📐 Calculus BC
Self-Study Notes
Self-Study Notes
20 Core Problems · Key Units · With Explanations
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① LIMITS & CONTINUITY
Limits are the FOUNDATION of all calculus. Master L'Hôpital's Rule and squeeze carefully — they appear EVERY year on AP exams.
PLUG → ZERO/ZERO? → Factor / Conjugate / L'Hôpital
∞/∞ or 0/0 = Indeterminate → must simplify
SQUEEZE: if g ≤ f ≤ h and g,h → L, then f → L
∞/∞ or 0/0 = Indeterminate → must simplify
SQUEEZE: if g ≤ f ≤ h and g,h → L, then f → L
📖 WORKED EXAMPLE
Find \(\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}\)Plug in: \(\frac{0}{0}\) ← indeterminate! Factor: \(\frac{(x-2)(x+2)}{x-2} = x+2\)
Answer: \(\lim_{x\to 2}(x+2) = \mathbf{4}\)
1
LIMITS
Evaluate \(\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{5x}\)
⚠️ Tricky: Don't just plug in! Use the standard trig limit.
💡 EXPLANATION
Use \(\lim_{u\to 0}\frac{\sin u}{u} = 1\). Rewrite: \(\frac{\sin(3x)}{5x} = \frac{3}{5} \cdot \frac{\sin(3x)}{3x}\). As \(x\to 0\), \(\frac{\sin(3x)}{3x}\to 1\), so the answer is \(\dfrac{3}{5}\).
2
LIMITS
Using L'Hôpital's Rule, find \(\displaystyle\lim_{x \to \infty} \frac{4x^3 - 2x}{7x^3 + x^2}\)
⚠️ Tricky: Many students apply L'Hôpital unnecessarily — just check leading terms!
💡 EXPLANATION
Divide numerator & denominator by \(x^3\): \(\frac{4 - 2/x^2}{7 + 1/x} \to \frac{4}{7}\) as \(x\to\infty\). Leading coefficient rule: same degree → ratio of leading coefficients = \(\dfrac{4}{7}\).
3
CONTINUITY
The function \(f(x) = \begin{cases} x^2 + k & x \lt 2 \\ 3x - 1 & x \ge 2 \end{cases}\) is continuous everywhere. Find \(k\).
⚠️ Tricky: Set LEFT limit = RIGHT value at the junction point.
💡 EXPLANATION
For continuity at \(x=2\): left limit = right value.Left: \(\lim_{x\to 2^-}(x^2+k) = 4+k\)
Right: \(3(2)-1 = 5\)
Set equal: \(4+k=5 \Rightarrow k=1\)
② DIFFERENTIATION
Chain Rule, Product Rule, Implicit Differentiation, Related Rates — these are the most-tested derivative topics. Don't skip implicit diff!
CHAIN: \(\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)\) → "outside·inside'"
PRODUCT: \((uv)' = u'v + uv'\) → "HI·dLO + LO·dHI"
IMPLICIT: every \(y\) → multiply by \(\frac{dy}{dx}\)
PRODUCT: \((uv)' = u'v + uv'\) → "HI·dLO + LO·dHI"
IMPLICIT: every \(y\) → multiply by \(\frac{dy}{dx}\)
📖 WORKED EXAMPLE — Implicit Differentiation
Find \(\frac{dy}{dx}\) if \(x^2 + y^2 = 25\)Differentiate both sides: \(2x + 2y\frac{dy}{dx} = 0\)
Solve: \(\frac{dy}{dx} = -\frac{x}{y}\)
4
CHAIN RULE
Find \(f'(x)\) if \(f(x) = \sin(e^{3x})\)
⚠️ Tricky: Three-layer chain — don't forget the innermost derivative!
💡 EXPLANATION
Layer 1 (outer): \(\frac{d}{dx}[\sin(\square)] = \cos(\square)\)Layer 2: \(\frac{d}{dx}[e^{3x}] = 3e^{3x}\)
Multiply: \(\cos(e^{3x}) \cdot 3e^{3x} = 3e^{3x}\cos(e^{3x})\)
5
IMPLICIT DIFF
Given \(x^3 + y^3 = 6xy\), find \(\dfrac{dy}{dx}\).
⚠️ Tricky: Product rule on the right side — students often forget!
💡 EXPLANATION
Differentiate: \(3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}\)Collect \(\frac{dy}{dx}\): \(3y^2\frac{dy}{dx} - 6x\frac{dy}{dx} = 6y - 3x^2\)
Factor: \(\frac{dy}{dx}(3y^2-6x) = 3(2y-x^2)\)
\(\frac{dy}{dx} = \dfrac{2y-x^2}{y^2-2x}\)
6
RELATED RATES
A spherical balloon is inflated so its volume increases at 20 cm³/sec. When the radius is 5 cm, how fast is the radius increasing? (\(V = \frac{4}{3}\pi r^3\))
⚠️ Tricky: Always differentiate with respect to TIME (t), not r!
