Essential Calculus Theorems and Formula Reference
Fundamental Calculus Definitions and Theorems
The Derivative and Integral Definitions
The Derivative Definition
The derivative of a function $f(x)$, denoted $f'(x)$, is defined using the limit of the difference quotient:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$
The Definite Integral (Riemann Sum)
The definite integral of $f(x)$ from $a$ to $b$ is defined as the limit of the Riemann sum:
$$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$$
Key Calculus Theorems
Mean Value Theorem
Read MoreAdvanced Trigonometric Identities and Ratio Calculations
1. Proving the Tangent Sum and Difference Identity
Identity to Prove:
\tan (x+y) + \tan (x-y) = \frac{\sin (2x)}{\cos (2y) – \cos (2x)}
Proof Steps
Start with the Left-Hand Side (LHS) using the definition of tangent:
\tan (x+y) + \tan (x-y) = \frac{\sin (x+y)}{\cos (x+y)} + \frac{\sin (x-y)}{\cos (x-y)}
Combine the fractions:
= \frac{\sin (x+y)\cos (x-y) + \sin (x-y)\cos (x+y)}{\cos (x+y)\cos (x-y)}
Using the Sine Addition Formula, $\sin(A+B) = \sin A \cos B + \cos A \sin B$:
= \frac{\sin [ (x+y) + (x-y)
Read MoreDevelopmental Stages in Mathematical Measurement and Number Sense
Learning Trajectory for Length Measurement
E1: Initial Unit Placement
Places the units from end to end. May not recognize the need for units of the same length or may not be able to measure if there are fewer units than necessary. Can use rulers with substantial guidance.
E2: Ordering and Seriation of Lengths
Orders lengths, marked from 1 to 6 units. Understands, at least intuitively, that any set of objects of different lengths can be placed in a series that is always increasing (or decreasing) in
Read MoreDiscrete Mathematics Formulas and Proof Techniques
Problem: what is the power set P(S) of S=(a,b,c) Solution: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}. |p ∨ (p ∧ q) ≡ p|, |p ∧ (p ∨ q) ≡ p|, |p → q ≡ ¬p ∨ q|, |p → q ≡ ¬q → ¬p|, |p ∨ q ≡ ¬p → q|, |p ∧ q ≡ ¬(p → ¬q)|, |¬(p → q) ≡ p ∧ ¬q|, |(p → q) ∧ (p → r) ≡ p → (q ∧ r)|, |(p → r) ∧ (q → r) ≡ (p ∨ q) → r|, |(p → q) ∨ (p → r) ≡ p → (q ∨ r)|, |(p → r) ∨ (q → r) ≡ (p ∧ q) → r|, |p ↔ q ≡
Read MoreEssential Physics Formulas: Kinematics and Dynamics
Cristián Arriagada: Essential Physics Formulas
This compilation provides fundamental equations covering kinematics, dynamics, work, energy, and rotational motion in classical mechanics. These formulas are crucial for solving problems involving motion and forces.
1. Kinematics (Equations of Motion)
1.1. Uniformly Accelerated Motion (MUA)
- Velocity: V = V0 + a(t – t0)
- Position: X = X0 + V0t + 1/2 at2
- Velocity-Position: V2 = V02 + 2a(X – X0)
- Time: t = (V – V0) / a
1.2. Uniform Rectilinear Motion (MRU)
- Velocity:
Understanding 10 Key Principles of Graphic Representation
Principle of Multiple Application
This patterning process involves the use of a simple figure to represent a variety of objects and body parts. With a limited graphic vocabulary, an artist can represent very different things. This process is useful for its economy of means and communicative effectiveness.
Principle of the Baseline
The baseline is a horizontal line that crosses the drawing near the bottom, serving as the support for characters, animals, plants, and objects. It is a very useful graphical
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