Source: AP Calculus BC Mock 3 (College Board 2008 PT)
Tags: differential equations, slope fields, Euler's method, separable equations, logistic growth, parametric equations, polar curves, polar tangent, acceleration vector, IVT, MVT, EVT, AP Calculus BC
Difficulty: Intermediate to Advanced Prerequisites: Derivatives and basic integration (see the Derivatives and Integration study notes). Familiarity with parametric and polar coordinate systems.
This topic collects several threads that run through BC but not AB: solving and approximating differential equations (slope fields, Euler's method, separable equations, logistic growth), working with parametric and polar curves (velocity, acceleration, tangent lines, area), and applying the major theorems of continuity and differentiability (IVT, MVT, EVT). These concepts appear regularly in both the multiple-choice and free-response sections, often as multi-part FRQs that combine a DE with an initial condition, Euler's method approximation, and a request for the exact solution.
Learn to read slope fields, execute Euler's method step by step, solve separable DEs by integration, and recognise the logistic model. For parametric motion, the key is computing dx/dt, dy/dt, and their derivatives. For polar curves, know how to find tangent slopes and set up area integrals.
Slope field (direction field)
A visual representation of a differential equation dy/dx = F(x, y), where short line segments at grid points show the slope at that point. Think of it as a roadmap that shows which way the solution curves flow.
Euler's method
A numerical technique for approximating the solution of a DE: from a known point (x_n, y_n), the next point is x_{n+1} = x_n + h, y_{n+1} = y_n + h * f(x_n, y_n). In simple terms, you follow the slope field in small straight-line steps.
Separable differential equation
A DE that can be written as g(y) dy = f(x) dx. Solve by integrating both sides. In simple terms, get all the y's on one side and all the x's on the other, then integrate.
Logistic differential equation
dP/dt = kP(M - P), where M is the carrying capacity and k is a growth constant. The population grows fastest at P = M/2 and levels off at P = M. Think of it as growth that slows down as the population approaches its ceiling.
Parametric equations
A pair x = f(t), y = g(t) that traces a curve in the xy-plane as t varies. The velocity vector is (dx/dt, dy/dt), the speed is sqrt[(dx/dt)^2 + (dy/dt)^2], and dy/dx = (dy/dt)/(dx/dt).
Acceleration vector (parametric)
The vector (d^2x/dt^2, d^2y/dt^2). For x = 4t + t^2 and y = 1/(3t+1), the acceleration at t = 1 is (2, 6/(3t+1)^3 evaluated at t=1) = (2, 6/64) = (2, 3/32). Wait: d^2y/dt^2 needs care. First dy/dt = -3/(3t+1)^2, then d^2y/dt^2 = 18/(3t+1)^3. At t = 1: 18/64 = 9/32.
Polar tangent slope
For a polar curve r = f(theta), the slope dy/dx = [r'sin(theta) + rcos(theta)] / [r'cos(theta) - rsin(theta)]. This comes from x = rcos(theta), y = rsin(theta) and the chain rule.
Mean Value Theorem (MVT)
If f is continuous on [a, b] and differentiable on (a, b), there exists c in (a, b) such that f'(c) = [f(b) - f(a)]/(b - a). In simple terms, somewhere the instantaneous rate equals the average rate.
Extreme Value Theorem (EVT)
If f is continuous on [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b]. In simple terms, a continuous function on a closed interval must hit its highest and lowest values.
L'Hopital's rule
If lim f(x)/g(x) gives 0/0 or infinity/infinity, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists. Apply repeatedly if needed.
A slope field for dy/dx = F(x, y) shows the slope at each grid point. To match a solution to an initial condition y(0) = 1, trace a curve through (0, 1) that follows the direction of the line segments.
For the slope field in the exam (question 15), the slopes are positive and steep for small x near x = 0 when y is near 1, and decrease as x moves away from 0 in either direction. Among the choices, y = sqrt(1 - x^2) passes through (0, 1), has slope 0 at x = 0, and curves downward symmetrically, matching the field.
Given dy/dx = x - y - 1, f(1) = -2, approximate f(1.4) with step size h = 0.2 (two steps).
