Mathematics 101 Midterm 1

Mathematics 101 Midterm 1 - topics and practice problems

Section 5.1:
- Sigma notation
- Evaluating sums (you should remember the formulas (a), (b), (d) from Theorem 1)
- Exercises: 1-29.
Section 5.2:
- Interpreting area under graph as a limit of Riemann sums
- Evaluating such sums and their limits in easy cases
- Exercises: 1-11; these are nice exercises in computing sums and limits, but have limited practical value for the purpose of integration
Section 5.3:
- Partitions, Riemann sums, definite integral, integrability
- Setting up and evaluating Riemann sums
- Terminology: limits of integration, integrand, etc.
- Exercises: 1-10, 16-19
Section 5.4:
- Properties of definite integrals: Theorem 3
- Using Theorem 3 and interpretation of integral as area to simplify and compute integrals
- Mean Value Theorem for integrals, average value of function
- Exercises: 1-38 (but do not spend too much time on this)
Section 5.5:
- Fundamental Theorem of Calculus
- Applications: integrating elementary functions, differentiating integrals, finding areas of planar regions
- Exercises: 1-46, 49-50. Note that in problems 21-32 it really is important to draw a picture.
Section 5.6:
- Elementary integrals, page 5.6. You should memorize the formulas 1-8 (note that 1-6 are special cases of 7), 9-12, 15-17, 19-20.
- Substitution in indefinite and definite integrals
- Trigonometric integrals. You should either memorize the formulas for integrals of tan x and cot x, or make sure that you can find them when needed (the derivation is very short and it is useful to remember it). You should also remember the tricks used to integrate powers of sine and cosine: substitution, converting even powers of sines to cosines and vice versa, and using the cos(2x) formulas.
- Exercises: 1-48.
Section 5.7:
- Finding areas of regions under graph of a function, or between two graphs
- Exercises: 1-25. (Some of these exercises may involve integration formulas from the textbook, including those that you do not need to memorize.)