Calculus: Finding Limits Graphically and Numerically

Calculus Maximus Notes 1.2: Finding Limits Graphically and Numerically

Graphical Analysis

• Open dots (holes) indicate a limit exists at that x-value.
• Vertical Asymptotes (VA) at $x = c$ mean the limit Does Not Exist (DNE).
• The limit as x approaches c from the left yields infinity: $\lim_{x \to c^-} f(x) = \infty$
• The limit as x approaches c from the right yields negative infinity: $\lim_{x \to c^+} f(x) = -\infty$
• Therefore, the overall limit at c Does Not Exist (DNE): $\lim_{x \to c} f(x) = DNE$
• Horizontal Asymptotes (HA) at $y=c$ exist when:
• The limit as x approaches infinity is c: $\lim_{x \to \infty} f(x) = c$
• The limit as x approaches negative infinity is c: $\lim_{x \to -\infty} f(x) = c$
• Oscillating Behavior means the limit DNE as the function doesn't approach a single value

Numerical Analysis

• Construct value tables as $x$ approaches $c$ from both sides.
• Calculators may provide a table function to generate values.

Special Trig Limits

• $\lim_{x \to 0} \frac{sin(x)}{x} = 1$
• $\lim_{x \to 0} \frac{1 - cos(x)}{x} = 0$

Example 1

• To find $\lim_{x \to 2} f(x)$ where $f(x) = \begin{cases} x^2, & x < 2 \ 7 - x, & x > 2 \end{cases}$
• Solve the limit from the left: $\lim_{x \to 2^-} f(x) = (2)^2 = 4$
• Solve the limit from the right: $\lim_{x \to 2^+} f(x) = 7 - 2 = 5$
• $\lim_{x \to 2} f(x)$ DNE because one-sided limits are unequal

Example 2

• Estimating with data $\lim_{x \to 0} \frac{e^x - 1}{x}$ yields 1

Chapter 4: Probability

Definitions

• A random experiment is a process that leads to an uncertain outcome.
• A sample space (S) includes all possible outcomes of a random experiment.
• An event comprises any subset of the sample space.
• Tossing a die yields S = {1, 2, 3, 4, 5, 6}, and an even number is Event A = {2, 4, 6}

Ways of Assigning Probability

• Classical: $P(Event) = \frac{Number , of , outcomes , in , the , event}{Total , number , of , possible , outcomes}$
• Applies with equally likely outcomes.
• Relative Frequency: $P(Event) = \frac{Number , of , times , the , event , occurred}{Total , number , of , observations}$
• Applies when prior data estimates outcome likelihood with repetitions.
• Subjective: Uses available information, like experience, for probability estimation.

Rules of Probability

• Probability Axioms:
• $P(A) \geq 0$ for any event A.
• $P(S) = 1$
• $P(A_1 \cup A_2 \cup A_3 \cup...) = \sum_i P(A_i)$ if events are mutually exclusive.
• Mutually Exclusive Events cannot occur simultaneously.
• Collectively Exhaustive Events encompass the entire sample space.
• Complement Rule: $P(A) = 1 - P(A^c)$
• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• $P(A \cup B) = P(A) + P(B)$ if A and B are mutually exclusive.

Conditional Probability

• $P(A|B) = \frac{P(A \cap B)}{P(B)}$, given $P(B) > 0$
• Television viewers are sampled in two cities, City 1 (150 viewers) and City 2 (200 viewers).
• City 1: 40 watch Program A, 110 don't.
• City 2: 80 watch Program A, 120 don't.
• Outcomes:
• The probability a viewer is from City 1: $P(C_1) = \frac{150}{350} = \frac{3}{7}$
• The probability a viewer watches Program A: $P(A) = \frac{120}{350} = \frac{12}{35}$
• The probability a viewer is from City 1 and watches Program A: $P(C_1 \cap A) = \frac{40}{350} = \frac{4}{35}$
• The probability a viewer is from City 1 or watches Program A: $P(C_1 \cup A) = \frac{23}{35}$
• The probability that a viewer from city 1 watches program A is 4/15.

Statistical Independence

• Two events A and B are statistically independent if and only if $P(A|B) = P(A)$ or $P(B|A) = P(B)$.
• If A and B are independent, then $P(A \cap B) = P(A) \cdot P(B)$.

Bayes' Theorem

• $P(B_i|A) = \frac{P(A|B_i)P(B_i)}{P(A|B_1)P(B_1) + P(A|B_2)P(B_2) +... + P(A|B_k)P(B_k)}$ where $B_1, B_2,..., B_k$ are mutually exclusive and collectively exhaustive events.
• A company sells basic (40%) and deluxe (60%) video camera models.
• 30% of basic models need warranty repair.
• 10% of deluxe models need warranty repair.
• Outcomes:
• The probability of buying a basic model needing repair: $P(B \cap R) = 0.12$
• The probability of needing a repair: $P(R) = 0.18$
• The probability of having bought a basic model given a repair: $P(B|R) = \frac{2}{3}$