Step 1: At (1, -2), slope = 1 - (-2) - 1 = 2. New y = -2 + 0.2(2) = -1.6. New x = 1.2.
Step 2: At (1.2, -1.6), slope = 1.2 - (-1.6) - 1 = 1.8. New y = -1.6 + 0.2(1.8) = -1.24. New x = 1.4.
So f(1.4) is approximately -1.24.
For dy/dx = 4x - 2xy with f(0) = 5: factor as dy/dx = 2x(2 - y). Separate: dy/(2 - y) = 2x dx. Integrate: -ln|2 - y| = x^2 + C. At x = 0, y = 5: -ln|2 - 5| = C, so C = -ln 3. Solve for y to get the particular solution.
Always apply the initial condition to find C before simplifying, because the algebra is cleaner.
dP/dt = kP(M - P). Given: M = 1200 (carrying capacity), P(t_0) = 200, dP/dt at that instant = 400 people per hour.
Substituting: 400 = k * 200 * (1200 - 200) = k * 200 * 1000 = 200000k. So k = 1/500.
The equation is dP/dt = (1/500) * P * (1200 - P), which can be rewritten as dP/dt = P(1200 - P)/500.
For x = 4t + t^2, y = 1/(3t + 1): dx/dt = 4 + 2t, d^2x/dt^2 = 2. dy/dt = -3/(3t+1)^2, d^2y/dt^2 = 18/(3t+1)^3.
At t = 1: acceleration vector = (2, 18/64) = (2, 9/32).
Speed at time t: sqrt[(4+2t)^2 + (3/(3t+1)^2)^2]. Tangent line at a point uses dy/dx = (dy/dt)/(dx/dt).
For dx/dt = (6/t - 3)^(1/3) and dy/dt = t*e^(-t), at t = 3 the particle is at (5, 4).
Speed at t = 3: dx/dt = (6/3 - 3)^(1/3) = (-1)^(1/3) = -1. dy/dt = 3e^(-3). Speed = sqrt(1 + 9e^(-6)).
The particle moves farthest right when dx/dt changes from positive to negative. Set dx/dt = 0: (6/t - 3)^(1/3) = 0, so 6/t = 3, t = 2. Check signs around t = 2.
For r = 2*theta at theta = pi/2: r = pi, r' = 2.
dy/dx = [r'sin(theta) + rcos(theta)] / [r'cos(theta) - rsin(theta)] = [2sin(pi/2) + picos(pi/2)] / [2cos(pi/2) - pisin(pi/2)] = [2 + 0] / [0 - pi] = -2/pi.
As x -> infinity, (x+1)/x^2 -> 0, so f(x) -> sin(0) = 0. Horizontal asymptote at y = 0.
As x -> 0, (x+1)/x^2 -> infinity (for x > 0) or -infinity (for x < 0). sin oscillates, so no horizontal asymptote at y = 1 and no vertical asymptote (the function oscillates, never blowing up).
Only statement I (horizontal asymptote at y = 0) is true.
If f is continuous on [a, b], the Extreme Value Theorem guarantees a max and min exist. There is a c in [a, b] with f(c) >= f(x) for all x in [a, b].
The Mean Value Theorem guarantees a c in (a, b) with f'(c) = [f(b) - f(a)]/(b - a). This is often used to bound or estimate f'.
Both f(a) < f(c) < f(b) (intermediate value) and f'(c) = 0 (if c is a max/min in the interior) are consequences students must distinguish between.
For lim as x -> 2 of f(x)/(x-2)^2 where f(2) = 0 and f'(2) = 0: this is 0/0, so apply L'Hopital's once: lim f'(x)/[2(x-2)], still 0/0. Apply again: lim f''(x)/2 = 5/2.
The limit exists because f''(2) = 5 is given and finite.
y_{n+1} = y_n + h \cdot f(x_n, y_n) \quad \text{(Euler's method)}\frac{dP}{dt} = kP(M - P) \quad \text{(logistic growth)}\text{Speed} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}, \quad \frac{dy}{dx} = \frac{dy/dt}{dx/dt}\text{Polar slope: } \frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}\text{MVT: } f'(c) = \frac{f(b)-f(a)}{b-a} \text{ for some } c \in (a,b)Students often forget to update both x and y at each Euler's method step. The slope at the new point uses the new x and new y, not the original values.
