Transcript Finding Solutions

What I wanted to do this week in BC Calc:
Differential Equations
Finding Solutions
Integration Methods
Separation of Variables
Rate of Change Proportional to
Exponential Growth
Logistic Growth
Estimating Solutions
Slope Fields
Taylor Approximations
Euler’s Method
Separation of Variables
1.
Rate of Change Proportional to
2.
Exponential Growth
3.
Logistic Growth
4. A single infected individual enters a community of n susceptible individuals. Let x be
the number of newly infected individuals at time t. The common epidemic model assumes that
the disease spreads at a rate proportional to the product of the total number infected and the
number not yet infected. Thus, the spread is modeled by
a) Find x(t), the solution to the differential equation, in terms of k and n.
b) If an infected person enters a community of 1500 susceptible individuals, and
100 are infected after 15 days, how many days will it take for 1000 people to be infected?
Slope Fields
5.
Be able to graph a slope field on the TI-89 calculator!
Mode Graph 6:Diff Equations
To enter equations use y1 for y and t for x
Taylor Approximations
6.(1995BC). Let f be a function that has derivatives of all orders for all real numbers.
Assume f(1) = 3, f′(1) = −2, f′′(1) = 2 and f′′′(1) = 4.
a) Write the second-degree Taylor polynomial for f at x = 1 and use it to approximate f(0.7).
b) Write the third-degree Taylor polynomial for f at x = 1 and use it to approximate f′(1.2).
c) Write the second-degree Taylor polynomial for f′ at x = 1 and use it to approximate f′(1.2)
Euler’s Method
HW 1:
HW 2:
HW 3:
HW 4:
HW 5:
HW 6:
HW 7: Sociologists sometimes use the phrase “social diffusion” to describe the way information
spreads through a population, such as a rumor, cultural fad, or news concerning a technological
innovation. In a sufficiently large population, the rate of diffusion is assumed to be proportional
to the number of people p who have the information times the number of people who do not.
Thus, if N is the population size, then
Suppose that t is in days,
and two people start a rumor at time t = 0 in a population of N = 1000 people. Find p(t) and
determine how many days it will take for half the population to hear the rumor.
HW 8: Let f be a function that has derivatives of all orders for all real numbers. Assume f(0) = 5,
f′(0) = −3, f′′(0) = 1, and f′′′(0) = 4.
a) Write the third-degree Taylor polynomial for f at x = 0 and use it to approximate f(0.2).
b) Write the fourth-degree Taylor polynomial for g, where g(x) = f(x2), at x = 0.
c) Write the third-degree Taylor polynomial for h, where
d) Let h be defined as in part (c). Given that f(1) = 3, either find the exact value of h(1)
or explain why it cannot be determined.