Mathematical Methods for Life Sciences

1. Functions in applied sciences. 1.1. The concept of function. 1.2. Linear functions. 1.3. Polynomial functions. 1.4. Rational functions and algebraic functions. 1.5. The exponential and logarithmic functions. 1.6. Malthusian Law. 1.7. Elementary trigonometric functions. 1.8. Exercises with solutions. 1.9. Suggested Exercises. 2. Limits and Derivatives. 2.1. Limits. 2.2. Properties of limits and standard limits. 2.3. Indeterminate forms. 2.4. Continuity. 2.5. Derivative of a function. 2.6. Derivability and Continuity. 2.7. De L'Hopital's Rule. 2.8. Derivative of the Inverse Function. 2.9. Exercises with solutions. 2.10. Suggested Exercises. 2.11. Appendix: Derivation rules. 2.12. Appendix: Derivatives. 2.13. Appendix: Theorems on limits. 3. Applications of the derivative. 3.1. The linear approximation. 3.2. The derivative as rate of change. 3.3. Local Maxima and Minima. 3.4. Graph sketching. 3.5. Optimization. 3.6. Exercises with SolutionsSuggested Exercises. 3.7. Appendix: Theorems of differential calculus. 4. Integrals. 4.1. The Definite Integral. 4.2. Properties of the definite integral. 4.3. The Fundamental Theorem of Calculus. 4.4. Integration by substitution. 4.5. Integration by parts. 4.6. Integration of rational functions. 4.7. Integration of trigonometric functions. 4.8. Applications. 4.9. Exercises with solutions. 4.10. Suggested Exercises. 4.11. Appendix: Indefinite integrals. 4.12. Appendix: Theorems on integral calculus. 5. First order differential equations. 5.1. First order equations. 5.2. The Cauchy problem. 5.3. Direction field. 5.4. Separable Equations. 5.5. Newton's law of cooling. 5.6. Linear equations. 5.7. Mixing problems. 5.8. Malthusian laws and population dynamics. 5.9. Homogeneous equations. 5.10. Autonomous differential equations. 5.11. The Logistics Model. 5.12. Solution of the logistic equation. 5.13. Exercises with solutions. 5.14. Suggested exercises. 6. Second order differential equations. 6.1. Cauchy's Theorem. 6.2. The Wronskian. 6.3. Homogeneous linear equations. 6.4. Linear equations. 6.5. Linear equations with constant coefficients. 6.6. Equations with constant coefficients: the general case. 6.7. Simple harmonic motion. 6.8. Harmonic motion with external force. 6.9. Damped harmonic motion. 6.10. Exercises with Solutions. 6.11. Suggested Exercises. 6.12. Appendix: Linear Systems. 7. Elementary Statistics. 7.1. Populations and Variables. 7.2. Absolute Frequencies and Percentages. 7.3. Graphical representation of data. 7.4. Mode, Average, and Median. 7.5. Variance and standard deviation. 7.6. Quartiles and Interquartile Range. 7.7. Normal Distribution. 7.8. Exercises with solutions. 7.9. Suggested Exercises. A. Solutions of some exercises.