Exam 8: First-Order Differential Equations

Match the appropriate slope field with the differential equation .

Use the following direction field to identify the stability of the equilibrium point (0.57, 0.14).

The conversion of sucrose (table sugar) to glucose and fructose is first order in the concentration of sucrose, which means that the rate of reaction is proportional to the concentration of the sucrose. The rate of disappearance of sucrose can be expressed as , where c represents the concentration of the sucrose, and k is called the rate constant and is mathematically identical to the negative of the decay constant. If the rate constant is 0.20 hour^-1, how long will it take for the sucrose concentration to diminish to 1/20 of its initial concentration?

Find and interpret all equilibrium points for the competing species model.

The population of New Zealand grew exponentially through the 20th century at a rate of 1.6% year^-1. If the population in 2000 was 3.9 million, when was the population 1.0 million?

Construct a direction field for the differential equation.

In 1965, Fairchild Semiconductor's R&D Director, Gordon Moore, noted that the maximum number of transistors that could be included cost effectively in one integrated circuit had been doubling yearly since 1959. While growth has not increased quite as quickly since 1965, it has still been exponential, and the continued exponential growth has become known as "Moore's Law". Considering that Intel's 8086 processor, introduced in 1978, had 29,000 transistors, and their Itanium processor, introduced in 2002, had 220,000,000 transistors, on average how long has it taken for processors to double the number of transistors they use.

Consider a chemical system containing species A, B, and C; and that A and B can react to make C in a bimolecular reaction with rate constant of k₁, and C can decompose to make A and B in a first order reaction with rate constant of k_-1. If the instantaneous amounts of A, B, and C are represented as a, b, and c, and the initial amounts are given as A₀, B₀, and C₀, the change in C can be represented with the differential equation . If A₀ = 0 , B₀ = 0, C₀ = 4, k₁ = 0.02 s ^-1, and k_-1 = 0.04 s ^-1, how much C is present after 15 seconds? [Note: c cannot be larger than C₀ plus the smaller of A₀ or B₀. Nor can it be smaller than 0.]

Find the solution to the following separable differential equation.

Solve the following initial value problem explicitly. .

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]