Then
(b) We must find N(8). Since t = 0 represents yesterday:
(c) The virus spreads fastest when 50,000/2 = 25,000 people have been infected.
Chapter Summary and Caution
In this chapter, we have considered some simple differential equations and ways to solve them. Our
methods have been graphical, numerical, and analytical. Equations that we have solved analytically—
by antidifferentiation—have been separable.
It is important to realize that, given a first-order differential equation of the type it is the
exception, rather than the rule, to be able to find the general solution by analytical methods. Indeed, a
great many practical applications lead to d.e.’s for which no explicit algebraic solution exists.
Practice Exercises
Part A. Directions: Answer these questions without using your calculator.
In Questions 1–10, a(t) denotes the acceleration function, v(t) the velocity function, and s(t) the
position or height function at time t. (The acceleration due to gravity is −32 ft/sec^2 .)
- If a(t) = 4t − 1 and v(1) = 3, then v(t) equals
(A) 2 t^2 − t
(B) 2 t^2 − t + 1
(C) 2 t^2 − t + 2
(D) 2 t^2 + 1
(E) 2 t^2 + 2 - If a(t) = 20t^3 − 6t, s (−1) = 2, and s(1) = 4, then v(t) equals
(A) t^5 − t^3
(B) 5 t^4 − 3t^2 + 1
(C) 5 t^4 − 3t^2 + 3
(D) t^5 − t^3 + t + 3
(E) t^5 − t^3 + 1 - Given a(t), s (−1), and s(1) as in Question 2, then s(0) equals
(A) 0
(B) 1