1. 1 st order linear differential equation: P, Q – continuous. Algorithm: Find I(x), s.t. by solving differential equation for I(x): then integrate both sides of the equation: Simplify the expression, if possible.

2. 2 nd order linear differential equation: P, Q, R, G – continuous. If G(x)  0 equation is homogeneous, otherwise – nonhomogeneous. 2 nd order linear homogeneous differential equation – 1). If y 1 and y 2 are solutions, then y=c 1 y 1 +c 2 y 2 (linear combination) is also a solution. 2). If y 1 and y 2 are linearly independent solutions and P  0, then the general solution is y=c 1 y 1 +c 2 y 2.

3. 2 nd order linear homogeneous differential equation with constant coefficients: Find r, s.t. y(x)=e rx is a solution (substitute into the equation): characteristic equation Case I. Two unequal real roots and Therefore, 2 linearly independent solutions General solution:

4. Case II. One real root Two linearly independent solutions are General solution: Case III. Two complex roots Two linearly independent solutions General solution:

6. Problems 2. Volumes by washer method (Sec. 6.2) and by cylindrical shells method (Sec 6.3): Find the volume of the solid obtained by rotating the region bounded by about x=-1. 1. Area between curves: Set up the area between the following curves: a) b) Sec. 6.1 5-26 3. Arclength (Sec. 8.1): Find the arclength of the curve 4. Approximate integration (Sec. 7.7). R n, L n, M n, T n, S n, formulae and error approximation: How large should we take n for Trapezoid / Midpoint / Simpson’s Rules in order to guarantee that the error of each method for would be within 0.001? Write down the expressions for R 5, L 5, M 5, T 5, S 5.

7. 5. L’Hospital Rule (Sec. 4.4): Find Justify every time you apply L’Hospital rule! 6. Improper integral (Sec. 7.8): type I, type II. 7. Differential equations: 1 st order separable equation (Sec. 9.3): 1 st order linear equation (Sec. 9.6): 2 nd order linear equation (Sec. 17.1): Initial Value Problem and Boundary Value Problem. 8. Modeling: mixing problem, fish growth problem, population growth: The population of the world was 5.28 billion in 1990 and 6.07 billion in 2000. Assuming that the population growth rate is proportional to the size of population, formulate and solve the corresponding differential equation. Predict world population in 2020. When will the world population exceed 10 billion? 9. Recall: graphs and derivatives of elementary functions, integration techniques.

8. Calculus (about limits) Differentiation Integration Inverse processes FTC Product rule Chain rule Implicit f-n Quotient rule Exact evaluationApproximationApplications By parts (follows from product rule) Substitution (follows from chain rule) Techniques of integration: Trigonometric integration Trigonometric substitution Rational functions Riemann’s sums Other (Taylor’s) RnLnMnRnLnMn TnSnTnSn AreaVolumeArc length under curve between curves washer method cylindrical shells Integral Definite number! Indefinite function! Improper (Type I, II) Convergent number! Divergent  ! OptimizationDifferential equations 1 st order2 nd order linear SeparableLinearHomogeneouos constant coefficients Non-homogeneouos Elementary f-ns: Polynomial Rational Algebraic Power Exponential Logarithmic Trigonometric Hyperbolic/Inverse Mean Value Th Intermediate Value Th Extreme Value Th Initial Value problem Boundary Value problem