Tópicos de Física

Mecânica

• Cinemática
• Movimento Uniforme (MU)
• Velocity is constant.
• $S = S_0 + v \cdot t$ helps find position.
• Graphs show position and velocity over time.
• Movimento Uniformemente Variado (MUV)
• Acceleration is constant.
• $S = S_0 + v_0 \cdot t + \frac{1}{2} a \cdot t^2$ and $v = v_0 + a \cdot t$ finds position and velocity over time.
• $v^2 = v_0^2 + 2 \cdot a \cdot \Delta S$ uses Torricelli's equation.
• Graphs show position, velocity, and acceleration.
• Lançamento Vertical e Queda Livre
• $g$: gravitational acceleration.
• Motion analysis happens vertically.
• Features time up, max height, and range.
• Movimento Circular Uniforme (MCU)
• $T$ (period) and $f$ (frequency).
• $\omega = \frac{2\pi}{T}$ and $v = \omega \cdot R$ define angular ($\omega$) and linear (v) speed.
• $a_c = \frac{v^2}{R}$ defines centripetal acceleration.
• Composição de Movimentos
• Motion of ballistic projectile.
• Movement is independent.
• Dinâmica
• Leis de Newton
• First Law: Inertia.
• Second Law: $F = ma$.
• Third Law: Action & Reaction.
• Aplicações das Leis de Newton
• Static and kinetic forces of friction.
• Force in inclined planes.
• Tracction forces.
• Trabalho e Energia
• $W = F \cdot d \cdot \cos(\theta)$ defines work
• $E_c = \frac{1}{2} m \cdot v^2$ is kinetic energy.
• $E_p = m \cdot g \cdot h$ is gravitational potential energy.
• $E_{pe} = \frac{1}{2} k \cdot x^2$ is elastic potential energy.
• Calculate with the work-energy theorem.
• Conservação de Energia
• Conservative and non-conservative systems.
• Find conservation of mechanical energy.
• Impulso e Quantidade de Movimento
• $I = F \cdot \Delta t$ calculates impluse
• $Q = m \cdot v$ calculates quantity of movement
• Calculate impulse-moment theorem.
• Colisões
• Define elastic, inelastic, and partially elastic collisions.
• Coefficient of restitution.
• Determine the motion of bodies after a collision.
• Estática
• Conditions for material point equilibrium of an extended body.
• Turning tendency of a force.
• Mass is at a fixed point in a body or system of bodies.
• Gravitação
• Lei da Gravitação Universal
• $F = G \cdot \frac{m_1 \cdot m_2}{r^2}$ gravitational force.
• $G$: Gravitational constant.
• Leis de Kepler
• First Law: Orbits
• Second Law: Areas
• Third Law: Periods
• Energia Potencial Gravitacional
• Gravitational potential energy of a 2-mass system.
• Define escape velocity.

Termologia

• Termometria
• Find the difference between temperature and heat, and equilibrate thermal systems.
• Different termometric scales and their conversinos are Celsius, Fahrenheit, and Kelvin.
• Expansion of linear, superficial and volumetric systems, define the expansion coefficients.
• Calorimetria
• $Q = m \cdot c \cdot \Delta T$ calculates the change in temperature of an object
• $Q = m \cdot L$calculates heat transfer during phase change
• $C = \frac{Q}{\Delta T}$ defines thermal capacity and the rate of heat flow.
• Thermal equilibria in closed systems and in calorimeters.
• Termodinâmica
• $PV = nRT$ is the ideal gas equation
• Thermodynamic transformations include:
• Isothermal
• Isobaric
• Isochoric
• Adiabatic
• $\Delta U = Q - W$ is the conservation of energy in thermodynamic systems, calculate from work and internal energy.
• Entropy and the irreversibility of all systems, Carnot Cycles use heat engines and refrigerators.
• Transmissão de Calor
• Heat transfer through a material is measured by Fourier's Law for conduction of heat transfer.
• Convection transfers heat through fluids by natural and forced actions.
• Transfer of heat through electromagnetic waves with the Stefan-Boltzmann law for black bodies.

Óptica

• Óptica Geométrica
• These principles apply to geometric optics:
• Propagation of rectilinear light.
• Light beams are reversible.
• The laws of reflection apply to plane and spherical mirrors.
• Law of Refraction describes how light passes through different reflective surfaces.
• Refraction index
• Laws include Snell's Law
• Lenses create images (Gauss Equation).
• Óptica Física
• Characterize the wave-particle duality within light theory, interferrence with Young's experiments.
• How light diffracts and polarizes.
• Electromagnetic spectrum shows radio to gamma waves.

Ondulatória

• Características das Ondas
• Sound and light and longitudinal, that share amplitude, frequency, period and wavelength.
• Relationship between wave speed, frequency and wavelength $v = \lambda \cdot f$
• Standing waves vibrate.
• These phenomenon are common to wave action in reflexions, refraction, diffraction interferenece and resonance.

Eletricidade

• Eletrostática
• Charge is transferred between materials with Coulumb's Lae.
• Electric field lines are measured by the magnitude of force a point charge recieves.
• The ability to do work depends on the electrical potential of capacitors.
• Eletrodinâmica
• Current is measured by current laws (Kirchoff).
• Ohm's law.
• Find the circuits of generators and power with electrical components.
• Eletromagnetismo
• There are many types of magnetic fields from wires to solenoids.
• Force on charges in moving conductors.
• Farady's law gives the electromotive force generated by transformers.

Física Moderna

• Relatividade
• Theory and equations relative to mass, energy, and momentum.
• There is equiavlence to $E=mc^2$.
• Física Quântica
• Energy as discrete value.
• Models include Bohr and Schrödinger.
• Understanding the atomic nucleus.
• Describe the interaction of nuclei with each other in radiation.

Diagram of a Car

• Engine: generates power.
• Fuel Tank: stores/supplies fuel.
• Steering Wheel: Directional control.
• Brakes: Safety/stopping.
• Battery: powers electrical system.