3rd Quarter Reviewer - StatProb - Grade 11 PDF

Statistics & Probability 3rd Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Introduction to Statistics and Probability - Lesson #1 Statistics - Branch of Science dealing with collection, presentation, analysis, and in...

Statistics & Probability 3rd Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Introduction to Statistics and Probability - Lesson #1 Statistics - Branch of Science dealing with collection, presentation, analysis, and interpretation of data ○ Applied Statistics - Procedural & Techniques Descriptive Statistics - Data w/o conclusion or inference (Statistical Treatment) Inferential Statistics - Data leads to conclusion, prediction, or inference ○ Mathematical Statistics - Theoretical & Mathematical Foundations Probability - Numerical measure of the likelihood of the occurrence of an event ○ Probability justifies statistics # 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 ○ 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑟 𝑃(𝑋) = # 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Discrete Random Variables Random Variable - Numerical Quantity assigned to the outcome of an experiment represented by a capital letter Discrete Random Variable - Countable (ex. Number of x) ○ Example: Number of even numbers (E) rolled in a set of 3 (three) die SS: {0, 1, 2, 3} Countable Random Variable - Measurable (ex. Length, Weight) ○ Example: Speeds driven by a car Discrete Random Distribution - Table giving a list of probability values along with their associated value wherein the following stipulations are true: ○ 0 < 𝑃(𝑋) ≤ 1 ○ 𝛴 𝑃(𝑋) = 1 Histogram - Graphing of the probability values and their associated values ○ Example: 4 coins are flipped, construct the discrete probability distribution & histogram for the variable (H) or the number of heads 𝑆𝑆 = {0,1,2,3,4} StatProb Reviewer by Ramon Caluag | 1 H 0 1 2 3 4 Outcomes TTTT TTTH HHTT HHHT HHHH TTHT HTTH HHTH THTT TTHH HTHH HTTT THHT THHH THTH HTHT Probability 1/16 4/16 6/16 4/16 1/16 Combination - Choosing/selecting 𝑟 number of objects from 𝑛 population ○ Example: 4 people are chosen out of a group of 10 girls and 7 boys. Construct the DPD for G (girls) G 0 1 2 3 4 P(Y) (10𝐶0)(7𝐶4) (10𝐶1)(7𝐶3) (10𝐶2)(7𝐶2) (10𝐶3)(7𝐶1) (10𝐶4)(7𝐶0) 17𝐶4 17𝐶4 17𝐶4 17𝐶4 17𝐶4 P(Y) 1/68 5/34 27/68 6/17 3/34 Statistics & Probability 3rd Quarter Reviewer —————————————————— Mean, Variance, and Standard Deviation - Lesson #2 StatProb Reviewer by Ramon Caluag | 2  Mean Mean - Average value of all outcomes ○ 𝜇 = 𝛴[𝑋 ⋅ 𝑃(𝑋)] Where 𝜇 is the mean, 𝑋 is the value of the outcome, and 𝑃(𝑋) is the probability of the outcome ○ Example: Values 0 1 2 Frequency 3 4 3 3 4 3 ○ 𝜇 = (0 × 10) + (1 × 10) + (2 × 10) ○ 𝜇 = 1, The average number is 1 Variance Variance - Average distance in units squared ○ 𝜎2 = 𝛴[𝑋 2 ⋅ 𝑃(𝑋)] − 𝜇 2 Where 𝜎2 is the variance, 𝑋 2 is the value of the outcome squared, 𝑃(𝑋) is the probability of the outcome, and 𝜇2 is the mean squared ○ Example: 𝜇 = 1 Values 0 1 2 Frequency 3 4 3 3 4 3 ○ 𝜎2 = [(02 × ) + (12 × ) + (22 × )] − 𝜇 2 10 10 10 ○ 𝜎2 = 3/5, The average distance squared is 1.4 units^2 Standard Deviation Standard Deviation - Average distance from the mean, square root of variance ○ 𝜎 = √𝜎2 or 𝜎 = √𝛴(𝑋2 ⋅ 𝑃(𝑋)) − 𝜇2 ○ Example: 𝜎2 = 3/5 ○ 𝜎2 = √3/5 √15 ○ 𝜎 = √15 5 , The average distance from the mean is units 5 StatProb Reviewer by Ramon Caluag | 3  —————————————————— Statistics & Probability 3rd Quarter Reviewer —————————————————— Normal Distribution - Lesson #3 Skewness - Measure of symmetry or asymmetry of the probability distribution ○ Negatively Skewed - Mean is less than the median ○ Normal Distribution - Mean is the equation to the median and the mode, perfectly symmetrical, asymptotic to the horizontal line, Area = 1 or 100% ○ Positively Skewed - Mean is greater than the median Probability of Continuous Random Variables 𝑥−𝜇 Convert Raw Score to Z-Score - 𝑧 = 𝜎 Examples: ○ 𝜇 = 150, 𝜎 = 25, 𝑥 < 100 100−150 Z-Score: −2 = 25 Z-Table: −2 = 0.0228 Area: 0.0228 × 100% = 2.28% Shaded to the left because the given is less than StatProb Reviewer by Ramon Caluag | 4  Note: Although the example on the right uses a 75 to 225 scale, for graphing a bell curve use the -3 to 3 scale with the corresponding z- score as the reference value ○ 𝜇 = 150, 𝜎 = 25, 𝑥 > 138 138−150 Z-Score: −0.48 = 25 Z-Table: −0.48 = 0.3156 0.3156 × 100% = 31.56% Because the given is greater than and the z-table provides probabilities for less than, we subtract the found value from 100% to get the probability of greater than Area: 100% − 31.56% = 68.44% ○ The average age of citizens in an area is 30 yrs old with a standard deviation of 5 years. If the area has 500,000 citizens, how many of them are older than 16 but younger than 25? 