AP Calculus BC Key Concepts

Average Rate of Change

• Definition: The average rate of change (slope of the secant line) over an interval

  • Given function: $f(x)$

  • Interval: $[a, b]$

  • Formula: $\text{Average R.o.C} = \frac{f(b) - f(a)}{b - a}$

• Application: If $v(t)$ is velocity, the formula gives average acceleration.

Differentiation and Integration Rules

Chain Rule

• Formula: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Product Rule

• Formula: $\frac{d}{dx}[f \cdot g] = f' \cdot g + f \cdot g'$

Quotient Rule

• Formula: $\frac{d}{dx}[\frac{f}{g}] = \frac{f' \cdot g - f \cdot g'}{g^2}$

Integration by Parts

• Formula: $\int u \cdot dv = u \cdot v - \int v \cdot du$

Fundamental Theorem of Calculus

Part 1

• Formula: $\int_a^b f'(x)\,dx = f(b) - f(a)$

  • Alternate forms:

  • $f(b) = f(a) + \int_a^b f'(x)\,dx$

  • $f(a) = f(b) - \int_a^b f'(x)\,dx$

Part 2

• Derivative of an integral: $\frac{d}{dx} \left( \int_a^{g(x)} f(t)\,dt \right) = f(g(x)) \cdot g'(x)$

Average Value Theorem

• If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, average value is given by:

  • $\text{Average value} = \frac{1}{b - a} \cdot \int_a^b f(x)\,dx$

  • If $f(x)$ is the velocity $v(t)$, the formula gives average velocity.

Mean Value Theorem

• If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that:

  • $f'(c) = \frac{f(b) - f(a)}{b - a}$

Definition of the Derivative

• Two forms:

  • $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

  • $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Basic Integral Formulas

• $\int dx = x + C$

• $\int x^n dx = \frac{1}{n+1} \cdot x^{n+1} + C$

• $\int \frac{1}{x} dx = \ln |x| + C$

• $\int e^x dx = e^x + C$

• $\int b^x dx = \frac{1}{\ln b} \cdot b^x + C$

• $\int \sin(x) dx = -\cos(x) + C$

• $\int \cos(x) dx = \sin(x) + C$

• $\int \frac{1}{\sqrt{1 - x^2}} dx = \arcsin(x) + C$

• $\int \frac{1}{1 + x^2} dx = \arctan(x) + C$

Intermediate Value Theorem

• If $f(x)$ is continuous on $[a, b]$ and f(a) < f(c) < f(b), then there exists some $c$ in $[a, b]$.

Derivative Formulas

• $\frac{d}{dx}[x^n] = nx^{n-1}$

• $\frac{d}{dx}[\sin(x)] = \cos(x)$

• $\frac{d}{dx}[\cos(x)] = -\sin(x)$

• $\frac{d}{dx}[\tan(x)] = \sec^2(x)$

• $\frac{d}{dx}[\cot(x)] = -\csc^2(x)$

• $\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$

• $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$

• $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$

• $\frac{d}{dx}[\log_b(x)] = \frac{1}{x \ln(b)}$

• $\frac{d}{dx}[e^x] = e^x$

• $\frac{d}{dx}[b^x] = b^x \cdot \ln(b)$

• $\frac{d}{dx} (f^{-1}(x)) = \frac{1}{f'(f^{-1}(x))}$

Extreme Value Theorem

• If $f(x)$ is continuous on $[a, b]$, then there exists an absolute maximum and minimum value on $[a, b]$.

L'Hôpital's Rule

• When evaluating $\lim_{x \to a} \frac{f(x)}{g(x)}$ with indeterminate forms (0/0 or ±∞), show each limit:

  • $\lim<em>{x \to a} \frac{f(x)}{g(x)} = \lim</em>{x \to a} \frac{f'(x)}{g'(x)}$

Volume of Solids of Revolution

• Disks: $V = \int \pi (f(x))^2 \,dx$

• Washers: $V = \pi \int{(f(x))^2 - (g(x))^2 \,dx}$

• Cross Sections: $V = \int A(x) \,dx$

U-Substitution in Integration

• Given $\int<em>{{x</em>0}}^{{x_1}} f(x) \, dx$:

  • Let $u$ be part of $f(x)$.

