MATH-108-E, Calculus 1 for Engineers. pptx

Calculus for 1st year civil engineering students

1. INTRODUCING YOURSELF - IN FRONT: Note: this will be graded 1). COMPLETE NAME 2). SCHOOL GRADUATED AND STRAND 3). RECEIVED HONOR OR AWARD 4). STATUS 5). EXPECTATION TO THIS SUBJECT 6). TALENT

2. MATH 108 E PRELIM COVERAGE

3. Importance of Calculus in Engineering Application: Calculus is a fundamental mathematical tool that plays a critical role in nearly all fields of engineering, as it provides the foundation for analyzing change, motion, forces, and energy in both theoretical and practical applications. Its principles are essential for developing models, solving equations, and designing efficient and innovative engineering systems.

4. 1. Modeling and Solving Real-World Problems Calculus allows engineers to model physical systems and solve problems involving change and motion. This includes predicting behavior over time and optimizing systems. Example: Modeling the flow of electricity, heat, fluids, or stress in materials.

5. 2. Design and Optimization Engineers use calculus to find maximum and minimum values essential for designing systems and structures that are both cost-efficient and safe. Example: Determining the optimal beam size to support a given load with minimal material.

6. 3. Analyzing Rates of Change Differential calculus helps engineers understand how a quantity changes with respect to another. Example: Calculating the acceleration of a moving vehicle or the rate of heat transfer.

7. 4. Integration in Engineering Analysis Integral calculus is used to compute total quantities from rates of change, such as area, volume, work, and energy. Example: Calculating the work done by a force or the volume of a tank.

8. 5. Control Systems and Signal Processing In electrical and mechanical engineering, calculus is essential in system dynamics, control theory, and signal processing. Example: Designing circuits or feedback systems using differential equations. 6. Engineering Software and Simulations Many engineering tools (like MATLAB, ANSYS, AutoCAD) rely on calculus-based algorithms to simulate real-world scenarios.

9. Review of Algebraic Functions (Preparation for Calculus)

10. LOGARITHMIC CONCEPTS Definition of Logarithm A logarithm is the inverse of an exponent. Example

11. Logarithmic Laws / Properties a. Product Rule: Example: b. Quotient Rule: Example:

12. c. Power Rule: Example: d. Change of Base Rule: Example:

13. NATURAL LOGARITHM (ln) The natural logarithm uses base 𝑒 ≈ 2.718 Example 1: Example 2:

14. ANALYTICGEOMETRY

15. TRIGONOMETRY

16. Function We say that a quantity y is a function of some other quantity x if the value of y is determined by the value of x. If f denotes the function, then we indicate the dependence of y on x by means of the formula y = f (x). The letter x is called the independent variable, and the letter y is called the dependent variable.

17. EXAMPLE: Given  

18. BOARDWORK  

19. Limit of a Function For example,  

20. Example

21. Theorems on Limits The following theorems are intuitively clear. Theorem 1: Theorem 2: Theorem 3:

22. Theorems on Limits Theorem 4: Theorem 5: Theorem 6:

23. Example

24. Example

25. Verify the following limit computations:

26. ONE-SIDED LIMITS

27. ONE-SIDED LIMITS

28. INFINITE LIMITS