𝜇 = 30, 𝜎 = 5,16 < 𝑥 < 25 Probability of 16 (Greater Than/Lower Limit) For word problems, when dealing with greater than add 0.5, for less than subtract 0.5. However, when dealing with “or equal to” do not add or subtract (16+0.5)−30 Z-Score: −2.7 = 5 Z-Table: −2.7 = 0.0035 Area: 0.35% Probability of 25 (Less Than/Upper Limit) (25−0.5)−30 Z-Score: −1.1 = 5 Z-Table: −1.1 = 0.1357 Area: 13.57% Subtract both probabilities 13.57% − 0.35% = 13.22% StatProb Reviewer by Ramon Caluag | 5  —————————————————— Statistics & Probability 3rd Quarter Reviewer —————————————————— Sampling - Lesson #4 Population - The subjects being studied ○ Example: The 110 Million Filipino population was the subject of a study on Political Polarization Sample - The respondents or participants selected from a population ○ Example: 1,000 Filipinos were chosen to participate in the study Parameter - Numerical measure based on the population ○ Example: Percent of Filipinos that agree that Political Polarization is present in the Philippines Statistic - Numerical Value of a Sample ○ Example: Percent of 1,000 Filipinos that agree that Political Polarization is present in the Philippines Sampling Techniques Random Sampling - Unbiased sampling, typically used in quantitative studies (To generalize) ○ Simple Random - Purely random sampling Example: Wheel of Names, Picking a Name from a Jar ○ Systematic - Every nth term in a list Example: Every Class Number divisible by 4 ○ Stratified - Splitting the population into different strata or groups for equal representation Example: A study on gender issues having strata by gender ○ Cluster - Used for large-scale studies (commonly nationwide), which splits a large area into smaller clusters for equal geographical representation Example: Exit-polling during elections Non-Random Sampling - Biased sampling, typically found in qualitative studies StatProb Reviewer by Ramon Caluag | 6 ○ Convinience - Incidental or ambush sampling, based on the convinience of the researchers Example: Interviewing whoever is closest/passes by ○ Purposive - Deliberate sampling, choosing who is best fit for the study based on certain criteria Example: A study on academic performance chooses students who are academically inclined as respondents ○ Snowball - Referral sampling, best used in studies with sensitive or confidential topics Example: Asking your respondents to refer others as your respondent ○ Quota - Stratified mixed with purposive, filling certain strata or groups to reach a quota Example: A study on the K-12 Program having a sample of 30 Students, 15 Teachers, 15 Parents, and 10 School Administrators StatProb Reviewer by Ramon Caluag | 7  General Biology 1 4th Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Transport Mechanisms - Lesson #1 Plasma Membrane - Cell membrane, separates the cell from its surroundings Passive Transport Passive Transport - Does not use ATP or energy ○ Higher concentration to lower concentration Diffusion - Movement of substance without help, spreading of a substance, hydrophobic molecules (O2 & CO2) Facilitated Transport - Differences in concentration, transport proteins (lock/key), hydrophilic molecules & ions Osmosis - Diffusion of water molecules based on tonicity ○ Tonicity - Ability of a surrounding solution to gain or lose water ○ Hypotonic Solution - Less solute, more water, swelling ○ Isotonic Solution - Equal solute/solvent ○ Hypertonic - More solute, less water inside, shrinkage General Biology 1 by Ramon Caluag | 1  Active Transport Active Transport - Uses ATP molecules and energy ○ Lower concentration to higher concentration Na-K (Sodium-Potassium) Pump - Expels 3 Na+ Ions to accept 2 K+ ions ○ Maintain proper concentrations ○ For muscles/cramping (electrolytes) ○ Lock & Key Mechanism Vesicle Transport - Movement of materials in and out of the cell ○ Endocytosis - Materials brought inside ○ Exocytosis - Materials brought outside ○ Phagocytosis - Cell engulfs food or material ○ Pinocytosis - Vesicles form around liquid (drinking) ○ Receptor-Mediated - Signals (Neurons) —————————————————— General Biology 1 4th Quarter Reviewer —————————————————— ATP-ADP Cycle- Lesson #2 Energy Flow - Energy flows as sunlight, leaves as heat, recycling of essential chemicals (PCHONS) ○ Phosphorus, Carbon, Hydrogen, Oxygen, Nitrogen, Sulfur Energy - Causes changes, Ability to do work (Movement) ○ Kinetic - Relative motion ○ Thermal - Movement of atoms & heat ○ Light - Sun & Photosynthesis General Biology 1 by Ramon Caluag | 2  ○ Potential - Energy at rest ○ Chemical - Potential energy released in a chemical reaction Laws of Energy Transformation Thermodynamics - Study of energy transfer in a system Open system - Organisms have energy flowing in and out 1st Law (Law of Conservation of Energy) - Energy of the universe is constant, it can be transferred & transformed but not created nor destroyed ○ Energy can be lost in the form of heat ○ Disorder of matter is measured through entropy (unavailability of a system’s thermal for conversion into mechanical work resulting in heat) ○ Disorder - Rearrangement of atoms/molecules ○ Entropy - Unavailability of a living system to do work Greater entropy = less able to do work 2nd Law - Every transfer of energy increases the energy of the universe ○ 10% Rule - On every trophic level in the food chain, only 10% of energy gets passed down to the next organism. The rest contributes to an increase of energy in the surroundings (Lost as heat) Exergonic - Energy is released, greater decrease = more work down Endergonic - Energy inward, plants store energy in glucose ( 