In the logistic equation dP/dt = kP(M - P), students sometimes confuse k with the maximum growth rate. The fastest growth occurs at P = M/2, not at P = 0.
When finding the tangent slope of a polar curve, students try to use dr/d(theta) as the slope. The slope dy/dx requires converting to Cartesian coordinates first using x = rcos(theta), y = rsin(theta).
Students confuse the IVT, MVT, and EVT. The IVT guarantees a value is hit; the MVT guarantees a slope is matched; the EVT guarantees a maximum and minimum exist. They have different hypotheses and conclusions.
⚠️ Euler's method appears in the FRQ section nearly every year. Practise it until you can do two or three steps quickly and accurately.
⚠️ Separable DEs with an initial condition are a staple of the free-response section. Show every step of the separation and integration.
⚠️ The logistic model is tested in the MC section. Recognise the standard form dP/dt = kP(M - P) and know that maximum growth is at P = M/2.
⚠️ Parametric motion (speed, acceleration, tangent line) appears in the FRQ section. The multi-part structure often builds from speed to tangent to behaviour as t increases.
⚠️ Polar area and tangent slope each appear roughly every other year. Set up carefully: the area formula uses (1/2) * integral of r^2, not integral of r.
True or false: In Euler's method, the step size h is always 1. (False, h can be any positive value)
Fill in the blank: The logistic equation dP/dt = kP(M - P) has the population growing fastest when P = ______. (M/2)
True or false: For parametric equations, speed equals dy/dx. (False, speed = sqrt[(dx/dt)^2 + (dy/dt)^2])
Fill in the blank: The slope of a polar curve r = f(theta) in Cartesian coordinates requires computing dy/dx = ______. ([r'sin(theta) + rcos(theta)] / [r'cos(theta) - rsin(theta)])
True or false: The MVT requires f to be continuous on the closed interval AND differentiable on the open interval. (True)
Q: Given dy/dx = x - y - 1, f(1) = -2, use Euler's method with h = 0.2 to approximate f(1.4).
A: -1.24. Step 1: slope = 2, new y = -1.6. Step 2: slope = 1.8, new y = -1.24.
Q: The rate of change of people on a beach follows a logistic model with carrying capacity 1200. At time t_0, P = 200 and dP/dt = 400. Which equation fits?
A: dP/dt = (1/500)P(1200 - P). Solve 400 = k2001000 to get k = 1/500.
Q: What is the slope of the tangent to the polar curve r = 2*theta at theta = pi/2?
A: -2/pi. Using the polar tangent formula with r = pi and r' = 2.
Q: For the parametric equations x = 4t + t^2, y = 1/(3t+1), what is the acceleration vector at t = 1?
A: (2, 9/32). d^2x/dt^2 = 2, d^2y/dt^2 = 18/(3t+1)^3 = 18/64 = 9/32.
Q: If f is continuous on [a, b], which must be true: there exists c in [a, b] with f(c) >= f(x) for all x in [a, b]?
A: Yes. This is the Extreme Value Theorem: a continuous function on a closed interval attains its maximum.
Q: lim as x -> 2 of f(x)/(x - 2)^2 where f(2) = 0, f'(2) = 0, f''(2) = 5?
A: 5/2. Apply L'Hopital's rule twice: lim f''(x)/2 = 5/2.
Differential equations connect to integration (solving a DE means antidifferentiating), to series (power series solutions of DEs), and to the derivative topics (slope fields visualise the derivative). Parametric and polar topics rely heavily on the chain rule from derivatives and on integration for arc length and area. The major theorems (IVT, MVT, EVT) underpin many existence and uniqueness arguments across the entire course.
slope field, direction field, Euler's method, numerical approximation, separable differential equation, logistic growth, carrying capacity, parametric equations, parametric velocity, parametric acceleration, parametric tangent, polar curve, polar tangent slope, polar area, IVT, intermediate value theorem, MVT, mean value theorem, EVT, extreme value theorem, L'Hopital's rule, indeterminate forms, AP Calculus BC differential equations, AP Calculus BC parametric polar