Introduction
One of the biggest challenges AP Calculus students face is remembering the huge list of formulas needed on test day. While the AP Exam provides a formula sheet, it’s limited — meaning you need to memorize and understand many more formulas to succeed.
This guide lays out the essential AP Calculus AB and BC formulas you need to know, explains when to use them, and shows how RevisionDojo can help you practice them until they become second nature.
Limits and Continuity
- limx→0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1
- limx→∞(1+1x)x=e\lim_{x \to \infty} \Big(1 + \frac{1}{x}\Big)^x = e
- Continuity: f(x)f(x) is continuous at cc if:
- limx→c−f(x)=limx→c+f(x)\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)
- f(c)f(c) is defined
Derivative Rules
- Power Rule: ddx[xn]=nxn−1\frac{d}{dx} \big[x^n\big] = nx^{n-1}
- Product Rule: (fg)’=f’g+fg’(fg)’ = f’g + fg’
- Quotient Rule: (fg)’=f’g−fg’g2\Big(\frac{f}{g}\Big)’ = \frac{f’g - fg’}{g^2}
- Chain Rule: (f(g(x)))’=f’(g(x))⋅g’(x)(f(g(x)))’ = f’(g(x)) \cdot g’(x)
- Derivatives of trig functions:
- (sinx)’=cosx(\sin x)’ = \cos x
- (cosx)’=−sinx(\cos x)’ = -\sin x
- (tanx)’=sec2x(\tan x)’ = \sec^2 x
Integrals (AB Must-Know)
- Power Rule for Integration: ∫xndx=xn+1n+1+C, n≠−1\int x^n dx = \frac{x^{n+1}}{n+1} + C, \ n \neq -1
- ∫1xdx=ln∣x∣+C\int \frac{1}{x} dx = \ln |x| + C
- Trigonometric Integrals:
- ∫sinxdx=−cosx+C\int \sin x dx = -\cos x + C
- ∫cosxdx=sinx+C\int \cos x dx = \sin x + C
- Fundamental Theorem of Calculus: ∫abf(x)dx=F(b)−F(a)\int_a^b f(x) dx = F(b) - F(a)
Applications of Derivatives
- Slope of tangent line: m=f’(x)m = f’(x)
- Linear Approximation: L(x)=f(a)+f’(a)(x−a)L(x) = f(a) + f’(a)(x-a)
- Related Rates: Differentiate both sides with respect to time tt.
- Optimization: Find critical points where f’(x)=0f’(x)=0, test for min/max.
Applications of Integrals
- Area Between Curves: A=∫ab[f(x)−g(x)]dxA = \int_a^b \big[f(x) - g(x)\big] dx
- Volume (Disk Method): V=π∫ab[R(x)]2dxV = \pi \int_a^b \big[R(x)\big]^2 dx
- Volume (Washer Method): V=π∫ab[R(x)2−r(x)2]dxV = \pi \int_a^b \big[R(x)^2 - r(x)^2\big] dx
AP Calculus BC Extensions
Sequences and Series
- Geometric Series: ∑n=0∞arn=a1−r,∣r∣<1\sum_{n=0}^\infty ar^n = \frac{a}{1-r}, \quad |r| < 1
- Taylor Series: f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n
Parametric & Polar
- Derivative (Parametric): dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}
- Arc Length (Parametric): L=∫ab(dxdt)2+(dydt)2 dtL = \int_a^b \sqrt{\Big(\frac{dx}{dt}\Big)^2 + \Big(\frac{dy}{dt}\Big)^2} \, dt
- Polar Area: A=12∫αβ[r(θ)]2dθA = \frac{1}{2}\int_\alpha^\beta \big[r(\theta)\big]^2 d\theta
Tips for Memorization
- Don’t just memorize — practice applying formulas in problems.
- Group formulas into categories (derivatives, integrals, applications).
- Create flashcards and test yourself daily.
- Use RevisionDojo’s formula practice sets to see how formulas appear on real AP-style questions.
Why This Formula Sheet Matters
The AP Calculus exam isn’t about regurgitating formulas — it’s about knowing when and how to use them. A formula sheet like this gives you a solid foundation, but only practice will make them second nature.
Frequently Asked Questions
Q: Does the AP Exam give me all the formulas I need?
A: No. The provided sheet has some but not all. You must memorize derivative rules, trig integrals, and BC formulas.
Q: How should I practice formulas?
A: Work through RevisionDojo’s step-by-step problem sets, which apply formulas in FRQ and MCQ formats.
Q: Which formulas are most commonly tested?
A: Fundamental Theorem of Calculus, area/volume formulas, derivative rules, and Taylor Series (for BC).
Q: How can I avoid confusing disk and washer method?
A: Sketch every problem. If there’s no hole → disk. If there’s an inner/outer radius → washer.