𝐶6𝐻12𝑂6) Equilibrium - No work is being done, isolated system, stagnant ○ Cell is dead when it reaches equilibrium as in a working system, all products should become reactants General Biology 1 by Ramon Caluag | 3  Adenosine Triphosphate ATP - Adenosine triphosphate, currency of energy for cell work ○ Chemical - Synthesis/breakdown ○ Transport - Active Transport ○ Mechanical - Misc. functions Hydrolysis - Addition of water to release energy for work ○ Water is added to ATP to break it down into ADP, Phosphate and Energy Dephosporilization - Removal of phosphate from ATP yielding energy, opposite of phosphorylation which requires energy ○ Coupled Reaction - Use ATP hydrolysis (exergonic) to fuel endergonic reactions Other exergonic reactions lead to ATP regeneration ATP released energy at a rate of 7.3 kcal/mole based on the 3rd Phosphate containing the most amount of energy out of the 3 phosphates General Biology 1 by Ramon Caluag | 4  General Biology 1 4th Quarter Reviewer —————————————————— Photosynthesis - Lesson #3 Conversion of solar energy to chemical energy, glucose 𝐶6𝐻12𝑂6 Photo = Photons = Light, Synthesis Chemical Reaction - 6𝐶𝑂2 + 12𝐻2𝑂 + 𝐿𝑖𝑔ℎ𝑡 → 𝐶6𝐻12𝑂6 + 6𝑂2 + 6𝐻2𝑂 ○ Simplified: 6𝐶𝑂2 + 6𝐻2𝑂 + 𝐿𝑖𝑔ℎ𝑡 → 𝐶6𝐻12𝑂6 + 6𝑂2 Occurs in the leaves ○ Stomata - Opens to absorb CO2 and release O2 ○ Chloroplast - Absorption of sunlight (Elodea cells) ○ Roots - Absorbs water Parts of a Chloroplast ○ Outer & Inner Membranes - Protective coverings ○ Stroma - Dense fluid, converts CO2 to sugar ○ Thylakoid - Flattened sacs, converts light energy to chemical energy ○ Grana - Green pigment, absorbs liquid Pigments - Substances that absorb visible light to capture the sun’s energy ○ Chloroplast is green due to chlorophyll, it can absorb almost all colors except green which it reflects Photosystems Group of pigments & proteins which absorb proteins & transfer energy/electrons General Biology 1 by Ramon Caluag | 5  Light Harvesting Complex - Antenna complex, captures & passes photons to the reaction center, Chlorophyll A & B, Carotenoids ○ Chlorophyll A - Blue-Violet & Orange-Red Light ○ Chlorophyll B - Blue Light ○ Carotenoids - Green Light Photosystem II - 1st in Function, 2nd in Discovery, P680 Wavelength Photosystem I - 2nd in Function, 1st in Discovery, P700 Wavelength Light-Dependent Reaction ETC - Electron Transport Chain NADP+ - Nicotinamide Adenine Dinucleotide Phosphate+ NADPH - Nicotinamide Adenine Dinucleotide Phosphate Hydrogen P680 - Chlorophyll A in Photosystem II P700 - Chlorophyll A in Photosystem I PSI - Photosystem I PSII - Photosystem II PEA - Primary Electron Acceptor Light Reaction - Uses sunlight, takes place in Thylakoids ○ Products - ATP, NADPH, Oxygen Steps 1 to 5 (Occurs in Photosystem II) ○ Step 1 - Photons absorbed by P680 (PS II), Light harvesting complex ○ Step 2 - PEA gets electrons from P680 which becomes positive + 𝑃680 cannot function and needs to gain an electron ○ Step 3 - Splitting of water, P680+ gets H+ electrons from water Input: H2O (from roots), By-Product: O2 ○ Step 4 - Carrying electrons from PEA to PSI via ETC Electrocarriers: Plastaquionne & Plastocyanin General Biology 1 by Ramon Caluag | 6  ○ Step 5 - Chemiosmosis happens, H+ exits through ATP synthase (ADP from Calvin cycle gets turned into ATP which just floats around) Product: ATP O2 is released by Stomata Steps 6 to 8 (Occurs in PSI) ○ Step 6 - Photons absorbed by PSI (P700) & PEA gets electrons from P700 which becomes positive + 𝑃700 gets electrons from PSII via ETC ○ Step 7 - Ferredoxin (electron carrier) gets electrons from PSII ○ Step 8 - Production of NADPH through NADP reductase H+ from stroma is transferred to NADP+ forming NADPH Calvin Cycle Calvin Cycle - Discovered by Melvin Calvin, Light-independent, dark reaction, takes place in stomata ○ Reactants: CO2, NADPH, ATP ○ Products: ADP, NADP+, Glucose (C6H12O6) ○ Needs to spin 3 times to make 1 molecule of G3P from 3CO2 G3P - Glyceraldehyde-3-phosphate ○ Sugar - Not six-carbon glucose but rather G3P (3-carbon) RuBP - Ribulose BiPhosphate Carbon Fixation - RuBP (five-carbon) and CO2 (one-carbon) is transformed (aided by Rubisco, an enzyme) into 2 molecules of 3-phosphoglycerate (three-carbon each for a total of six) Reduction General Biology 1 by Ramon Caluag | 7  ○ Each 3-phosphoglycerate is broken down into 1,3 - 1,3-biphosphoglycerate which in turn transforms ATP into ADP (ADP goes to PSII, ATP synthase) ○ The two 1,3-biphosphoglycerate molecules get transformed into G3P which in turn transforms NADPH into NADP+ (Goes to PSI, NADP+ reductase) ○ Products: ADP (Goes to PSII, ATP synthase), NADP+ (Goes to PSI, NADP+ Reductase), 2 G3P Molecules Regeneration of RuBP - Cycle repeats 3 times to produce a total of 6 G3P molecules ○ 5 G3P molecules are used to produce 3 RuBP to continue the process ○ The remaining one G3P combines with another G3P produced from another repeat of the Calvin cycle to produce 1 Glucose molecule ○ 2 G3P : 1 Glucose —————————————————— General Biology 1 4th Quarter Reviewer —————————————————— Cellular Respiration - Lesson #4 Metabolism, processing food to energy, occurs in the mitochondria Anabolism - Build-up of molecules Catabolism - Breakdown of glucose (CUT-abolism) Chemical Formula: 𝐶6𝐻12𝑂6 + 6𝑂2 → 6𝐶𝑂2 + 6𝐻2𝑂 + 𝐴𝑇𝑃 ○ Reverse of Photosynthesis General Biology 1 by Ramon Caluag | 8  Aerobic - Uses oxygen Anaerobic - Does not use oxygen Aerobic Respiration Stages - Glycolysis, Krebs Cycle, Electron Transport Chain ○ Glycolysis - 1st Step, Breaks glucose into 2 pyruvic acid molecules Products: 4 ATP (used 2 ATP at the start), 6 ATP from NADH, 2 pyruvic acid ○ Transition Reaction - 2 NADH are used to produce 6 ATP ○ Krebs Cycle - Explained by Sir Hans Krebs Tricarboxylic/citric acid, occurs in Mitochondria Produces ATP, NADH (Nicotinamide Adenine Dinucleotide Hydrogen), FADH (Flavin Adenine Dinucleotide Hydrogen) Products: 8 NADH (2 from transition reaction, 6 from Glycolysis), 2 FADH2, 2 ATP ○ Electron Transport Chain - Last step, series of redox reactions (reduction/oxidation) 10 NADH + 2FADH2 are converted into ATP 1 NADH = 3 ATP 1 FADH2 = 2 ATP Total: 10 NADH = 30 ATP, 2 FADH = 4 ATP, 4 ATP = 4 ATP Total ATP Produced: 38 ATP Anaerobic Respiration Alcoholic Fermentation - Uses glycolysis ○ Products: ATP, NADH, pyruvic acid ○ Pyruvic acid is converted into Ethanol & NAD+ ○ Examples: Beer & Wine General Biology 1 by Ramon Caluag | 9  Lactic Acid Fermentation - Glucose is broken down into 2 molecules of lactic acid ○ Products: NADH, ATP, Pyruvic Acid ○ Example: Muscles using lactic acid & Dairy products General Biology 1 by Ramon Caluag | 10  Basic Calculus 4th Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Slope of a Tangent Line - Lesson #1 Secant Line Secant Line - A line touching a curve at exactly two points Slope - Steepness or gradient of a line ○ Represented by m or monter meaning “to climb” 𝑟𝑖𝑠𝑒 ∆𝑦 (𝑦2−𝑦1) 𝑓(𝑥+ℎ)−𝑓(𝑥) ○ Formula: 𝑚 = 𝑟𝑢𝑛 = ∆𝑥 = (𝑥2−𝑥1) = ℎ Tangent Line Tangent Line - A line touching a curve at exactly one point 𝑓(𝑥+ℎ)−𝑓(𝑥) Formula for Slope: 𝑚 = lim ℎ ℎ→0 ○ H or the distance between the two points touching the curve is 0 (it is the same singular point under a tangent line) thus the value of H approaches 0 ○ The formula is used in the increment method for looking for the derivative of the function or the formula used to find the slope, however, it is extremely time-consuming Point-Slope Form - 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 2 ○ Example: 𝑓(𝑥) = 2𝑥 | 𝑃(2, 8) | 𝑚 = 8 𝑦 − 8 = 8(𝑥 − 2) 𝑦 − 8 + 8 = 8𝑥 − 16 + 8 𝑦 = 8𝑥 − 8 BasCal Reviewer by Ramon Caluag | 1  Basic Calculus 4th Quarter Reviewer —————————————————— Derivative of a Function - Lesson #2 Algebraic Functions The slope formula is the derivative of a function Denoted by: ○ 𝑓'(𝑥) - f prime of x ○ 𝑦' - y prime 𝑑𝑦 ○ 𝑑𝑥 - derivative of y with respect to x 𝑑𝑦 ○ 𝑑𝑥 [𝑓(𝑥)] or 𝑑𝑥[𝑓(𝑥)] Differentiation - Process of finding derivatives 𝑓(𝑥+∆𝑥)−𝑓(𝑥) 𝑓(𝑥+ℎ)−𝑓(𝑥) ○ 𝑓'(𝑥) = 𝑚 = lim ∆𝑥 = lim ℎ ∆𝑥 → 0 ℎ→0 Derivative of a Constant - 𝑑(𝑐) = 0 ○ Example: 𝑓(𝑥) = 7, 𝑓'(𝑥) = 0 𝑛 𝑛−1 Power Rule - 𝑑(𝑥 ) = 𝑛𝑥 & 𝑑(𝑐𝑢) = 𝑐𝑑𝑢 ○ Example: 𝑓(𝑥) = 𝑥, 𝑓'(𝑥) = 1 5 4 𝑓(𝑥) = 7𝑥 , 𝑓'(𝑥) = 35𝑥 −7 −8 7 𝑓(𝑥) = 𝑥 , 𝑓'(𝑥) =− 7𝑥 =− 8 𝑥 Sum of Terms - 𝑑(𝑢 ± 𝑣) = 𝑑𝑢 ± 𝑑𝑣 ○ Example: 4 2 3 𝑓(𝑥) = 12𝑥 + 3𝑥 − 6𝑥 + 1, 𝑓'(𝑥) = 48𝑥 + 6𝑥 − 6 Product of Terms - 𝑑(𝑢 * 𝑣) = 𝑢(𝑑𝑣) + 𝑣(𝑑𝑢) ○ Example: 𝑓(𝑥) = (𝑥 + 3)(2𝑥 + 7) 𝑢 = (𝑥 + 3), 𝑑𝑢 = 1 𝑣 = (2𝑥 + 7), 𝑑𝑣 = 2 BasCal Reviewer by Ramon Caluag | 2  2(𝑥 + 3) + 1(2𝑥 + 7) 2𝑥 + 6 + 2𝑥 + 7 𝑓'(𝑥) = 4𝑥 + 13 𝑢 𝑣𝑑𝑢−𝑢𝑑𝑣 Rational Function - 𝑑( 𝑣 ) = 2 𝑣 ○ Example: 2 𝑥 −4 𝑓(𝑥) = 𝑥+5 2 𝑢 = 𝑥 − 4, 𝑑𝑢 = 2𝑥 𝑣 = 𝑥 + 5, 𝑑𝑣 = 1 2 2𝑥(𝑥+5)−1(𝑥 −4) 2 (𝑥+5) 2 2 2𝑥 +10𝑥−𝑥 +4 2 (𝑥+5) 2 𝑥 +10𝑥+4 𝑓'(𝑥) = 2 (𝑥+5) 𝑛 𝑛−1 Polynomial raised to a Constant - 𝑑(𝑢 ) = 𝑛𝑢 𝑑𝑢 ○ Example: 2 𝑓(𝑥) = (𝑥 + 3) 𝑢 = 𝑥 + 3, 𝑑𝑢 = 1 2−1 𝑓'(𝑥) = 2(𝑥 + 3) *1 𝑓'(𝑥) = 2(𝑥 + 3) * 1 𝑓'(𝑥) = 2𝑥 + 6 𝑑𝑢 Derivative of a Root Function - 𝑑( 𝑢) = 2 𝑢 ○ Example: 2 𝑓(𝑥) = 𝑥 −3 2 𝑢 = 𝑥 − 3, 𝑑𝑢 = 2𝑥 2𝑥 𝑓'(𝑥) = 2 2 𝑥 −3 𝑥 𝑓'(𝑥) = 2 𝑥 −3 𝑐 −𝑐𝑑𝑣 Constant over a Polynomial - 𝑑( 𝑣 ) = 2 𝑣 ○ Example: 5 𝑓(𝑥) = 2 𝑥 −9 BasCal Reviewer by Ramon Caluag | 3  2 𝑐 = 5, 𝑣 = 𝑥 − 9, 𝑑𝑣 = 2𝑥 −5(2𝑥) 𝑓'(𝑥) = 2 2 (𝑥 −9) −10𝑥 𝑓'(𝑥) = 2 2 (𝑥 −9) Exponential & Logarithmic Functions 𝑢 𝑢 𝑢 Exponential Function - 𝑑(𝑎 ) = 𝑎 𝑙𝑛𝑎 * 𝑑𝑢 = 𝑑𝑢 * 𝑎 𝑙𝑛𝑎 ○ Example: (3𝑥+3) 𝑓(𝑥) = 4 𝑎 = 4, 𝑢 = 3𝑥 + 3, 𝑑𝑢 = 3 (3𝑥+3) 𝑓'(𝑥) = 4 𝑙𝑛4 * 3 (3𝑥+3) 𝑓'(𝑥) = 3 * 4 𝑙𝑛4 2 (2𝑥 +4) 𝑓(𝑥) = 2 2 𝑎 = 2, 𝑢 = 2𝑥 + 4, 𝑑𝑢 = 4𝑥 2 (2𝑥 +4) 𝑓'(𝑥) = 4𝑥 * 2 𝑙𝑛2 𝑢 𝑢 𝑢 Euler’s Number raised to a Polynomial - 𝑑(𝑒 ) = 𝑒 * 𝑑𝑢 = 𝑑𝑢 * 𝑒 ○ Example: 2 (𝑥 +4) 𝑓(𝑥) = 𝑒 2 𝑢 = 𝑥 + 4, 𝑑𝑢 = 2𝑥 2 (𝑥 +4) 𝑓'(𝑥) = 𝑒 * 2𝑥 2 (𝑥 +4) 𝑓'(𝑥) = 2𝑥 * 𝑒 1 𝑑𝑢 Logarithmic Function - 𝑑(𝑙𝑜𝑔𝑎𝑢) = 𝑢(𝑙𝑛𝑎) * 𝑑𝑢 = 𝑢(𝑙𝑛𝑎) ○ Example: 3 𝑓(𝑥) = 𝑙𝑜𝑔9𝑥 3 2 𝑎 = 9, 𝑢 = 𝑥 , 𝑑𝑢 = 3𝑥 2 3𝑥 𝑓'(𝑥) = 3 𝑥 (𝑙𝑛9) 1 𝑑𝑢 Natural Log - 𝑑(𝑙𝑛𝑢) = 𝑢 * 𝑑𝑢 = 𝑢 ○ Example: BasCal Reviewer by Ramon Caluag | 4  3 𝑓(𝑥) = 𝑙𝑛(𝑥 − 4𝑥) 3 2 𝑢 = 𝑥 − 4𝑥, 𝑑𝑢 = 3𝑥 − 4 2 3𝑥 −4 3 𝑥 −4𝑥 Trigonometric Functions Sine Function - 𝑑(𝑠𝑖𝑛𝑢) = (𝑐𝑜𝑠𝑢) * 𝑑𝑢 ○ 𝑓(𝑥) = 2𝑠𝑖𝑛5𝑥 𝑢 = 5𝑥, 𝑑𝑢 = 5 𝑓'(𝑥) = 2𝑐𝑜𝑠5𝑥 * 5 𝑓'(𝑥) = 10𝑐𝑜𝑠5𝑥 Cosine Function - 𝑑(𝑐𝑜𝑠𝑢) =− (𝑠𝑖𝑛𝑢) * 𝑑𝑢 ○ 𝑓(𝑥) = 10𝑐𝑜𝑠5𝑥 𝑢 = 5𝑥, 𝑑𝑢 = 5 𝑓'(𝑥) =− 10𝑠𝑖𝑛5𝑥 * 5 𝑓'(𝑥) =− 50𝑠𝑖𝑛5𝑥 2 Tangent Function - 𝑑(𝑡𝑎𝑛𝑢) = (𝑠𝑒𝑐 𝑢) * 𝑑𝑢 2 ○ 𝑓(𝑥) = 3𝑡𝑎𝑛(𝑥 + 2) 2 𝑢 = 𝑥 + 2, 𝑑𝑢 = 2𝑥 2 2 𝑓'(𝑥) = 3𝑠𝑒𝑐 (𝑥 + 2) * 2𝑥 2 2 𝑓'(𝑥) = 6𝑥𝑠𝑒𝑐 (𝑥 + 2) Secant Function - 𝑑(𝑠𝑒𝑐𝑢) = (𝑠𝑒𝑐𝑢)(𝑡𝑎𝑛𝑢) * 𝑑𝑢 2 ○ 𝑓(𝑥) = 𝑠𝑒𝑐(𝑥 + 3𝑥) 2 𝑢 = 𝑥 + 3𝑥, 𝑑𝑢 = 2𝑥 + 3 2 2 𝑓'(𝑥) = 𝑠𝑒𝑐(𝑥 + 3𝑥)𝑡𝑎𝑛(𝑥 + 3𝑥) * 2𝑥 + 3 Cosecant Function - 𝑑(𝑐𝑠𝑐𝑢) =− (𝑐𝑠𝑐𝑢)(𝑐𝑜𝑡𝑢) * 𝑑𝑢 ○ 𝑓(𝑥) = 2𝑐𝑠𝑐(𝑥 − 4) 𝑢 = 𝑥 − 4, 𝑑𝑢 = 1 𝑓'(𝑥) = 2 *− 𝑐𝑠𝑐(𝑥 − 4) * 𝑐𝑜𝑡(𝑥 − 4) * 1 𝑓'(𝑥) =− 2[𝑐𝑠𝑐(𝑥 − 4)][𝑐𝑜𝑡(𝑥 − 4)] 2 Cotangent Function - 𝑑(𝑐𝑜𝑡𝑢) =− (𝑐𝑠𝑐 𝑢) * 𝑑𝑢 ○ 𝑓(𝑥) = 3𝑐𝑜𝑡5𝑥 BasCal Reviewer by Ramon Caluag | 5  𝑢 = 5𝑥, 𝑑𝑢 = 5 2 𝑓'(𝑥) =− 3𝑐𝑠𝑐 5𝑥 * 5 2 𝑓'(𝑥) =− 15𝑐𝑠𝑐 5𝑥 Basic Calculus 4th Quarter Reviewer —————————————————— Applications of Differentiation - Lesson #3 Rectilinear Motion Rectilinear Motion - Motion of an object along a straight line ○ 𝑠 = 𝑓(𝑡) - s represents the position of the object Velocity - Rate of change of the position of an object with respect to time (𝑚/𝑠) 𝑑𝑠 ∆𝑠 ○ Formula: 𝑣 = 𝑑𝑡 = 𝑓'(𝑡) = ∆𝑡 Acceleration - Rate of change of the velocity of an object, derivative of the instantaneous velocity v with respect to time 2 (𝑚/𝑠 ) 𝑑𝑣 ○ Acceleration: a= 𝑑𝑡 = 𝑓''(𝑡) Example: 2 ○ 𝑓(𝑡) = 4𝑡 − 5𝑡 + 7 Find the Position at 3 seconds: 2 𝑓(3) = 4(3) − 5(3) + 7 𝑓(3) = 4(9) − 15 + 7 𝑓(3) = 36 − 15 + 7 𝑠 = 28𝑚 Velocity at 3 seconds: 2−1 1−1 𝑓'(𝑡) = 2(4𝑡) − 1(5𝑡) +0 𝑓'(𝑡) = 8𝑡 − 5 𝑓'(3) = 8(3) − 5 𝑓'(3) = 24 − 5 BasCal Reviewer by Ramon Caluag | 6  𝑓'(3) = 19 𝑣 = 19𝑚/𝑠 Acceleration at 3 seconds: 𝑓''(𝑡) = 8 𝑓''(3) = 8 2 𝑎 = 8 𝑚/𝑠 When is the Object at rest? 𝑣=0 0 = 8𝑡 − 5 5 = 8𝑡 5 8 =𝑡 5 𝑡= 8 𝑠 Projectile Motion Projectile - An object where the only force acting upon it is gravity 1 2 ○ Formula: 𝑠 = 𝑓(𝑡) =− 2 𝑔𝑡 + 𝑣0𝑡 + ℎ𝑜 Where 𝑠 is the distance of the object from the ground, 𝑔 is the acceleration due to gravity or 2 2 9. 8𝑚/𝑠 / 32𝑓𝑡/𝑠 , 𝑡 is time, 𝑣𝑜 is the initial velocity, and ℎ𝑜 is the initial height Example: ○ ℎ𝑜 = 0 𝑓𝑡. 𝑣𝑜 = 64 𝑓𝑡/𝑠 1 2 𝑓(𝑡) =− 2 (32)𝑡 + 64𝑡 + 0 2 𝑓(𝑡) =− 16𝑡 + 64𝑡 ○ When does it reach its Highest Peak: 𝑓'(𝑡) =− 32𝑡 + 64 0 =− 32𝑡 + 64 32𝑡 = 64 𝑡 = 2𝑠 ○ Maximum Height: BasCal Reviewer by Ramon Caluag | 7  2 𝑓(2) =− 16(2) + 64(2) 𝑓(2) = 64 𝑠 = 64𝑓𝑡 ○ Velocity at t = 3 seconds: 𝑓'(3) =− 32(3) + 64 𝑓'(3) =− 32 𝑣 =− 32 𝑓𝑡/𝑠 ○ Object hits the ground: 2 0 =− 16𝑡 + 64𝑡 2 16𝑡 = 64𝑡 𝑡 = 4𝑠 ○ Speed when it hits the ground: |𝑓'(4)| = | − 32(4) + 64| |𝑓'(4)| = | − 128 + 64| |𝑓'(4)| = | − 64| |𝑣| = 64 𝑓𝑡/𝑠 Critical Numbers Values of x that give us a relative maximum or relative minimum First Derivative Test: ○ Case 1 (+,-) - Relative maximum ○ Case 2 (-,+) - Relative minimum ○ Case 3 (-,-) or (+,+) - Neither maximum nor minimum Second Derivative Test: ○ Case 1: 𝑓''(𝑥) > 0 local minimum (positive) ○ Case 2: 𝑓''(𝑥) < 0 local maximum (negative) ○ Case 3: 𝑓''(𝑥) = 0 Neither maximum nor minimum Example: 3 2 ○ 𝑓(𝑥) = 𝑥 − 9𝑥 + 24𝑥 2 Find the Derivative - 3𝑥 − 18𝑥 + 24 2 Equate to 0 - 0 = 3𝑥 − 18𝑥 + 24 Find the Critical Numbers: 2 3𝑥 −18𝑥+24 0 3 = 3 BasCal Reviewer by Ramon Caluag | 8  2 𝑥 − 6𝑥 + 8 = 0 (𝑥 − 4) = 0, (𝑥 − 2) = 0 𝑥 = 4, 𝑥 = 2 Find Coordinates: 3 2 𝑓(4) = 4 − 9(4) + 24(4) 𝑓(4) = 16, (4, 16) 3 2 𝑓(2) = 2 − 9(2) + 24(2) 𝑓(2) = 20, (2, 20) Find Second Derivative: 𝑓''(𝑥) = 6𝑥 − 18 𝑓''(4) = 6(4) − 18 𝑓''(4) = 24 − 18 = 6 LOCAL MINIMUM 𝑓''(2) = 6(2) − 18 𝑓''(2) = 12 − 18 =− 6 LOCAL MAXIMUM ○ The sum of 2 numbers is 22. What is the largest possible sum of the squares of the numbers? 2 2 Function - 𝑥 + 𝑦 = 22 | 𝑥 + 𝑦 = 𝑠 𝑦 = 22 − 𝑥 2 2 𝑥 + (22 − 𝑥) = 𝑠 2 2 𝑥 + 𝑥 − 44𝑥 + 484 = 𝑠 2 2𝑥 − 44𝑥 + 484 = 𝑠 Find the Derivative - 𝑠' = 4𝑥 − 44 Equate to 0 - 0 = 4𝑥 − 44 Find the Critical Numbers: 4𝑥 = 44 4𝑥 44 4 = 4 𝑥 = 11 Find Y-value: 𝑦 = 22 − 𝑥 𝑦 = 22 − 11 𝑦 = 11 Find the Sum: 2 2 𝑠 = 11 + 11 𝑠 = 121 + 121 BasCal Reviewer by Ramon Caluag | 9  𝑠 = 242 BasCal Reviewer by Ramon Caluag | 10  Statistics & Probability 4th Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Sampling Distribution - Lesson #1 Sample Distribution - Analysis from 1 sample Sampling Distribution - Analysis from the mean of multiple samples Sample (n) - Selected from population Population (N) - Entirety of the subjects being studied Statistic - Numerical value of the sample Parameter - Numerical value of the population Sampling Distribution Means Steps in determining Sampling Distribution Means: ○ List all possible samples ○ Compute the mean of each sample ○ Create frequency distribution ○ Create a probability distribution table ○ Draw histogram Sampling Mean - The mean of sampling distribution is equal to the population mean ○ X̄ or µ - Sampling Mean χ ○ µ - Population Mean ̄ = Σ(𝑋) = µ = Σ[𝑋 * 𝑃(𝑋)] ○ X 𝑁 Standard Deviation - Sampling SD is not equal to Population SD and uses a different formula, Measure of how much variance there is between the samples ○ s - Sampling Standard Deviation 2 ○ 𝑠 - Sampling Variance 2 2 Σ(𝑋−µχ) ○ 𝑠= 𝑠 = 𝑁 StatProb Reviewer by Ramon Caluag | 1  Standard Error - Likelihood that the Sampling Distribution is different from the Population Distribution ○ σχ - Standard Error σ ○ σχ = 𝑛 Example: Random sample of sizes 𝑛 = 3 are drawn from numbers 5, 6, 7, 8, 9 ○ List all possible samples & Compute the mean of each sample Sample Mean 567 6 568 6.3 569 6.7 578 6.7 579 7 589 7.3 678 7 679 7.3 689 7.7 789 8 ○ Create frequency distribution & Probability distribution table X 6 6.3 6.7 7 7.3 7.7 8 Freq. 1 1 2 2 2 1 1 P(X) 1/10 1/10 2/10 2/10 2/10 1/10 1/10 ○ Mean: 5+6+7+8+9 µχ = 5 = 7, The average value is 7. StatProb Reviewer by Ramon Caluag | 2  ○ Variance: 2 2 2 2 2 2 (5−7) +(6−7) +(7−7) +(8−7) +(9−7) 𝑠 = 5 = 2, The average. distance squared from the mean is 2 units squared ○ Standard Deviation: 𝑠 = 2, The average distance from the mean is 2 units ○ Standard Error: 2 6 σχ = = 3 , The standard error of the sampling 3 6 distribution is 3 —————————————————— Statistics & Probability 4th Quarter Reviewer —————————————————— Hypothesis Testing (Z-Test) - Lesson #2 Making generalizations about an entire population by testing using a sample statistic. ○ Testing a hypothesis applying to a population parameter using a sample statistic. ○ The probability (confidence level) that a hypothesis is true Steps in Hypothesis Testing ○ State Null & Alternative Hypothesis 𝐻𝑂 Null Hypothesis - Value assumed to be true on the parameter (ALWAYS equal to) e.g.: The average value is equal to 5. (µ = 5) 𝐻𝑎 Alternative Hypothesis - Negation of the Null Hypothesis StatProb Reviewer by Ramon Caluag | 3  Directional - 𝐻𝑎 applies to only one side (Less than or greater than) e.g.: The average value is less than 5. (µ < 5) ○ Left-tailed e.g.: The average value is greater than 5. (µ > 5) ○ Right-tailed Non-Directional - 𝐻𝑎 applies to both sides (Is not equal to) e.g.: The average value is not equal to 5. (µ ≠ 5) ○ Set the Standard - Critical Region, Tail, Bell Curve Based on z-table & alpha-value (degree of confidence) Usually α = 0. 01, 0. 05, 𝑜𝑟 0. 1 (1%, 5%, 10%) If α is not given, assume α = 0. 05 χ−µ ○ Compute for the Test Statistic - 𝑧 = σ 𝑛 χ−µ CASE 1 - 𝑧 = σ , 𝑛 < 30, Population SD Known 𝑛 Where 𝑧 is the Test Statistic, χ is the sampling statistic, µ is the population parameter, σ is the Population SD, and 𝑛 is the sample size χ−µ CASE 2 - 𝑧 = 𝑠 𝑜𝑟 σ , 𝑛 ≥ 30 𝑛 Where 𝑧 is the Test Statistic, χ is the sampling statistic, µ is the population parameter, 𝑠 is the Sampling SD, and 𝑛 is the sample size Either the sample or population SD may be used. According to the Central Limit Theorem, with a sample greater than or equal to 30 a normal distribution in the sampling distribution can be assumed CASE 3 - 𝑛 < 30, Population SD Unknown Use T-Test ○ Make the Decision Critical/Shaded region = Rejection region for null hypothesis StatProb Reviewer by Ramon Caluag | 4  Base conclusion on the accepted hypothesis Example: The manufacturer of a certain brand of wristwatch claims that the mean life expectancy of the battery of the watch is 48 months. A researcher wanted to validate this claim, so he chose a sample of 64 watches and noted that their mean life expectancy is 45 months with a standard deviation of 2 months. At a confidence level of 95%, can you conclude that the mean life expectancy is less than 48 months? ○ State Null & Alternative Hypothesis 𝐻𝑜: The mean life expectancy of the watch battery is 48 months 𝐻𝑜: µ = 48 𝐻𝑎: The mean life expectancy of the watch battery is less than 48 months 𝐻𝑎: µ < 48 ○ Set the Standard Confidence Level: 95%, Alpha Value: α = 0. 05 Less Than = Left-Tailed Critical Region: − 1. 645 ○ Compute for the Test Statistic 𝑛 > 30, Case #2 45−48 𝑧= 2 64 𝑧 =− 12 ○ Make the Decision Reject 𝐻𝑜, Accept 𝐻𝑎 CONCLUSION: In conclusion, the mean life expectancy of the watch battery is less than 48 months StatProb Reviewer by Ramon Caluag | 5  Statistics & Probability 4th Quarter Reviewer —————————————————— Hypothesis Testing (T-Test) - Lesson #3 T-Test - Is used in Case #3 when the sample is less than 30 but the population sd or variance is unknown ○ Sample SD is used in this case ○ T-Test only differs in Step #2 Set The Standards (T-Test) ○ Find the degree of freedom (df) 𝑑𝑓 = 𝑛 − 1 ○ Identify alpha-value and directionality ○ Use T-table instead of z-table to set the critical region Example: A researcher wants to determine if review sessions affect the performance of students in exams. A review session is administered to 25 students and a mean of 43 is recorded with a standard deviation of 8. From the previous exams, the population mean of the same exam was 40. Can the professor conclude the effectiveness of review sessions in improving exam performance? Use alpha at 0.10 ○ State Null & Alternative Hypothesis 𝐻𝑜: The mean exam score of the students is 40. 𝐻𝑜: µ = 40 𝐻𝑎: The mean exam score of the students is not 40. 𝐻𝑎: µ ≠ 40 ○ Set the Standard 𝑑𝑓 = 25 − 1 = 24 α = 0. 10 Two-tailed Critical Region: ± 1. 711 StatProb Reviewer by Ramon Caluag | 6  ○ Compute for Test Statistic 43−40 𝑡= 8 24 𝑡 = 1. 88 ○ Make the Decision 𝐻𝑜 is rejected, 𝐻𝑎 is accepted CONCLUSION: In conclusion, the mean exam score of the students is not 40. —————————————————— Statistics & Probability 4th Quarter Reviewer —————————————————— Hypothesis Testing (Population Proportion) - Lesson #4 Population Proportion - Fraction of a population falling under a characteristic ○ Same process with as Z-Test differing only in the formula used 𝑋−𝑛𝑝𝑂 ○ 𝑧= 𝑛𝑝𝑜𝑞𝑜 𝑋 - Number of Success in the Sample (Note: The value of X is the direct Number not a Percentage) 𝑛 - Sample Size 𝑝𝑜 - Percent success 𝐻𝑜 𝑞𝑜 - Percent fail 𝐻𝑜 (𝑞𝑜 = 100% − 𝑝𝑜) Example: It is generally assumed that 40% of students are in Non-STEM strands. A researcher tested this assumption using a sample size of 50 students. His results showed that 16 out of the 50 students are studying in Non-STEM strands. Using this data can he conclude that 40% of students are studying in Non-STEM strands? StatProb Reviewer by Ramon Caluag | 7 ○ State your Null & Alternative Hypothesis: 𝐻𝑜: The percentage of students studying in Non-STEM strands is 40%. 