Barrons AP Calculus - David Bock

FIGURE N9–2b (b) Since we already know that, if then we are assured of having found the correct general solution in (a). In Figu ...

FIGURE N9–3c FIGURE N9–3d SOLUTIONS: (A) goes with Figure N9–3c. The solution curves in the family y = sin x + C are quite obvio ...

SOLUTIONS: (a) By differentiating equation x^2 + y^2 = r^2 implicitly, we get 2x + 2y from which which is the given d.e. (b) x^2 ...

FIGURE N9–5 C. EULER’S METHOD BC ONLY In §B we found solution curves to first-order differential equations graphically, using sl ...

y 3 = y 2 + Δy = 2.5 + 0.75 = 3.25, P 3 = (2.5, 3.25), and so on. The table summarizes all the data, for the four steps specifie ...

FIGURE N9–6b We observe that, since y ′′ for 3 ln x equals the true curve is concave down and below the Euler graph. The last co ...

P 4 0.4 0.064 (0.464)(0.1) = 0.046 P 5 0.5 0.110 A Caution: Euler’s method approximates the solution by substituting short line ...

A first-order d.e. in x and y is separable if it can be written so that all the terms involving y are on one side and all the te ...

When x = e^4 , we have thus ln^2 y = 9 and ln y = 3 (where we chose ln y > 0 because y > 1), so y = e^3. EXAMPLE 11 Find t ...

exponential growth or decay. The length of time required for a quantity that is decaying exponentially to be reduced by half is ...

where Q 0 is the initial amount and k is the (negative) factor of proportionality. Since it is given that when t = 1612, equatio ...

Each of the above four quantities (5 through 8) is a function of the form ce−kt (k > 0). (See Figure N9–7b.) FIGURE N9–7a FIG ...

Q(t) = Q 0 e−0.00012t (where Q 0 is the original amount). We are asked to find t when Q(t) = 0.25Q 0. Rounding to the nearest 50 ...

where (in both) f (t) is the amount at time t and k and A are both positive. We may conclude that (a) f (t) is increasing (Fig. ...

EXAMPLE 18 According to Newton’s law of cooling, a hot object cools at a rate proportional to the difference between its own tem ...

We now seek t when N = 2700: Applications of Restricted Growth (1) Newton’s law of heating says that a cold object warms up at a ...

Shortly we will solve specific examples of the logistic d.e. (1), instead of obtaining the general solution (2), since the latte ...

(1) Some diseases spread through a (finite) population P at a rate proportional to the number of people, N(t), infected by time ...

and, finally (!), Please note that this is precisely the solution “advertised” in equation (2), with A equal to 1000. Now, using ...

FIGURE N9–10 Figure N9–10 shows the slope field for equation (3), with k = 0.00179, which was obtained by solving equation (5) a ...