L'Hospital's Rule and Improper Integrals: Differential Equations and Mixing Problems

1. L’Hospital’s Rule: f and g – differentiable, g(a)  0 near a (except possibly at a). If (indeterminate form of type ) or (indeterminate form of type ) then (if this limit exists).

2. Indeterminate Product If (indeterminate form of type ) then use L’Hospital Rule for Indeterminate Difference If (indeterminate form of type ) then try to convert into quotent and use L’Hospital Rule. Indeterminate Power If (indeterminate form of type ) or (indeterminate form of type ) or (indeterminate form of type ) then use

3. Improper Integrals Type II: Discontinuous Integrand Type I: Infinite Interval Type I Improper Integral: If limit exists then this integral is convergent, otherwise – divergent.

4. Type II Improper Integral: a). If f – continuous on [a,b) and discontinuous at b b). If f – continuous on (a,b] and discontinuous at a c). If f – discontinuous at c(a,b) and If limit exists then this integral is convergent, otherwise – divergent. Comparison Th for Improper Integrals: fand g – continuous, f(x)g(x) 0 for xa.

5. Differential equation – Equation that contains an unknown function and some of its derivatives. Order of differential equation = order of highest derivative in equations. f(x) – solution of differential equation, if it satisfies the equation for all x in some interval. Solve differential equation = find ALL possible solutions. Condition of the form y(t0)=y0 – initial condition. Initial-value Problem (IVP) – find solution that satisfies the given initial condition. Separable equation – 1st order differential equation in which expression for dy/dx can be factored as a function of x times a function of y:

6. Mixing problem: A tank contains 20 kg of salt dissolved in 5000L of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25L/min. The solution is kept throughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour?