Calculus AB Main Points

Unit 1 Main Points

Two-Sided Limits

• Existence of Two-Sided Limits: For the limit $ext{lim}<em>{x o a} f(x) = L$ to exist, the left-hand limit and the right-hand limit must be equal:   - $ext{lim}</em>{x o a^-} f(x) = L$ and $ext{lim}_{x o a^+} f(x) = L$

Value and Limit of a Function

• The value of a function at a point $a$ is independent of the limit as $x$ approaches $a$.
    - For example, if $f(a) = L$, it does not imply that $ext{lim}_{x o a} f(x) = L$ holds.
• A function can be undefined at $x = a$ while $ext{lim}_{x o a} f(x)$ can still exist.

Limit Laws

• For a constant $c$ and limits $ext{lim}<em>{x o a} f(x)$ and $ext{lim}</em>{x o a} g(x)$, the following limit laws apply:
    - Sum Rule: $ext{lim}<em>{x o a} [f(x) + g(x)] = ext{lim}</em>{x o a} f(x) + ext{lim}<em>{x o a} g(x)$   - Constant Multiple Rule: $ext{lim}</em>{x o a} [c f(x)] = c ext{lim}<em>{x o a} f(x)$   - Difference Rule: $ext{lim}</em>{x o a} [f(x) - g(x)] = ext{lim}<em>{x o a} f(x) - ext{lim}</em>{x o a} g(x)$
    - Product Rule: $ext{lim}<em>{x o a} [f(x) g(x)] = ext{lim}</em>{x o a} f(x) imes ext{lim}<em>{x o a} g(x)$   - Quotient Rule: $ext{lim}</em>{x o a} \frac{f(x)}{g(x)} = \frac{ ext{lim}<em>{x o a} f(x)}{ ext{lim}</em>{x o a} g(x)}$ if $ext{lim}<em>{x o a} g(x) \neq 0$   - Power Rule: $ext{lim}</em>{x o a} [f(x)]^n = ( ext{lim}_{x o a} f(x))^n$

Methods for Finding Limits

Direct Substitution

• Direct substitution can be applied when the function is continuous at the point approaching.
• Usable for the following types of functions:
    1. Polynomials
    2. Rational functions
    3. Trigonometric functions

Example Scenarios

1. Existence of Limit:
      - For $ext{lim}_{x o 2} \frac{ ext{√}x - 2}{x - 2}$, secures that its limit does exist.
2. Non-Existence of Limit:
      - For $ext{lim}_{x o 2} \frac{ ext{√}x - 2}{x-2}$, depending on conditions, the limit might be DNE (Does Not Exist).

Indeterminate Forms

Rewriting Techniques

• Four algebra techniques allow rewriting functions in indeterminate forms:

1. Factoring and Canceling: Find common factors to simplify.
2. Common Denominators: Change expressions to a common denominator.
3. Multiplying By the Conjugate: Apply conjugate pairs to eliminate square roots.
4. Multiply Out and Simplify: Distributing and combining like terms appropriately.

End Behavior and Asymptotes

Horizontal Asymptotes

• To determine horizontal asymptotes:
    - If $ext{lim}_{x o ext{infinity}} f(x) = c$, then $y = c$ is a horizontal asymptote.

Vertical Asymptotes

• Existence indicated when substituting a point $x = a$ leads the denominator to zero:
    - If $ext{lim}_{x o a} f(x) = ext{±} ext{∞}$, then there is a vertical asymptote at $x=a$.

Types of Discontinuities

Forms of Discontinuity

• Removable Discontinuity (Hole): Exists at $x=c$ if making $f(c)$ continuous “fills in the hole” (extending $f$).
• Jump Discontinuity: Exists if left and right-hand limits at $x=a$ both exist but differ.
• Infinite Discontinuity: If approaching $a$ drives $f(a)$ to infinity (vertical asymptote).

Continuity at a Point

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

Unit 2 Main Points

Definition of the Derivative

• For function $f(x)$ with respect to variable $x$, the derivative $f'(x)$ is defined as:
  $f'(x) = ext{lim}_{h o 0} \frac{f(x + h) - f(x)}{h}$ provided that limit exists.

