Barrons AP Calculus

We observe that, since y′′ for 3 ln x equals , the true curve is concave down and below the Euler graph. ...

EULER APPROXIMATION ...

ERROR 4 3.85 0.554 10 3.505 0.209 20 3.398 0.102 40 3.346 0.050 80 3.321 0.025 Doubling the number of steps cuts the e ...

the domain of the solution. The accompanying graph (Figure N9–7) shows that this is nowhere near the solutio ...

BC ONLY Always pay attention to the domain of any particular solution. Figure N9–7 D. SOLVING FIRST-ORDER DIFFERENTIA ...

Separating Variables ...

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

The general solution is Example 8 __ Solve the d.e. , given the initial condition y(0) = 2. SOLUTION: We r ...

then integration yields Example 10 __ If (ln y) , and y = e when x = 1, find the value of y greate ...

We now apply the method of separation of variables to three classes of functions associated with different ra ...

Exponential growth where k > 0 if y is increasing and k < 0 if y is decreasing. From (1) it fol ...

Then If we are given an initial amount y, say y 0 at time t = 0, then y 0 = c · e k · 0 = c ...

(a) (b) (a) (b) The bacteria in a certain culture increase continuously at a rate proportional to the numbe ...

We now have Q = Q 0 e−0.00043t. (2) When one quarter of Q 0 has decayed, three quarters of the in ...

(1) (2) (3) (4) (5) (6) (7) (8) Applications of Exponential Growth A colony of bacteria may grow at a rate prop ...

Figure N9–7a Figure N9–7b Example 15 __ At a yearly rate of 5% compounded continuously, how long does it t ...

One important method of dating fossil remains is to determine what portion of the carbon content of a fossil i ...

with solution Q(t) = Q 0 e−0.00012t (where Q 0 is the original amount). We are asked to find t whe ...

(a) (b) (a) (b) Then P(t) = (3.5 × 10^9 )e 0.019t. The question is: for what t does P(t) equal 5.5 × ...

Figure N9–8a Figure N9–8b Here is how we solve the d.e. for Case II(a), where A − y > 0. If the q ...