Lecture_1 - CALCULUS.pdf - CHAPTER 1 CALCULUS Topics...

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1 CHAPTER 1 CALCULUS Topics covered: Derivatives (various rules of differentiation), Integration (one-and multi-dimensional), Power series, Convergence, Taylor Expansion, Mean-Value Theorem, Fundamental Theorems of Calculus, Min-Max-Theorems, Multivariate calculus the basics. The Greek Alphabet ? ? alpha Ι ? iota ? ? rho ? ? beta Κ ? kappa Σ 𝜎 sigma Γ ? gamma Λ ? lambda Τ ? tau Δ ? delta Μ ? mu Υ ? upsilon Ε ? epsilon Ν ? nu Φ 𝜙 phi Ζ ? zeta Ξ ? xi Χ ? chi Η ? eta Ο ? omicron Ψ ? psi Θ ? theta Π ? pi Ω ? omega
2 Functions A function ? is a rule that assigns each element ? in a set ? exactly one element ? = ?(?) in another set ? . Definition: The range of ? is the set of all possible values of ?(?) , and the domain is the set of ? ’s where the function ? is defined. Example 1 ?(?) = √? + 2 The domain consist of all ? ’s for which ? + 2 ≥ 0 or ? ≥ −2 . The range is the set of all possible values of ?(?): [0, ∞) . Example 2 ?(?) = 1 ?−2 The domain is: ? ≠ 2 . The range is: (−∞, 0) and (0, ∞) . Various functional forms 1. Linear functions: ?(?) = ?? + ? . Here the domain is (−∞, +∞) ≡ ℝ and the range is also . ? is the ? -intercept and ? is the slope of the line. 2. A quadratic function: ?(?) = ?? 2 + ?? + ?. The graph is a parabola and it may have 0,1, or 2 ? -intercepts. If ? > 0 then the parabola opens upward and if ? < 0 then it opens downward. The solutions of ?(?) = 0 are: −?±√? 2 −4?? 2? if ? 2 − 4?? ≥ 0. In case if ? 2 − 4?? < 0 then there are no real solutions to ?(?) = 0. 3. Polynomials: ?(?) = ? 𝑛 ∙ ? 𝑛 + ? 𝑛−1 ∙ ? 𝑛−1 + ⋯ + ? 1 ∙ ? + ? 0 . If ? 𝑛 ≠ 0 then this polynomial is of degree ? . A polynomial of degree ? can have no more than ? zeros. Power functions: ?(?) = ? 𝑛 is a special case of a polynomial. For the case when ? is an even integer, the graph of ?(?) = ? 𝑛 is similar to the parabola of ? = ? 2 ,
3 and for the cases of an odd value of ? graphs of ?(?) = ? 𝑛 are similar to the graph of ? = ? 3 . 4. Exponential functions ?(?) = ? ? , where ? > 0 is a constant. If 0 < ? < 1 , then the function is a decreasing function for all ? . If ? > 1 , then the function is an increasing function for all ? . A special case: ?(?) = ? ? . This is an important function with special properties. The approximate value of ? is ? ≈ 2.718282 . The log function (or the natural logarithm-Ln): ? = ln ? . This function is the inverse of the exponential function, ? = ? ? . 5. Bell-shaped curves ?(?) = 1 √2𝜋 ∙𝜎 ∙ ? (?−𝜇) 2 2𝜎 2 , for constants ? and 𝜎; and ? ∈ ℝ. These functions are used in probability and statistics as the density function of normally distributed random variables. Some properties ? ?+? = ? ? ∙ ? ? ? ?−? = ? ? ? ? (? ? ) ? = ? ?∙? ? ln ? = ? ln(??) = ln ? + ln ? ln ( ? ? ) = ln ? − ln ? ln(?) ? = ? ∙ ln ?
4 ln(? ? ) = ? ??? ? ? = ??? ? ? /??? ? ? Exercises