Calculus – James Steward (1)

In Exercise 9.3.56 you are asked to derive a differential equation that models the thickness of sea ice as it changes over time. © Alexey Seafarer / Shutterstock.com Differential Equations PERHAPS THE MOST IMPORTANT of all the applications of calculus is to differential equations. When physical scientists or social scientists use calculus, more often than not it is to analyze a differential equation that has arisen in the process of modeling some phenomenon that they are studying. Although it is often impossible to find an explicit formula for the solution of a differ- ential equation, we will see that graphical and numerical approaches provide the needed information.

https://www.jamarana.com https://t.me/universitarios_info https://t.me/universitarios https://t.me/universitarios_info https://www.jamarana.com https://t.me/universitarios CHAPTER 9 Differential Equations Modeling with Differential Equations In describing the process of modeling in Chapter 1, we talked about formulating a mathematical model of a real-world problem either through intuitive reasoning about the phenomenon or from a physical law based on evidence from experiments. The mathematical model often takes the form of a differential equation, that is, an equation that contains an unknown function and some of its derivatives.

This is not surprising because in a real-world situation we often notice that changes occur and we want to predict future behavior on the basis of how current values change. We begin by examin- ing several examples of how differential equations arise when we model physical phenomena. ■ Models for Population Growth One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population.

That is a reasonable assumption for a population of bacteria or animals under ideal conditions (unlimited environment, ade­quate nutrition, absence of predators, immunity from disease). Let’s identify and name the variables in this model: t −time sthe independent variabled P −the number of individuals in the population sthe dependent variabled The rate of growth of the population is the derivative dPydt. So our assumption that the rate of growth of the population is proportional to the population size is written as the equation dP dt −kP where k is the proportionality constant.

Equation 1 is our first model for population growth; it is a differential equation because it contains an unknown function P and its derivative dPydt. Having formulated a model, let’s look at its consequences. If we rule out a population of 0, then Pstd . 0 for all t.

So, if k . 0, then Equation 1 shows that P9std . 0 for all t. This means that the population is always increasing. In fact, as Pstd increases, Equation 1 shows that dPydt becomes larger.

Distance and Midpoint Formulas Distance between P1sx1, y1d and P2sx2, y2d: d −ssx2 2 x1d2 1 sy2 2 y1d2 Midpoint of P1P2: S x1 1 x2 2 , y1 1 y2 2 D Lines Slope of line through P1sx1, y1d and P2sx2, y2d: m −y2 2 y1 x2 2 x1 Point-slope equation of line through P1sx1, y1d with slope m: y 2 y1 −msx 2 x1d Slope-intercept equation of line with slope m and y-intercept b: y −mx 1 b Circles Equation of the circle with center sh, kd and radius r: sx 2 hd2 1 sy 2 kd2 −r 2 ALGEBRA Arithmetic Operations asb 1 cd −ab 1 ac a b 1 c d −ad 1 bc bd a 1 c b −a b 1 c b a b c d −a b 3 d c −ad bc Exponents and Radicals x mx n −x m1n x m x n −x m2n sx mdn −x mn x2n −1 x n sxydn −x nyn S x yD n −x n yn x 1yn −s n x x myn −s n x m −(s n x ) m s n xy −s n x s n y Î n x y −s n x s n y Factoring Special Polynomials x 2 2 y2 −sx 1 ydsx 2 yd x 3 1 y3 −sx 1 ydsx 2 2 xy 1 y2d x 3 2 y3 −sx 2 ydsx 2 1 xy 1 y2d Binomial Theorem sx 1 yd2 −x 2 1 2xy 1 y2 sx 2 yd2 −x 2 2 2xy 1 y2 sx 1 yd3 −x 3 1 3x 2y 1 3xy2 1 y3 sx 2 yd3 −x 3 2 3x 2y 1 3xy2 2 y3 sx 1 ydn −x n 1 nx n21y 1 nsn 2 1d 2 x n22y2 1 …

1S n kDx n2kyk 1 … 1 nxyn21 1 yn where S n kD −nsn 2 1d … sn 2 k 1 1d 1 ? 2 ? 3 ? … ? k Quadratic Formula If ax 2 1 bx 1 c −0, then x −2b 6 sb 2 2 4ac 2a . Inequalities and Absolute Value If a , b and b , c, then a , c.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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