Final Exam Review

1. Final Exam Review Final Exam: Krieger 205 on Thursday, May 7 from 9am-12noon. It will be a comprehensive exam covering the entire semester’s material • No notes or calculators • No makeup exams

2. Calculation of limits • Rules: sum, difference, product, quotient of limits, powers of limits • Algebraic manipulations, e.g. limx3(x2-6x+9)/(x-3) • Rational limits at infinity (only top order terms count)

3. Continuity • Definition: f is continuous at x=a if: (1) f is defined at a(2) limxa f(x)=f(a) • Rules for continuous functions: sum, difference, product, quotient, powers • Right, left continuity • Continuity on open or closed intervals

4. Derivative • Derivative of f(x) is instantaneous rate of change of f with respect to x • Derivative f (x) = slope of tangent line at x (limit of slopes of secant lines) • Formal definition:

5. Calculating Derivatives • Product rule: (fg)=f g + f g  • Quotient rule: (f/g)  =[gf  - f g ]/g2 • Reciprocal rule: (1/g)  = - g  /g2

6. Derivatives of all trig functions

7. Chain rule

8. Log & Exponential Functions • Logarithm is an exponent: bx and logb x are inverses of each other • ln x = logex, (d/dx) ln x = 1/x • (d/dx) ex = ex • logarithmic differentiation • e.g. compute derivative of xx

9. Implicit Differentiation • Model problems:

10. Derivatives of Inverse Trig Functions

11. Related rates • Draw picture; label quantities that vary • Identify known rate of changes and one to be found • Find equation relating them • Differentiate • Model Example: Go over examples from class notes

12. First and second derivatives • Sign of 1st derivative implies increasing/decreasing • Sign of 2nd derivative implies concave up/down • Stationery point: 1st derivative vanishes • Inflection point: Change of concavity

13. 1st and 2nd derivative test • 1st derivative test: Relative max. (resp. min.) if 1st derivative changes from + to - (resp. - to +) • Second Derivative test: If f (x)=0 and f (x) < 0 then rel. max.; If f (x)=0 and f (x) > 0 then rel. min. • Relative extrema must occur at critical points

14. Absolute Max/Min • Continuous functions on closed bounded intervals have absolute max. & min. • These must occur at critical points • Functions with exactly one relative extremum must have an absolute extremum at that point

15. Indefinite integral

16. Integration formulae

17. Fundamental Theorem of Calculus

18. Applications of integration • Area between curves • Volumes: Volume of rotating y=f(x) around the x-axis:

19. Integration by substitution

20. Substitution for definite integrals • Similar to indefinite case except need to deal with limits of integration • Two choices • Method 1: Make substitution, evaluate definite integral, convert back to x-valued function and use x-limits of integration • Method 2: Make substitution for integrand as well as limits of integration

21. Integration by parts

22. Improper integrals • Integrals over intervals of the form (-∞,a] or [a,+∞): Replace -∞, +∞ by l and take limits of the integrals over [l,a], [a,l], respectively • Integrals over [a,b] where f is not continuous at a: Take integrals over [c,b] and then take limits as c→a+ • Similar procedure if f not continuous at b or at some point c in [a,b]

23. Approximations • Linear approximation to f at a: L(x) = f(a) + f (a) (x – a) • Taylor approximation of order n