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Math Refresher for STAT 391

Math Refresher for STAT 391. Bhushan Mandhani, TA. Topics Covered. Differential Calculus Integral Calculus Univariate optimization. Elementary Functions. Polynomials Exponential and Logarithmic functions. Trigonometric and Inverse Trigonometric.

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Math Refresher for STAT 391

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  1. Math Refresher for STAT 391 Bhushan Mandhani, TA

  2. Topics Covered • Differential Calculus • Integral Calculus • Univariate optimization

  3. Elementary Functions • Polynomials • Exponential and Logarithmic functions. • Trigonometric and Inverse Trigonometric. • Sums, products and compositions of the above.

  4. Derivative of a Function • It represents the slope of the function. • Its inverse is the antiderivative or integral.

  5. Some Common Derivatives

  6. Cases of the power function • Note the following: • y =1/x2 can be written as y = x-2 can be written y = x1/2 The respective derivatives are given by: dy/dx = -2x-3 And dy/dx = .5x-1/2 =

  7. Products and Quotients • Suppose y = f(x) g(x). • Suppose y = f(x)/g(x).

  8. Chain Rule • Suppose we have • The derivative of y is

  9. Other Techniques • Implicit Differentiation • Product of multiple functions.

  10. Integral Calculus

  11. Properties of Integrals

  12. Some Common Integrals

  13. Trigonometric Integrals

  14. Techniques for Integration • By Substitution. • By Parts • LIATE rule. • By Splitting into Partial Fractions.

  15. Minimizing Univariate Functions • Determine the zeros of the derivative. • For each of them, determine whether local minima by checking second derivative. • Check local minima and relevant endpoints to determine global minimum. • Maximizing is analogous.

  16. Find the absolute minimum value of g on the closed interval [–5,4].

  17. Note that (c) Find the absolute minimum value of g on the closed interval [–5,4]. g decreases on [–5,– 4], increases on [– 4,3], decreases on [3,4], so candidates for location of minimum are x = – 4, 4.

  18. Acknowledgements • This talk borrows some examples and figures from: • Anthony Tovar, Eastern Oregon University. • David Bressoud, Macalester College.

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