𝐻𝑜: 𝑝 = 40% 𝐻𝑎: The percentage of students studying in Non-STEM strands is not 40%. 𝐻𝑎: 𝑝 ≠ 40% ○ Set the Standard: Two-Tailed α = 0. 05 Critical Region: ± 1. 96 ○ Compute for the Test Statistic 𝑋−𝑛𝑝𝑜 𝑧= 𝑛𝑝𝑜𝑞𝑜 𝑋 = 16, 𝑛 = 50, 𝑝𝑜 = 0. 4,𝑞𝑜 = 0. 6 16−50(0.4) 𝑧= 50(0.4)(0.6) 𝑧 =− 1. 15 ○ Make the Decision Accept 𝐻𝑜, Reject 𝐻𝑎 In conclusion, the percentage of students studying in Non-STEM strands is 40%. StatProb Reviewer by Ramon Caluag | 8  Basic Calculus 3rd Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Limits - Lesson #1 Solving for Limits Limit - The intended value of a function ○ The value a function/graph approaches as x approaches but does not land on a certain value on both sides of the graph ○ Example: 𝑙𝑖𝑚 𝑓(𝑥) 𝑥→2 The limit here is the y-value that 𝑓(𝑥) gets close to on both sides (left and right) as the x-value gets closer and closer to 2 ○ Example: 𝑙𝑖𝑚 𝑥 2 + 4 𝑥→2 2 2 + 4 = 8, In this function, the limit can be found through substitution as the function is a polynomial function ○ Example: 𝑙𝑖𝑚 𝑓(𝑥), refer to the graph on the right 𝑥→0 The value of the limit of 𝑓(𝑥) as 𝑥 approaches 0 is 0 as when the value of 𝑥 is approaching 0 the y-value is also approaching 0 Indeterminate - Functions that when evaluated result in a 0/0 (undefined) answer 𝑥 2 −4 ○ Example: 𝑙𝑖𝑚 𝑥→2 𝑥−2 22 −4 4−4 0 = = 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 2−2 2−2 0 0 ○ Indeterminate or a limit that result in a answer 0 when evaluated is not a valid answer for limits as the value of limits can be different from the actual value and thus it must be solved Table of Values - Generally considered as the most reliable method of determining the value of a limit especially in indeterminate functions, done by substituting 𝑥 with values that get exponentially close to the approached value BasCal Reviewer by Ramon Caluag | 1  𝑥 2−4 ○ 𝑙𝑖𝑚 𝑥→2 𝑥−2 𝑥 1.9 1.99 2 2.01 2.1 𝑦 3.9 3.99 0 4.01 4.1 0 The value of the limit is 4 as the limit gets closer and closer to 4 from both the left and right There are situations where the value of a limit is not equal to the value of its function especially seen in piecewise functions ○ Example: 𝑙𝑖𝑚 {1, 𝑥 ≠ 1; 0, 𝑥 = 1} 𝑥→1 Table of Values - The limit is equal to 0 𝑥 0.9 0.99 1 1.01 1.1 𝑦 1 1 0 1 1 Although when directly substituted the limit is equal to 0, as seen in the table of values and on the graph on the right the approached value is 1 as the value of the limit is the approached or intended value not the actual value One-Sided Limits DNE (Does Not Exist) - Used for limits that do not exist primarily because the value approached by the function differs on either side of the approached x-value |𝑥−1| ○ Example: 𝑙𝑖𝑚 𝑥→1 𝑥−1 Table of Values 𝑥 0.9 0.99 1 1.01 1.1 𝑦 −1 −1 0 1 1 0 BasCal Reviewer by Ramon Caluag | 2  As the limit approaches different values of both the left and right side seen both in the table of values and in the graph on the right, the limit Does Not Exist (DNE) as regular limits need to approach the same value on both sides One Sided Limit - Limits that approach a certain value on either the left or right side |𝑥−1| ○ Example: 𝑙𝑖𝑚− 𝑥→1 𝑥−1 The negative exponent signifies “from the left” Based from the above table of values and graph, the value of the limit from the left is equal to −1 |𝑥−1| ○ Example: 𝑙𝑖𝑚+ 𝑥→1 𝑥−1 The positive exponent signifies “from the right” Based from the above table of values and graph, the value of the limit from the left is equal to 1 Basic Calculus 3rd Quarter Reviewer —————————————————— Limits of Algebraic Functions - Lesson #2 Limit Laws Limit of a Constant - 𝑙𝑖𝑚 𝑘 = 𝑘 𝑥→𝑐 Limit of an Identity Function - 𝑙𝑖𝑚 𝑥 = 𝑐 𝑥→𝑐 𝑛 𝑛 Power Rule - 𝑙𝑖𝑚 𝑥 = 𝑐 𝑥→𝑐 Radical Rule - 𝑙𝑖𝑚 𝑛√𝑥 = 𝑛√𝑐 𝑥→𝑐 BasCal Reviewer by Ramon Caluag | 3  Sum Rule - 𝑙𝑖𝑚 [𝑓(𝑥) + 𝑔(𝑥)] = 𝑙𝑖𝑚 𝑓(𝑥) + 𝑙𝑖𝑚 𝑔(𝑥) 𝑥→𝑐 𝑥→𝑐 𝑥→𝑐 Difference Rule - 𝑙𝑖𝑚 [𝑓(𝑥) − 𝑔(𝑥)] = 𝑙𝑖𝑚 𝑓(𝑥) − 𝑙𝑖𝑚 𝑔(𝑥) 𝑥→𝑐 𝑥→𝑐 𝑥→𝑐 Limit of a Constant Multiple - 𝑙𝑖𝑚 𝑘(𝑓(𝑥)) = 𝑘(𝑙𝑖𝑚 𝑓(𝑥)) 𝑥→𝑐 𝑥→𝑐 Product Rule - 𝑙𝑖𝑚 [𝑓(𝑥) × 𝑔(𝑥)] = 𝑙𝑖𝑚 𝑓(𝑥) × 𝑙𝑖𝑚𝑔(𝑥) 𝑥→𝑐 𝑥→𝑐 𝑥→𝑐 Quotient Rule - 𝑙𝑖𝑚 [𝑓(𝑥) ÷ 𝑔(𝑥)] = 𝑙𝑖𝑚 𝑓(𝑥) ÷ 𝑙𝑖𝑚 𝑔(𝑥) 𝑥→𝑐 𝑥→𝑐 𝑥→𝑐 𝑛 𝑛 Power Function Rule - 𝑙𝑖𝑚 [𝑓(𝑥)] = [𝑙𝑖𝑚𝑓(𝑥)] 𝑥→𝑐 𝑥→𝑐 Radical Function Rule - 𝑙𝑖𝑚 𝑛√𝑓(𝑥) = 𝑛√𝑙𝑖𝑚𝑓(𝑥) 𝑥→𝑐 𝑥→𝑐 Direct Substituition - 𝑙𝑖𝑚 𝑓(𝑥) = 𝑓(𝑥) 𝑥→𝑐 ○ Applicable if 𝑓(𝑥) is a polynomial, rational, and radical function and 𝑐 is in the domain of the function Examples: ○ 𝑙𝑖𝑚 𝑥 2 + 2𝑥 𝑥→2 22 + 2(2) = 8 ○ 𝑙𝑖𝑚 𝑥 2 (𝑥 + 1) 𝑥→3 𝑙𝑖𝑚 𝑥 2 × 𝑙𝑖𝑚 (𝑥 + 1) 𝑥→3 𝑥→3 𝑙𝑖𝑚 𝑥 2 × (𝑙𝑖𝑚 𝑥 + 𝑙𝑖𝑚 1) 𝑥→3 𝑥→3 𝑥→3 2 3 × (3 + 1) 9×4 36 Limits in Indeterminate Form 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 The value of a limit must be in the form of for a valid answer, thus 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 0 indeterminate limits or must be solved through other means instead of direct 0 substitution to find the approached value ○ Table of values can be used and is generally the