AP® Calculus BC Cheat Sheet

This AP Calculus BC cheat sheet provides a quick reference for key concepts like limits, derivatives, integrals, sequences, series, and parametric equations, helping students prepare efficiently for the AP exam.

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Unit 1: Limits & Continuity

• Algebraic simplifications for limits: Completing the square, rationalization, factoring.
• Intermediate Value Theorem (IVT): If f(x) is continuous on [a,b], and f(c) is between f(a and f(b), then there is a c in (a,b) such that f(c) = 0.
• Limits formulas:
• \(\lim_{x \to c} [af(x)] = a \cdot \lim_{x \to c} f(x)\)
• \(\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)\)
• \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \quad \text{(L’Hopital’s Rule)}\)

Unit 2: Differentiation: Definition and Fundamental Properties

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes at any given point. Essentially, it measures how a function’s output value changes as its input value changes. The derivative of a function at a particular point provides the slope of the tangent line to the curve at that point.

In mathematical terms, if y = f(x), the derivative of f(x) with respect to x, denoted by f′(x) or \(\frac{dy}{dx}\), is the rate of change of y with respect to x.

• Power Rule: \(\frac{d}{dx}(x^n) = n x^{n-1}\)
• Sum/Difference Rule: \(\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)\)
• Product Rule: \(\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)\)
• Quotient Rule: \(\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2}\)
• Chain Rule: \(\frac{d}{dx} f(g(x)) = g'(x) \cdot f'(g(x))\)
• Implicit Differentiation: Differentiate both sides with respect to the variables.
• Inverse Trig Functions:
• \(\frac{d}{dx} (\sin^{-1}(x)) = \frac{1}{\sqrt{1 – x^2}}\)
• \(\frac{d}{dx} (\cos^{-1}(x)) = – \frac{1}{\sqrt{1 – x^2}}\)

Unit 3: Composite, Implicit, & Inverse Functions

Chain Rule for Composite Functions:

• The derivative of a composite function f(g(x)) is: \(\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\)
• This applies to any combination of nested functions. For example: \(\frac{d}{dx} \sin(2x) = \cos(2x) \cdot 2\)

Derivatives of Inverse Trigonometric Functions:

• \(\left( f^{-1} \right)'(x) = \frac{1}{f'(f^{-1}(x))}\)
• \(\frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 – x^2}}\)
• \(\frac{d}{dx} \cos^{-1}(x) = -\frac{1}{\sqrt{1 – x^2}}\)
• \(\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1 + x^2}\)
• \(\frac{d}{dx} \cot^{-1}(x) = -\frac{1}{1 + x^2}\)
• \(\frac{d}{dx} \sec^{-1}(x) = \frac{1}{|x| \sqrt{x^2 – 1}}\)
• \(\frac{d}{dx} \csc^{-1}(x) = -\frac{1}{|x| \sqrt{x^2 – 1}}\)

Unit 4: Contextual Applications of Differentiation

Particle Motion:

• Position: s(t)
• Velocity: v(t) = s′(t)
• Acceleration: a(t) = v′(t) = s′′(t)
• If velocity is negative, the particle is moving to the left.
• If velocity is positive, the particle is moving to the right.
• If velocity and acceleration have the same sign, the particle is speeding up.
• If velocity and acceleration have different signs, the particle is slowing down.

Steps for Related Rates:

1. Draw a picture and label it, assigning variables.
2. List known and unknown values.
3. Write an equation to model the situation.
4. Differentiate both sides with respect to time (use d/dt).
5. Plug in known values and solve for the desired value. Don’t forget units!

Linearization:

• Linear approximation of f(x) at x = a is L(x) = f(a) + f′(a) (x−a).

L’Hopital’s Rule:

• Use when \( \frac{f(x)}{g(x)} \) is indeterminate (0/0 or ∞/∞).
• \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\)

Unit 5: Analytical Applications of Differentiation

• Mean Value Theorem (MVT):
  • If f(x) is continuous on [a,b] and differentiable on (a,b), there is a ccc in (a,b) such that:
  \(f'(c) = \frac{f(b) – f(a)}{b – a}\)
• Extreme Value Theorem (EVT):
  • If f(x) is continuous on [a,b], there exists at least one local maximum and one local minimum on [a,b].
• Critical Points:
  • Occur where f′(x) = 0 or does not exist.
• First Derivative Test:
  • If f′(x) changes from positive to negative at c, f(x) has a local maximum at c.
  • If f′(x) changes from negative to positive at c, f(x) has a local minimum at c.
• Concavity:
  • f′′(x) > 0: Concave up.
  • f′′(x) < 0: Concave down.
  • f′′(x) = 0: Possible inflection point.
• Second Derivative Test:
  • If f′(x) = 0 and f′′(x) > 0, then f(x) has a local minimum.
  • If f′(x)=0 and f′′(x)<0, then f(x) has a local maximum.
• Steps for Optimization:
  1. Draw and label a picture.
  2. Assign variables and write an equation.
  3. Find relationships among the variables.
  4. Differentiate and find extrema (min/max).

Unit 6: Integration & Accumulation of Change

• Fundamental Theorem of Calculus (FTC): \(\int_a^b f(x) dx = F(b) – F(a)\) where F(x) is an antiderivative of f(x).
• Integration by Parts: ∫u dv = uv −∫vdu
• Riemann Sum: Approximation of area under a curve using left, right, midpoint, or trapezoidal sums.

Unit 7: Differential Equations

• Logistic Differential Equation: \(\frac{dP}{dt} = kP \left( 1 – \frac{P}{L} \right)\), where P is the population, L is the carrying capacity, and k is a constant.
• Slope Fields: Graphical representation of a differential equation \( \frac{dy}{dx} = f(x, y) \)
• Euler’s Method: Used to numerically approximate solutions of differential equations.

Unit 8: Applications of Integration

Volumes:

• Washer Method: \(V = \pi \int_a^b \left( R(x)^2 – r(x)^2 \right) dx\)
• Disc Method: \(V = \pi \int_a^b \left( R(x)^2 \right) dx\)

Arc Length:

• Parametric: \( L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt \)
• Polar: \(\int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} d\theta/)

Area under a Polar Curve: \(A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta\)

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

• Parametric Equations: x = f(t) ,y = g(t)
• Second Derivative: \( \frac{d^2 y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right) \)
• Area under Polar Curves: \(A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta\)

Unit 10: Infinite Sequences and Series

Taylor Series:

• \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x – c)^n\)

Power Series: Converges when p>1; diverges otherwise.

Convergence Tests:

• Nth-Term Test: Series diverges if \(\lim_{n \to \infty} a_n\)0
• Ratio Test: Converges if \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n} < 1\)