💡 EXPLANATION
\(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\)Plug in: \(20 = 4\pi(25)\frac{dr}{dt}\)
\(\frac{dr}{dt} = \frac{20}{100\pi} = \frac{1}{5\pi}\) cm/sec
7
DERIVATIVES
What is \(\dfrac{d}{dx}\left[\ln(\cos x)\right]\)?
⚠️ Tricky: \(\frac{d}{dx}[\ln u] = \frac{u'}{u}\) — don't forget the chain rule!
💡 EXPLANATION
\(\frac{d}{dx}[\ln(\cos x)] = \frac{-\sin x}{\cos x} = -\tan x\)Memory trick: derivative of \(\ln(\cos x)\) = \(-\tan x\). This is a standard result worth memorizing!
③ INTEGRATION TECHNIQUES
Integration by parts, u-substitution, trig integrals, and partial fractions. These are the big four techniques for Calc BC!
u-SUB: let u = inside function, find du
BY PARTS: \(\int u\,dv = uv - \int v\,du\) → LIATE order
(Log, Inverse trig, Algebraic, Trig, Exponential)
BY PARTS: \(\int u\,dv = uv - \int v\,du\) → LIATE order
(Log, Inverse trig, Algebraic, Trig, Exponential)
📖 WORKED EXAMPLE — Integration by Parts
Evaluate \(\int x e^x\,dx\)LIATE → \(u = x,\; dv = e^x dx\) → \(du = dx,\; v = e^x\)
\(\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C\)
8
U-SUBSTITUTION
Evaluate \(\displaystyle\int \frac{2x}{x^2 + 1}\,dx\)
💡 EXPLANATION
Let \(u = x^2+1\), then \(du = 2x\,dx\).\(\int \frac{2x}{x^2+1}dx = \int \frac{du}{u} = \ln|u| + C = \ln|x^2+1|+C\)
9
BY PARTS
Evaluate \(\displaystyle\int x\ln x\,dx\)
⚠️ Tricky: Use LIATE — Logarithm comes FIRST as u!
💡 EXPLANATION
LIATE: \(u = \ln x,\; dv = x\,dx \Rightarrow du = \frac{dx}{x},\; v = \frac{x^2}{2}\)\(\int x\ln x\,dx = \frac{x^2}{2}\ln x - \int \frac{x^2}{2}\cdot\frac{1}{x}\,dx = \frac{x^2\ln x}{2} - \frac{1}{2}\int x\,dx = \frac{x^2\ln x}{2} - \frac{x^2}{4} + C\)
10
DEFINITE INTEGRAL
The graph of \(f(x)\) is given. If \(f(x) = 3x^2 - 2\), find \(\displaystyle\int_0^2 f(x)\,dx\).
⚠️ Tricky: Don't mix up antiderivative and original function!
💡 EXPLANATION
\(\int_0^2(3x^2-2)\,dx = [x^3 - 2x]_0^2 = (8-4) - (0-0) = 4\)
④ FUNDAMENTAL THEOREM OF CALCULUS
FTC Part 1: \(\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)\)
FTC Part 2: \(\int_a^b f(x)\,dx = F(b) - F(a)\)
KEY: FTC Part 1 uses CHAIN RULE on the upper limit!
FTC Part 2: \(\int_a^b f(x)\,dx = F(b) - F(a)\)
KEY: FTC Part 1 uses CHAIN RULE on the upper limit!
11
FTC PART 1
Find \(\dfrac{d}{dx}\displaystyle\int_1^{x^2} \sqrt{t^3+1}\,dt\)
⚠️ Tricky: Upper limit is \(x^2\), not just \(x\) — use chain rule!
💡 EXPLANATION
FTC Part 1 + Chain Rule:Substitute upper limit \(x^2\) into integrand: \(\sqrt{(x^2)^3+1} = \sqrt{x^6+1}\)
Multiply by derivative of upper limit: \(\cdot 2x\)
Answer: \(2x\sqrt{x^6+1}\)
12
MVT FOR INTEGRALS
Find the average value of \(f(x) = x^2\) on \([0, 3]\).
⚠️ Tricky: Average value = \(\frac{1}{b-a}\int_a^b f(x)\,dx\) — students forget the \(\frac{1}{b-a}\)!
💡 EXPLANATION
\(f_{avg} = \frac{1}{3-0}\int_0^3 x^2\,dx = \frac{1}{3}\left[\frac{x^3}{3}\right]_0^3 = \frac{1}{3}\cdot 9 = 3\)
⑤ INFINITE SERIES (BC Only!)
Series is the hardest BC topic. Taylor/Maclaurin series, convergence tests, radius of convergence — all heavily tested!
