1 st order linear differential equation : P, Q – continuous.

1. 1st 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. 2nd order linear differential equation: P, Q, R, G – continuous. If G(x)0 equation is homogeneous, otherwise – nonhomogeneous. 2nd order linear homogeneous differential equation – 1). If y1and y2are solutions, then y=c1y1+c2y2 (linear combination) is also a solution. 2). If y1and y2are linearly independent solutions and P0, then the general solution is y=c1y1+c2y2.

3. 2nd order linear homogeneous differential equation with constant coefficients: Find r, s.t. y(x)=erx 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 1. Area between curves: Set up the area between the following curves: a) b) Sec. 6.1 5-26 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. 3. Arclength (Sec. 8.1): Find the arclength of the curve 4. Approximate integration (Sec. 7.7). Rn, Ln, Mn, Tn, Sn, 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 R5, L5, M5, T5, S5.

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: 1st order separable equation (Sec. 9.3): 1st order linear equation (Sec. 9.6): 2nd 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. Product rule Chain rule Implicit f-n Quotient rule Calculus(about limits) FTC Differentiation Integration Inverse processes Differential equations Optimization Elementary f-ns: Polynomial Rational Algebraic Power Exponential Logarithmic Trigonometric Hyperbolic/Inverse Exact evaluation Approximation Applications By parts (follows from product rule) Substitution (follows from chain rule) Riemann’s sums Other (Taylor’s) Area Arc length Volume under curve between curves washer method cylindrical shells Tn Sn Rn Ln Mn Techniques of integration: Trigonometric integration Trigonometric substitution Rational functions Integral Definite number! Indefinite function! Improper(Type I, II) Differential equations Initial Value problem Boundary Value problem Convergent number! Divergent ! 1st order 2nd order linear Mean Value Th Intermediate Value Th Extreme Value Th Separable Linear Homogeneouos constant coefficients Non-homogeneouos