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Differential Equations, Series & BC Extensions — Visual Lab
Module 5. The capstone of the calculus year, and the heart of Calculus BC: equations that describe how things change, curves traced by time, and the radical idea that a polynomial — built one term at a time — can become a transcendental function. Add terms below and watch a Taylor polynomial reach out and grip the curve.
A Taylor series takes everything calculus knows about a function at a single point — its value, its slope, its concavity, and every higher derivative — and packs it into an infinite polynomial \(\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}\,(x-a)^n\). Centered at \(a=0\) it's called a Maclaurin series. Truncate it after a few terms and you get a partial sum that already hugs the curve near the center; add more terms and the interval of agreement spreads outward. Build one below, term by term, and watch a polynomial learn to imitate \(e^x\), \(\sin x\), or \(\cos x\).
Taylor Series Builder
Pick a function, then add terms \(1 \to N\). The faded curve is the true function; the solid curve is your degree-\(N\) Maclaurin polynomial. Watch the green band — the region where the two agree to within a hair — widen with every term. The current partial-sum formula renders live.
What you're seeing
- The faded curve is the truth. It's the real \(e^x\), \(\sin x\), or \(\cos x\). The solid curve is your polynomial — so the gap between them is your error.
- Each term adds one power of \(x\). The degree-\(N\) partial sum uses derivatives up to order \(N\). More terms = a higher-degree polynomial that can bend in more places.
- They always agree at the center first. Every Maclaurin polynomial is exact at \(x=0\); accuracy then radiates outward as you add terms.
- The factorial in the denominator tames the powers. The coefficient of \(x^n\) is \(\frac{f^{(n)}(0)}{n!}\). The \(n!\) is why \(e^x\), \(\sin x\), and \(\cos x\) converge for every \(x\) (infinite radius of convergence).
- Sine is odd, cosine is even. Notice \(\sin x\) uses only odd powers \(x, x^3, x^5,\dots\) and \(\cos x\) only even powers \(1, x^2, x^4,\dots\) — the symmetry of the function shows up in the symmetry of its series.
Try this
- Start with \(e^x\) and one term. The polynomial is just the constant \(1\) — a flat line touching the curve at \((0,1)\). Add a term: now it's \(1+x\), the tangent line. Each term improves the fit.
- Climb to \(N=6\). Watch the green agreement band stretch from a sliver near \(0\) to most of the screen. Where the solid curve peels away from the faded one is where you'd need more terms.
- Switch to \(\sin x\) and add terms one at a time. Predict before each click: only odd powers appear, and the signs alternate \(+,-,+,\dots\). Confirm it in the live formula.
- Compare \(\cos x\) at \(N=2\) vs \(N=4\). Degree 2 gives the parabola \(1-\tfrac{x^2}{2}\) — right near \(0\), but it dives off too fast. The \(+\tfrac{x^4}{24}\) term pulls it back up. That's a higher-order correction at work.
Worked Examples
One separable differential equation (Unit 7), one Maclaurin series built from scratch (Unit 10) — the two pillars of this module.
Solve \(\dfrac{dy}{dx} = \dfrac{2x}{y}\), given \(y(0) = 3\).
- Separate the variables. Move every \(y\) to the left and every \(x\) to the right: \(y\,dy = 2x\,dx\). (This step is exactly where the most common mistake happens — you may only separate when the equation factors into an \(x\)-part times a \(y\)-part.)
- Integrate both sides. \(\displaystyle\int y\,dy = \int 2x\,dx \;\Rightarrow\; \frac{y^2}{2} = x^2 + C\). Write the constant \(+C\) once, on the side that's most convenient.
- Apply the initial condition now (before simplifying further). With \(x=0,\ y=3\): \(\frac{3^2}{2} = 0 + C\), so \(C = \frac{9}{2}\).
- Solve for \(y\) and pick the right branch. \(y^2 = 2x^2 + 9\), so \(y = \pm\sqrt{2x^2+9}\). Since \(y(0)=3>0\), keep the positive root.
Find the Maclaurin series of \(f(x) = \cos x\) (the curve the lab draws).
- List the derivatives, cycling every four. \(f=\cos x,\ f'=-\sin x,\ f''=-\cos x,\ f'''=\sin x,\ f^{(4)}=\cos x,\dots\)
- Evaluate each at the center \(a=0\). \(f(0)=1,\ f'(0)=0,\ f''(0)=-1,\ f'''(0)=0,\ f^{(4)}(0)=1,\dots\) — the odd derivatives vanish.
- Drop them into \(\sum \frac{f^{(n)}(0)}{n!}x^n\). Only even \(n\) survive: \(1 + 0\cdot x - \frac{1}{2!}x^2 + 0\cdot x^3 + \frac{1}{4!}x^4 - \cdots\)
- Write the general term. The signs alternate and only even powers appear, so \(\displaystyle\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\,x^{2n}\), valid for all real \(x\).
Key Vocabulary
The precise words a mathematician uses for what the lab — and this module — is doing.
An equation relating a function to its derivatives, e.g. \(\frac{dy}{dx}=ky\). Its solution is a family of functions; an initial condition selects one.
A differential equation that can be written \(g(y)\,dy = f(x)\,dx\), so each side integrates independently.
A grid of short segments showing \(\frac{dy}{dx}\) at each point — a visual map of every solution curve at once.
A step-by-step numerical estimate of a solution: from a point, follow the slope a short step \(h\), then repeat.
A sequence is a list of terms \(a_n\); a series is their running sum \(\sum a_n\). A sequence can converge while its series diverges (e.g. the harmonic series).
\(\sum r^n\) converges iff \(|r|<1\); \(\sum \frac{1}{n^p}\) converges iff \(p>1\). The two benchmark series every test compares against.
A series \(\sum c_n (x-a)^n\) in a variable \(x\). It converges on an interval of convergence with a radius \(R\) found via the ratio test.
\(\sum \frac{f^{(n)}(a)}{n!}(x-a)^n\) — the power series matching a function's derivatives at \(a\). Maclaurin is the special case \(a=0\).
Curves given by \(x(t), y(t)\), by \(r(\theta)\), or by a vector \(\langle x(t), y(t)\rangle\) — calculus extended to motion and rotation.
AP Calculus BC — Units 7, 9, 10
Module 5 carries the three BC-only units plus the collegiate Calculus II sequence: differential equations and modeling, the calculus of parametric/polar/vector functions, and the full theory of infinite sequences and series — culminating in the Taylor series this lab builds.
Unit 7 slope fields, Euler's method, separable equations, exponential & logistic models · Unit 9 derivatives & integrals of parametric, polar, and vector-valued functions · Unit 10 convergence tests (\(n\)th-term, geometric, \(p\)-series, integral, comparison, ratio, alternating), power series, intervals of convergence, and Taylor/Maclaurin series.
Ready for the full course map? Head back to Calculus, read the Student Support, or check the Pacing Guide to see where Module 5 sits in the year.
Course framework follows the College Board AP Calculus BC Course and Exam Description (Units 7, 9, 10) and the standard collegiate Calculus II sequence. AP® is a trademark of the College Board, which was not involved in the production of, and does not endorse, this material.