RATIO TEST: \(L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\): L<1 converges, L>1 diverges
GEOMETRIC: \(\sum ar^n\) converges if \(|r|<1\), sum = \(\frac{a}{1-r}\)
Taylor: \(f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)
GEOMETRIC: \(\sum ar^n\) converges if \(|r|<1\), sum = \(\frac{a}{1-r}\)
Taylor: \(f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)
📖 KEY MACLAURIN SERIES TO MEMORIZE
\(e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}\) |
\(\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\) |
\(\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\) |
\(\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n\) for \(|x|<1\)
13
SERIES CONVERGENCE
Does the series \(\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{n^3 + 1}\) converge or diverge?
⚠️ Tricky: Use Limit Comparison Test — compare to \(\frac{1}{n}\)!
💡 EXPLANATION
Compare to \(b_n = \frac{1}{n}\) (harmonic, diverges).\(\lim_{n\to\infty}\frac{a_n}{b_n} = \lim_{n\to\infty}\frac{n^2}{n^3+1}\cdot n = \lim\frac{n^3}{n^3+1} = 1 \ne 0,\infty\)
Since \(\sum\frac{1}{n}\) diverges and limit = 1, the series diverges.
14
TAYLOR SERIES
What is the first three non-zero terms of the Maclaurin series for \(f(x) = e^{-x^2}\)?
⚠️ Tricky: Substitute \(-x^2\) into the \(e^x\) series — don't derive from scratch!
💡 EXPLANATION
\(e^u = 1 + u + \frac{u^2}{2!} + \cdots\)Substitute \(u = -x^2\):
\(e^{-x^2} = 1 + (-x^2) + \frac{(-x^2)^2}{2!} + \cdots = 1 - x^2 + \frac{x^4}{2} - \cdots\)
15
RADIUS OF CONVERGENCE
Find the radius of convergence of \(\displaystyle\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}\)
⚠️ Tricky: The ratio test with \(n!\) always gives \(R = \infty\)!
💡 EXPLANATION
Ratio Test: \(L = \lim_{n\to\infty}\left|\frac{(2x)^{n+1}}{(n+1)!}\cdot\frac{n!}{(2x)^n}\right| = \lim\frac{|2x|}{n+1} = 0\) for all \(x\).Since \(L = 0 < 1\) for ALL \(x\), the series converges everywhere: \(R = \infty\).
⑥ APPLICATIONS OF INTEGRALS
DISK: \(V = \pi\int_a^b [f(x)]^2\,dx\)
WASHER: \(V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx\)
ARC LENGTH: \(L = \int_a^b\sqrt{1+[f'(x)]^2}\,dx\)
WASHER: \(V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx\)
ARC LENGTH: \(L = \int_a^b\sqrt{1+[f'(x)]^2}\,dx\)
16
AREA BETWEEN CURVES
Find the area between \(y = x^2\) and \(y = x + 2\).
⚠️ Tricky: Always find intersection points first! Which function is on top?
💡 EXPLANATION
Intersections: \(x^2 = x+2 \Rightarrow x^2-x-2=0 \Rightarrow (x-2)(x+1)=0\), so \(x=-1, 2\).Top: \(y = x+2\), Bottom: \(y = x^2\)
\(A = \int_{-1}^{2}(x+2-x^2)\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2} = \frac{9}{2}\)
17
VOLUME (DISK)
Find the volume of the solid obtained by rotating \(y = \sqrt{x}\) on \([0, 4]\) about the x-axis.
💡 EXPLANATION
Disk method: \(V = \pi\int_0^4(\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot 8 = 8\pi\)
18
PARAMETRIC
For parametric equations \(x = t^2,\; y = t^3\), find \(\dfrac{dy}{dx}\) at \(t = 2\).
⚠️ Tricky: \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\) — divide by dx/dt, NOT multiply!
💡 EXPLANATION
\(\frac{dx}{dt} = 2t,\quad \frac{dy}{dt} = 3t^2\)\(\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}\)
At \(t = 2\): \(\frac{dy}{dx} = \frac{3(2)}{2} = 3\)
19
DIFFERENTIAL EQUATIONS
Solve the differential equation \(\dfrac{dy}{dx} = 2xy\) with initial condition \(y(0) = 3\).
⚠️ Tricky: Separate variables FIRST before integrating. Don't forget the constant!
💡 EXPLANATION
Separate: \(\frac{dy}{y} = 2x\,dx\)Integrate: \(\ln|y| = x^2 + C\)
\(y = Ae^{x^2}\)
Apply IC: \(3 = Ae^0 = A\), so \(y = 3e^{x^2}\)
20
IMPROPER INTEGRAL
Evaluate \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\)
⚠️ Tricky: Replace \(\infty\) with a limit variable \(b\), evaluate, then take \(b\to\infty\)!
💡 EXPLANATION
\(\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b}+1\right) = 0 + 1 = 1\)p-series rule: \(\sum\frac{1}{n^p}\) converges for \(p>1\) ✓ (similarly for integrals)
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Calculus BC Self-Study Notes · All 20 Problems