Mathematics 36, section 1 -- AP Analysis

Information on Final Exam

December 8, 1997

Background

The final exam for this section of AP Analysis will be given at 8:30 a.m. on Tuesday, December 16, in our regular classroom -- Swords 359. The exam will be comprehensive, covering all the material we have studied since the beginning of the semester. I will write the exam to be roughly twice the length of the in-class midterms. However, you will have the full three-hour period 8:30 - 11:30 a.m. to work on the exam if you need that much time. The questions will be similar to those on the 3 inclass exams. I may include some questions asking for definitions of terms or statements of theorems.

Topics to be Covered

  1. Functions defined by graphs, tables, formulas, the ``library of functions''
  2. Derivatives (know the definition): know how to sketch the graph of the derivative of a function defined by a graph, how to approximate values of the derivative given a table for the function, how to use the derivative rules on functions defined by formulas.
  3. The meaning of the signs of f'(x), f''(x).
  4. Differentiability = local linearity, tangent lines, etc.
  5. Taylor polynomials, approximations, the error bound
  6. Taylor series,
  7. Total change of a function, Riemann sums, the definite integral (know the definition), antiderivatives
  8. The Fundamental Theorem of Calculus, and,
  9. Integration by substitution, by parts, using the table. A copy of the table of integrals from the text will be provided with the exam; any integral you need to compute will be do-able by some combination of substitution, integration by parts, and/or consultation with the table).
  10. Applications of integration (setting up problems via Riemann sums; in limit a definite integral is obtained) -- volumes by slices, arclengths, physical examples like work, mass from non-constant density functions, etc.
  11. Differential equations -- slope fields and solutions
  12. Solving differential equations via separation of variables and integration, including population growth models.
  13. Phase-plane analysis for systems of differential equations and parametric curves.

Suggestion on How to Study

Begin by reviewing the class notes and the discussion write-ups and lab reports from your groups' work. Look at the in-class midterm problems and try to work out solutions for those again, without refering to your previous work. Look over some of the problems from the text from the previous review sheets. Then take a couple of hours and try the practice exam problems below.

Practice final exam:
page 1
page 2
page 3