2020 AP Calculus BC Formula List

Definition of the Derivative (Alternative Form)

• The derivative of a function $f$ at a point $c$ is defined as:
  $f'(c) = ext{lim}_{h o 0} \frac{f(c+h) - f(c)}{h}$

Definition of Continuity

• A function $f$ is continuous at $c$ if and only if:
    1. $f(c)$ is defined;
    2. $ext{lim}<em>{x o c} f(x)$ exists;   3. $ext{lim}</em>{x o c} f(x) = f(c)$.

Mean Value Theorem

• If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one number $c$ in $(a, b)$ such that:
  $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Intermediate Value Theorem

• If $f$ is continuous on $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ between $a$ and $b$ such that:
  $f(c) = k$.

Definition of a Definite Integral

• The definite integral of $f$ from $a$ to $b$ is defined as:
  $ext{lim}<em>{n o ext{∞}} \frac{1}{n} ext{sum}</em>{i=1}^{n} f(x_i^<em>) imes ext{Δ}x$ where $ext{Δ}x = \frac{b-a}{n}$ and $x_i^</em>$ is a sample point in each subinterval.

Definition of a Critical Number

• A number $c$ is a critical number of the function $f$ if it is defined at $c$ and either:
    1. $f'(c)$ is undefined, or
    2. $f'(c) = 0$.

First Derivative Test

• Let $c$ be a critical number of $f$ which is continuous on an interval $I$ containing $c$:
    1. If $f'(x)$ changes from negative to positive at $c$, then $f(c)$ is a relative minimum.
    2. If $f'(x)$ changes from positive to negative at $c$, then $f(c)$ is a relative maximum.

Second Derivative Test

• If $f''(c)$ exists, then:
    1. If f''(c) > 0, then $f(c)$ is a relative minimum.
    2. If f''(c) < 0, then $f(c)$ is a relative maximum.

Definition of Concavity

• A function $f$ is:
    - Concave upward on an interval $I$ if $f'$ is increasing on that interval.
    - Concave downward on an interval $I$ if $f'$ is decreasing on that interval.

Test for Concavity

• The graph of $f$ is:
    1. Concave upward on $I$ if f''(x) > 0 for all $x$ in $I$.
    2. Concave downward on $I$ if f''(x) < 0 for all $x$ in $I$.

Definition of an Inflection Point

• A function $f$ has an inflection point at $c$ if:
    1. $f''(c)$ does not exist, or
    2. $f''(x)$ changes sign from positive to negative or from negative to positive at $x = c$.

First and Second Fundamental Theorem of Calculus

• First Fundamental Theorem of Calculus:
    If $F$ is an antiderivative of $f$ on $[a, b]$, then:
  $ext{If } f(x) = F'(x), ext{ then } ext{ } ext{ } ext{ } \boxed{F(b) - F(a) = ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }F(x)= ext{u'}=f(x) }$
• Second Fundamental Theorem of Calculus:
    If $f$ is continuous on [a, b], then the function defined by $F(x) = ext{if } ext{ exists; } f(x)= ext{integral form};$ can be represented as:
  $ext{ } F'(x) = f(x)$.

Chain Rule

• If $y = f(g(x))$, then:
  $\frac{dy}{dx} = f'(g(x)) imes g'(x)$.

Average Rate of Change of $f(x)$ on $[a, b]$

• The average rate of change is given by:
  $ext{Average Rate = } \frac{f(b) - f(a)}{b - a}$.

