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

Learn how to apply L'Hospital's Rule to solve indeterminate forms and use it to evaluate improper integrals. Additionally, understand the concept of differential equations and solve a mixing problem involving salt solution.

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

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  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?

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