  • $du = \text{(derivative)} \cdot dx$

  • Change limits: $u(x<em>0)$ and $u(x</em>1)$.

  • Integral becomes $\int<em>{{u</em>0}}^{{u_1}} f(u) \, du$

Particle Motion in One Dimension

• Position: $x(t)$ or $y(t)$ or $s(t)$

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = v'(t) = s''(t)$

• Speed: $|v(t)|$

• Displacement: $\int<em>{t</em>0}^{t_1} v(t) \, dt$

• Total Distance: $\int<em>{t</em>0}^{t_1} |v(t)| \, dt$

Solving Differential Equations

• Given: $\frac{dy}{dt} = \text{(expression)}$ and initial condition:

  1. Separate $dy$ and $dt$ to opposite sides.

  2. Integrate both sides.

  3. Add integration constant $C$.

  4. Use initial condition to solve for $C$.

  5. Isolate $y$.

Taylor Series

• Approximation of a function:

  • $f(x) \approx f(a) + f'(a)(x - a) + \frac{f''(a)(x - a)^2}{2!} + \cdots + \frac{f^{(n)}(a)(x-a)^n}{n!}$

• Maclaurin Series: Special case where $a = 0$.

Euler's Method

• Given: $\frac{dy}{dx} = f'(x, y)$ and initial point $(x<em>0, y</em>0)$:

  1. Approximate the solution:

  2. Use tangent line:
     $y(t) = \frac{dy}{dx}(x<em>0, y</em>0)(x - x<em>0) + y</em>0$

  3. Find the next point using the tangent line equation.

  4. Repeat until the solution point is reached.

Series to Memorize

• $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

• $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

• $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

• $\frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots$

Radius and Interval of Convergence

• Radius of Convergence: Use Ratio Test.

• Interval of Convergence: Check endpoints after finding radius.

Particle Motion in Two Dimensions

• Position: $\langle x(t), y(t) \rangle$

• Velocity: $\langle x'(t), y'(t) \rangle$

• Acceleration: $\langle x''(t), y''(t) \rangle$

• Speed: $\sqrt{(x'(t))^2 + (y'(t))^2}$

• Distance traveled: $\int<em>{t</em>0}^{t_1} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$

Logistic Curves

• Differential equation: $\frac{dP}{dt} = kP(L - P)$

  • Parameters:

  • $t$: time

  • $P$: population size

  • $L$: carrying capacity

  • $k$: constant of proportionality

• Population growth: Fastest growth when $P = \frac{1}{2} L$.

• Limits:

  • $\lim_{t \to \infty} P(t) = L$

  • $\lim_{t \to 0^+} P(t) = 0$.

Series Error Bounds

• The difference between actual value and calculated series is bounded:

  • For Alternating Series: Error bound $\leq |a_{n+1}|$ (next neglected term).

  • Lagrange Error Bound: $\leq \frac{\max_{a \leq z \leq x} |f^{(n+1)}(z)|}{(n + 1)!} \cdot (x-a)^{n+1}$

Rectangular Arc Length of a Function

• Arc length formula: $\int_a^b \sqrt{1 + (f'(x))^2} \, dx$

Polar Derivatives

• Derivative in polar coordinates: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$

  • Using polar form:

  • $\frac{dy}{d\theta}(r \sin(\theta))$

  • $\frac{dx}{d\theta}(r \cos(\theta))$

Area Calculations

• Area of a single curve: $\frac{1}{2} \int<em>{\theta</em>1}^{\theta_2} (f(\theta))^2 \, d\theta$

• Area between two curves: $\frac{1}{2} \int<em>{\theta</em>1}^{\theta_2} [g(\theta)^2 - f(\theta)^2] \, d\theta$

Absolute and Conditional Convergence

• A series $\sum a_n$ is:

  • Absolutely Convergent if $\sum |a_n|$ converges.

  • Conditionally Convergent if $\sum |a<em>n|$ does not converge, but $\sum a</em>n$ does.

Parametric Functions

First Derivative

• Derivative: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Second Derivative

• Second derivative: $\frac{d^2y}{dx^2} = \frac{d}{dt} \left( \frac{dy}{dx} \right) \cdot \frac{dt}{dx}$

Arc Length

• Arc length formula: $\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$