This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Exercises 3.2
16. By deﬁnition, p(t + h) − p(t) . h→0 h p (t) = lim Replacing h by −h in the above equation, we obtain p (t) = lim p(t − h) − p(t) p(t) − p(t − h) = lim . h→0 h→0 −h h Adding the previous two equations together yields 2p (t) = lim p(t + h) − p(t) p(t) − p(t − h) + h→0 h h p(t + h) − p(t − h) = lim . h→0 h p(t + h) − p(t − h) . 2h Thus p (t) = lim
h→0 19. This problem can be regarded as a compartmental analysis problem for the population of ﬁsh. If we let m(t) denote the mass in million tons of a certain species of ﬁsh, then the mathematical model for this process is given by dm = increase rate − decrease rate. dt The increase rate of ﬁsh is given by 2m million tons/yr. The decrease rate of ﬁsh is given as 15 million tons/yr. Substituting these rates into the above equation we obtain dm = 2m − 15, dt m(0) = 7 (million tons). This equation is linear and separable. Using the initial condition, m(0) = 7 to evaluate the arbitrary constant we obtain 1 15 m(t) = − e2t + . 2 2 Knowing this equation we can now ﬁnd when all the ﬁsh will be gone. To determine when all the ﬁsh will be gone we set m(t) = 0 and solve for t. This gives 1 15 0 = − e2t + 2 2 111 ...
View
Full
Document
This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

Click to edit the document details