Average Value of $f(x)$ on $[a, b]$

• The average value is given by:
  $ext{Average Value = } \frac{1}{b - a} ext{ } imes ext{ } ext{ } ext{ }$ ext{integral form} = ext{ }(f(x)dx)

Particle Motion

• If an object moves along a straight line with position function s(t), then:
    - Velocity: v(t) = s'(t).
    - Acceleration: a(t) = v'(t) = s''(t).
    - Displacement from t_0$to$t_1 is:
       ext{Displacement} = s(t_1) - s(t_0).
    - Total Distance traveled:
       ext{Total Distance} = ext{ } s(t) - s(t)
    - An object is at rest when v(t) = 0.
    - An object is moving left (down) when v(t) < 0.   - An object is moving right (up) when v(t) > 0.
    - An object changes direction when velocity changes signs.
    - An object is speeding up when velocity and acceleration have the same sign.
    - An object is slowing down when velocity and acceleration have different signs.

Rate In/Rate Out

• If rate in > rate out, then amount is increasing.
• If rate out > rate in, then amount is decreasing.
• When finding absolute maximum and/or absolute minimum, use the candidates test. To find critical values, set rate in = rate out.

L'Hôpital's Rule

• Suppose that:
    If f$and$g are differentiable functions and that:
      1. ext{lim}{x o a} f(x) = 0$and$ ext{lim}{x o a} g(x) = 0
      OR
      2. ext{lim}{x o a} f(x) = ext{ } ext{ } ext{ }${} ext{hll}; ext{that,s,t}$      ext{lim}{x o a} f(x) = ext{ } ext{ } ext{ }ny }+,
• Then, ext{lim}{x o a} rac{f(x)}{g(x)} = ext{lim}{x o a} rac{f'(x)}{g'(x)} if limits exist.

Integration by Parts

• Indefinite: ext{ } oxed{ ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }{ ext{integral form}}

• Definite: ext{ } ext{ } ext{ } ext{ }d~u~{v} = uv- ext{u.d} ext{v}

Area Under a Curve

• Formula: ext{Area} = ext{integral from } a ext{ to } b y f(x) dx.

Volume

• Disk Method (no hole):
    - A. Horizontal Axis of Rotation:
      V = ext{π} imes ext{(radii)}ootnotesize{dx};
    - B. Vertical Axis of Rotation:
      dy = x.

• Washer Method (with hole):
    - A. Horizontal Axis of Rotation:
      V = ext{π}( ext{R}^2 - ext{r}^2) ext{ }dy;
    - B. Vertical Axis of Rotation:
      dy = ( ext{ )(Fg,h})/ ext{ }{$V = rac{1}{2}∫{a}^{b} dy}=A{lower}dy.

Length of a Curve

• The length L$of a curve from$x=a$to$x=b is given by:
     L = ext{lim}{n o ext{∞}} ext{sum}{i=1}^n ext{ } ext{ of the integrals } . \n

Definition of a Taylor Polynomial

• If a function f$has$n$derivatives at$c$, then the polynomial$P_n(x) = f(c) + f'(c)(x-c) + rac{f''(c)}{2!}(x-c)^2 + ackslash + rac{f''''(c)}{3!}(x-c)^3$is called the$n^{th}$Taylor polynomial for$f$at$c.

Tests for Convergence

• Nth Term Test for Divergence: This test cannot be used to show convergence.

• Geometric Series: If |r| < 1$, the geometric series converges to$rac{1}{1-r}.

• P-Series: A series of the form rac{1}{n^p}$converges if$p > 1$and diverges if$p ext{ } ext{!} ext{ } ext{!}

• Alternating Series Test (Leibniz's Test): An alternating series converges if:
    1. The absolute value of the terms decreases (i.e., |a_{n+1}| < |a_n|)
    2. The limit of the terms is 0 (i.e., $ext{lim}_{n o ext{∞}} a_n$ = 0).

• Remainder Estimate: For a convergent series, the remainder $R_n$ when approximating with $n$ terms can be estimated with:
  $|R_n| ext{ } ext{!} ext{ }|a_{n+1}|$.

Conclusion

• Familiarizing yourself with these definitions, theorems, and formulas is crucial for succeeding in AP Calculus BC, as they represent key concepts that will appear on exams and in practical applications in calculus and higher mathematics.