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Engineering Mathematics Class # 1: Review

Engineering Mathematics Class # 1: Review. Sheng-Fang Huang. Derivatives. Definition of Derivative. The Basic Concept . Definition of Derivative. Definition Let y  =  f ( x ) be a function. The derivative of f is the function whose value at x is the limit :

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Engineering Mathematics Class # 1: Review

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  1. Engineering Mathematics Class #1: Review Sheng-Fang Huang

  2. Derivatives

  3. Definition of Derivative • The Basic Concept

  4. Definition of Derivative • Definition • Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit: provided this limit exists. • The derivative of f is also written as f’, or df/dx. • If this limit exists for each x in an open interval I (e.q. from -∞ to + ∞), then we say that f is differentiable on I.

  5. Standard Derivative

  6. Standard Derivative

  7. Exercises • Compute dy/dx

  8. Functions of a function (1) • sinx is a function of x since the value of sinx depends on the value of the angle x. • Simiarly, sin(2x+5) is a function depends on ________. • Since (2x+5) is also a function of x, we say sin(2x+5) is a function of a function of x. • By Chain Rule: • Let y = sin(2x+5), u = 2x+5.

  9. Functions of a function (2) • Products • Compute the derivative of e2xln5x. • Quotients • Compute the derivative of .

  10. Logarithmic Differentiation • The rule for differentiating a product or a quotient is used when there are only two factors, i.e. uv or u/v. Where there are more than two functions, the derivative is best found by ‘logarithmic differentiation’. • Let , where u, v, and w are functions of x. • Take logs to the base e on both sides:

  11. Exercises • Solve the derivative of . • y = x4e3xtanx.

  12. Implicit Functions • Explicit function: • If y is completely defined in terms of x, y is called an explicit. • E.g. y = x2 – 4x + 2 • Implicit function • E.g. x2 + y2 = 25, or x2 + 2xy + 3y2 = 4. • The differentiation of an implicit function:

  13. Partial Differentiation • The volume V of a cylinder of radius r and height h is given by • V depends on two quantities, r and h. • Keep r constant, V increases as h increases. • Keep h constant, V increases as r increases. • is called the partial derivative of V with respect to h where r is constant. • Let z = 3x2 + 4xy – 5y2. • Compute and .

  14. Integrals

  15. Standard Integrals (1)

  16. Function of a linear function of x • We know that , but how about ? • Solution: • Let z = 5x-4. Change the original equation into the following form: • We have • Now, try to solve and .

  17. and • Formula: • How to prove? • Exercises:

  18. Integration of products • How to integrate x2lnx? • Integration by parts: • Solve .

  19. Integration by partial fractions • Example: • Rules: • degree of numerator < degree of denominator • If not, first of all, divide out by the denominator. • Denominator must be factorized into prime factors (important!). • (ax+b) gives partial fractions A/(ax+b) • (ax+b)2 gives partial fractions A/(ax+b) + B/(ax+b)2 • (ax+b)3 gives partial fractions A/(ax+b) + B/(ax+b)2 + C/(ax+b)3 • ax2+bx+c gives partial fractions (Ax+B)/(ax2+bx+c)

  20. Integration by partial fractions • Example:

  21. Integration of Trigonometrical Functions • Basic trigonometrical formula:

  22. Integration of Trigonometrical Functions

  23. Multiple Integration • Double integral • A double integral can be evaluated from the inside outward. • Key point: when integrating with respect to x, we consider y as constant. 2 1

  24. Multiple Integration • A double integral can also be sometimes expressed as:

  25. Multiple Integration • Example • If . Compute V.

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