Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146 The security staff at a rock concert found a dead body in a mezzanine restroom, the apparent victim of a fatal shooting. They alert the police who arrive at precisely 12 midnight. At that instant, the body’s temperature is 91º F; by 1:30 a.m., 90 minutes later, the body’s temperature has dropped to 82º F. Noting that the thermostat in the restroom was set to maintain a constant temperature of 69º F, and assuming the the victim’s temperature was 98.6º F when she was shot, determine the time, to the nearest minute, that the fatal shooting occurred. Assume that the victim died instantly and that Newton’s Law of Cooling holds. Show all appropriate evidence to support your solution.

Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146

Monday, March 21MAT 146 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?

Monday, March 21MAT 146 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?

Thursday, Oct 22, 2015MAT 146 Spreading a Rumor: Suppose that y represents the number of people that know a rumor at time t and that there are M people in the population. For these parameters, one model for the spread of the rumor is that “the rate at which the rumor is spread is proportional to the product of those who have heard the rumor and those who have not heard it.”

Monday, March 21MAT 146 Details... details... details! package-differential-equations

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146 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:

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146 Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for

Thursday, Oct 22, 2015MAT 146 Create a 3rd-degree polynomial approximator for

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 146

Thursday, Oct 22, 2015MAT 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?

Thursday, Oct 22, 2015MAT 146 We start with a sequence {a n }, n going from 1 to ∞, and define {s i } as shown. The {s i } are called partial sums. These partial sums themselves form a sequence. An infinite series is the summation of an infinite number of terms of the sequence {a n }.

Thursday, Oct 22, 2015MAT 146 Our goal is to determine whether an infinite series converges or diverges. It must do one or the other. If the sequence of partial sums {s i } has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges.

Thursday, Oct 22, 2015MAT 146 A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge?

Thursday, Oct 22, 2015MAT 146 The harmonic series is the sum of all possible unit fractions.

Thursday, Oct 22, 2015MAT 146 A telescoping sum can be compressed into just a few terms.

Thursday, Oct 22, 2015MAT 146