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Calculus II (MAT 146) Dr. Day Wednesday March 26, 2014. Solutions of Differential Equations (9.1) Slope Fields: Graphical Solutions to DEs (9.2 - I) Euler’s Method: Numerical Solutions to DEs (9.2 - II)
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Calculus II (MAT 146)Dr. Day Wednesday March 26, 2014 • Solutions of Differential Equations (9.1) • Slope Fields: Graphical Solutions to DEs (9.2 - I) • Euler’s Method: Numerical Solutions to DEs (9.2 - II) • Separation of Variables: One Method for Determining an Exact (Analytical) Solution to a DE (9.3) • Applications of DEs MAT 146
Separable Differential Equations MAT 146
Applications! Rate of change of a population P, with respect to time t, is proportional to the population itself. MAT 146
Rate of change of the population is proportional to the population itself. Slope Fields Euler’s Method Separable DEs MAT 146
Slope Fields MAT 146
Slope Fields MAT 146
Euler’s Method MAT 146
Separable DEs MAT 146
Applications! The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (A) with respect to time (t) is proportional to the amount present (A). MAT 146
Exponential Decay The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (A) with respect to time (t) is proportional to the amount present (A). Carbon-14 has a half-life of 5730 years Write and solve a differential equation to determine the function A(t) to represent the amount, A, of carbon-14 present, with respect to time (t in years), if we know that 300 grams were present initially. Use A(t) to determine the amount present after 250 years. MAT 146
Applications! Mixtures A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 20 minutes? MAT 146
Applications! Mixtures A tank contains 2000 L of brine with 30 kg of dissolved salt. A solution enters the tank at a rate of 20 L/min with 0.25 kg of salt per L . The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 60 minutes? MAT 146
Applications! MAT 146
Applications! Known information: A= 20° C and (0,95) and (20,70) MAT 146
Applications! MAT 146
Applications! MAT 146
Polynomial Approximators Our goal is to generate polynomial functions that can be used to approximate other functions near particular values of x. The polynomial we seek is of the following form: MAT 146
Polynomial Approximators Goal: Generate polynomial functionsto approximate other functions near particular values of x. Create a third-degree polynomial approximator for MAT 146
What is a Sequence? MAT 146
Sequence Characteristics Convergence/Divergence: As we look at more and more terms in the sequence, do those terms have a limit? Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence. Boundedness: Are there values we can stipulate that describe the upper or lower limits of the sequence? MAT 146
Assignments WebAssignments • DE Applications • 11.1: Sequences • 11.2 Series MAT 146