Slope Interpretations

• Average Rate of Change: Represents the slope of the secant line.
• Instantaneous Rate of Change: Represents the slope of the tangent line at a specific point.

Differentiability and Continuity

• If a function is differentiable at a point, it must be continuous there, but continuity alone doesn't guarantee differentiability.
    - Example functions highlight continuous yet non-differentiable (e.g., $f(x) = |x|$ at $x=0$).

Rules of Derivatives

Rules of Differentiation

1. Constant Function: $\frac{d}{dx}(c) = 0$
2. Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
3. Constant Multiple Rule: $\frac{d}{dx}[c f(x)] = c \frac{d}{dx}[f(x)]$
4. Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
5. Difference Rule: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$
6. Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
7. Quotient Rule: $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Derivatives of Trigonometric Functions

\ - $\frac{d}{dx}( ext{sin } x) = ext{cos } x$
\ - $\frac{d}{dx}( ext{cos } x) = - ext{sin } x$
\ - $\frac{d}{dx}( ext{tan } x) = ext{sec}^2 x$
\ - $\frac{d}{dx}( ext{sec } x) = ext{sec } x an x$
\ - $\frac{d}{dx}( ext{csc } x) = - ext{csc} x ext{cot } x$
\ - $\frac{d}{dx}( ext{cot } x) = - ext{csc}^2 x$

Higher Order Derivatives

• Second Derivative: $f''(x)$ or $\frac{d^2y}{dx^2}$
• Third Derivative: $f'''(x)$ or $\frac{d^3y}{dx^3}$
• nth Derivative: $f^{(n)}(x)$ or $\frac{d^ny}{dx^n}$

The Chain Rule

• The chain rule is expressed in multiple forms:
    1. $\frac{dy}{dx} = f'(u) \frac{du}{dx}$
    2. If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) g'(x)$.

Particle Motion

• Displacement: Calculated as $s(b) - s(a)$
• Average Velocity: $\frac{s(b) - s(a)}{b - a}$
• Instantaneous Velocity: $v(t) = s'(t)$
• Acceleration: $a(t) = v'(t) = s''(t)$

Motion Dynamics

• Condition defining movement direction:
    - Moving left if $v(t) < 0$   - Moving right if $v(t) > 0$
• Condition for speeding up and slowing down is dependent on the sign of velocity and acceleration.
  

Implicit Differentiation

1. Differentiate both sides of an equation w.r.t. $x$.
2. Collect terms containing $y$ on one side.
3. Factor out $y'$ and solve for it.

Unit 6 AP Calculus AB Main Points

Area Between Two Curves

• Area A between curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$:
  $A = ext{Area} = ext{lim} o ext{volume} = ext{integral function} = ext{function bounds}$
• The formula for area is expressed as:
  $A = ext{Area} = ext{∫ from a to b of} (f(x) - g(x)) ext{d}x$ for continuous and defined between $a$ and $b$.

Average Value of a Function

• The average value of a function $f$ on interval $[a,b]$ is given by:
  $f_{ ext{avg}} = \frac{1}{b - a} ext{∫ from a to b of } f(x) ext{d}x$

Unit 7 AP Calculus AB Main Points

Volumes of Solids Using Disk and Washer Methods.

• Volume of Solid (disk method - horizontal axis of revolution):
  $V = ext{π} ext{∫ from a to b of } [f(x)]^2 ext{d}x$
• Washer Method (area between functions):
  $V = ext{π} ext{∫ from a to b of } [R(x)^2 - r(x)^2] ext{d}x$

Solids of Known Cross-sections

• Volume for cross-sections perpendicular to the x-axis:
  $V = ext{∫ from a to b of } A(x) ext{d}x$
• Volume for cross-sections perpendicular to the y-axis:
  $V = ext{∫ from c to d of } A(y) ext{d}y$

Differential Equations and Population Growth

• For equations of the form $\frac{dy}{dt} = ky$, the solution is exponential:
  $y(t) = y_0 e^{kt}$, where $y_0$ is the initial value.