. 10.1. Introduction. 10.2. Basic Notions about Series. 10.3. Theorems and Applications. 10.4. Convergence Criteria for Non-Negative Series. 10.5. Non-Positive and Alternating Series. 10.6. Function Series. 10.7. Power Series. 10.8. Taylor Series. 10.9. Concluding Remarks.