most reliable, but is extremely time consuming Solving thru Factoring - Factorable functions can be solved through factoring to achieve a non-indeterminate answer ○ Examples: 𝑥 3−27 𝑙𝑖𝑚 𝑥→3 𝑥 2 −9 BasCal Reviewer by Ramon Caluag | 4  (𝑥−3)(𝑥 2+3𝑥+9) 𝑙𝑖𝑚 𝑥→3 (𝑥−3)(𝑥+3) (𝑥 2 +3𝑥+9) 𝑙𝑖𝑚 𝑥→3 (𝑥+3) (32 +3(3)+9) 𝑙𝑖𝑚 𝑥→3 (3+3) (9+9+9) 𝑙𝑖𝑚 = 4.5 𝑥→3 (6) 𝑓(𝑥) Solving thru Division - Function can be simplified by dividing 𝑔(𝑥) with 𝑔(𝑥) 𝑓(𝑥) ○ Examples: 𝑥 2−9 𝑙𝑖𝑚 𝑥→3 𝑥−3 𝑥 − 3 = 0, 𝑥 = 3 3 1 0 -9 Synthetic Division - 𝑥 + 3 3 9 Substitute: 3 + 3 = 6 13 0 Rationalization - Used for terms with radicals by multiplying the function to its conjugate (Rationalizing term) ○ Examples: 𝑥−9 𝑙𝑖𝑚 𝑥→9 √𝑥−3 𝑥−9 √𝑥+3 (𝑥−9)(√𝑥+3) × = √𝑥−3 √𝑥+3 (𝑥−9) √𝑥 + 3 = √9 + 3 = 3 + 3 = 6 𝑓(𝑥) L’hopital’s Rule - When the limit is indeterminate and 𝑔′(𝑥) ≠ 0, then 𝑙𝑖𝑚 = 𝑥→𝑐 𝑔(𝑥) 𝑓′(𝑥) 𝑙𝑖𝑚 meaning that the derivative of 𝑓(𝑥) divided by the derivative of 𝑔(𝑥) is 𝑥→𝑐 𝑔′(𝑥) equal to the value of the limit ○ Note: L’hopital’s Rule is not included in the discussed lessons and is just added in this reviewer as an extra method for calculating limits (The best method imo) ○ Examples: 𝑥 2−9 𝑙𝑖𝑚 𝑥→3 𝑥−3 𝑓′(𝑥) = 2𝑥 2−1 , 𝑔′(𝑥) = 𝑥1−1 𝑓′(𝑥) = 2𝑥, 𝑔′(𝑥) = 1 2𝑥 2(3) = =6 1 1 Derivation may also be done through your calculator (thus solving the limit automatically) by pressing the “𝑑/𝑑𝑥” button or its equivalent in your specific type of calculator BasCal Reviewer by Ramon Caluag | 5  Basic Calculus 3rd Quarter Reviewer —————————————————— Limits of Transcendental Functions - Lesson #3 Transcendental Function - Transcends the 4 basic arithmetic operations Exponential Functions Exponential Growth - Suppose the equation 𝑦 = 𝑏 𝑥 where 𝑏 > 1, the value of the function exponentially grows thus: ○ 𝑙𝑖𝑚 𝑏 𝑥 = +∞ 𝑥→+∞ ○ 𝑙𝑖𝑚 𝑏 𝑥 = 0 𝑥→−∞ Exponential Decay - Suppose the equation 𝑦 = 𝑏 𝑥 where 0 < 𝑏 < 1, the value of the function exponentially decreases thus: ○ 𝑙𝑖𝑚 𝑏 𝑥 = 0 𝑥→+∞ ○ 𝑙𝑖𝑚 𝑏 𝑥 = +∞ 𝑥→−∞ Substitution - Where 𝑙𝑖𝑚 𝑏 𝑥 = 𝑏𝑐 𝑥→𝑐 𝑥 ○ Example: 𝑙𝑖𝑚 5 𝑥→3 3 5 = 125 Euler’s Number - Approx. 2.7183 (𝑒) ○ Suppose 𝑏 = 𝑒 thus: 𝑙𝑖𝑚 𝑒 𝑥 = +∞ 𝑥→+∞ BasCal Reviewer by Ramon Caluag | 6  𝑙𝑖𝑚 𝑒 𝑥 = 0 𝑥→−∞ 𝑒 𝑥−1 ○ Suppose the limit 𝑙𝑖𝑚 , its limit is equal to 1 𝑥→0 𝑥 𝑒 9𝑥−1 Example: 𝑙𝑖𝑚 𝑥→0 𝑥 As the coefficients of the two x-variables need to be equal, 9 multiply the function by 9 𝑒 9𝑥−1 9 𝑒 9𝑥−1 9 𝑙𝑖𝑚 × = × 𝑥→0 𝑥 9 9𝑥 1 1×9=9 Logarithmic Functions Logarithmic Functions - Inverse of exponential functions Exponential Logarithmic Domain {𝑥|𝑥 ∈ 𝑅} {𝑦|𝑦 > 0} Range {𝑦|𝑦 > 0} {𝑥|𝑥 ∈ 𝑅} ○ There are no negative 𝑥 values in logarithmic functions ○ 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 or 𝑏 𝑦 = 𝑥 Exponential Growth - Suppose the equation 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 where 𝑏 > 1, the value of the function exponentially grows thus: ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑏 𝑥 = +∞ 𝑥→+∞ ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑏 𝑥 = −∞ 𝑥→0+ Exponential Decay - Suppose the equation 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 where 0 < 𝑏 < 1, the value of the function exponentially decreases thus: ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑏 𝑥 = −∞ 𝑥→+∞ ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑏 𝑥 = +∞ 𝑥→0+ Substitution - Where 𝑙𝑖𝑚 𝑙𝑜𝑔𝑏 𝑥 = 𝑙𝑜𝑔𝑏 𝑐 𝑥→𝑐 ○ Example: 𝑙𝑖𝑚 𝑙𝑜𝑔3 𝑥 𝑥→27 BasCal Reviewer by Ramon Caluag | 7  𝑙𝑜𝑔3 27 = 3 Natural Logarithm - 𝑙𝑛𝑥 = 𝑙𝑜𝑔𝑒 𝑥 therefore: ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑒 𝑥 = +∞ 𝑥→+∞ ○ 𝑙𝑖𝑚 𝑙𝑜𝑔𝑒 𝑥 = −∞ 𝑥→0+ Trigonometric Functions Substitution - Solving trigonometric functions by evaluation ○ Example: 𝑙𝑖𝑚𝜋𝑐𝑜𝑠𝑥 𝑥→ 2 𝜋 180∘ Convert Rad to Deg: × = 90∘ 2 𝜋 𝑐𝑜𝑠90∘ = 0 𝑠𝑖𝑛𝑥 𝑥 Sine - 𝑙𝑖𝑚 = 1, 𝑙𝑖𝑚 =1 𝑥→0 𝑥 𝑥→0 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛9𝑥 ○ Example: 𝑙𝑖𝑚 𝑥→0 𝑠𝑖𝑛7𝑥 𝑠𝑖𝑛9𝑥 𝑠𝑖𝑛9𝑥 1 = × 𝑠𝑖𝑛7𝑥 1 𝑠𝑖𝑛7𝑥 𝑠𝑖𝑛9𝑥 9𝑥 1 7𝑥 × × × 1 9𝑥 𝑠𝑖𝑛7𝑥 7𝑥 𝑠𝑖𝑛9𝑥 7𝑥 9𝑥 × × 9𝑥 𝑠𝑖𝑛7𝑥 7𝑥 9 9 1×1× = 7 7 1−𝑐𝑜𝑠 2𝑥 ○ Example: 𝑙𝑖𝑚 𝑥→0 𝑠𝑖𝑛5𝑥 Pythagorean Identity: 𝑠𝑖𝑛 2 + 𝑐𝑜𝑠 2 = 1, 𝑠𝑖𝑛 2 = 1 − 𝑐𝑜𝑠 2 𝑠𝑖𝑛2 𝑥 (𝑠𝑖𝑛𝑥)(𝑠𝑖𝑛𝑥) 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥 1 𝑙𝑖𝑚 = = × = × × 𝑥→0 𝑠𝑖𝑛5𝑥 𝑠𝑖𝑛5𝑥 𝑠𝑖𝑛5𝑥 1 1 1 𝑠𝑖𝑛5𝑥 𝑠𝑖𝑛𝑥 𝑥 𝑠𝑖𝑛𝑥 𝑥 1 5𝑥 × × × × × 1 𝑥 1 𝑥 𝑠𝑖𝑛5𝑥 5𝑥 𝑠𝑖𝑛𝑥 𝑥 𝑠𝑖𝑛𝑥 𝑥 5𝑥 1 1 × × × × × =1×𝑥×𝑥× 𝑥 1 𝑥 1 𝑠𝑖𝑛5𝑥 5𝑥 5𝑥 𝑥 1 1×𝑥× =1×𝑥× 5𝑥 5 1 1×0× =0 5 1−𝑐𝑜𝑠𝑥 Cosine - 𝑙𝑖𝑚 =0 𝑥→0 𝑥 1−𝑐𝑜𝑠5𝑥 ○ Example: 𝑙𝑖𝑚 𝑥→0 3𝑥 1−𝑐𝑜𝑠5𝑥 5 1−𝑐𝑜𝑠5𝑥 5 1−𝑐𝑜𝑠5𝑥 5 × = × = × 3𝑥 5 15𝑥 1 5𝑥 3 BasCal Reviewer by Ramon Caluag | 8  5 0× =0 3 BasCal Reviewer by Ramon Caluag | 9  General Chemistry 1 2nd Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Chemical Reactions - Lesson #1 A process where a substance or a combination of substances undergo a change in properties characterized by ○ Evolution of heat/light ○ Evolution of gas ○ Formulation of precipitate ○ Production of mechanical and electrical energy ○ Change in color and taste Synthesis Reaction - A combination reaction where two substances is combined to form a combined compound ○ General Format - 𝐴 + 𝐵 → 𝐴𝐵 ○ 24 grams of Solid carbon reacts with oxygen gas to form how much Carbon dioxide gas 𝐶(𝑠) + 𝑂2(𝑔) → 𝐶𝑂2(𝑔) ↑ Molar Mass of Solid Carbon: 12 g/mol Molar Mass of Oxygen Gas: 32 g/mol Molar mass of Carbon Dioxide: 44 g/mol 𝑚𝑜𝑙 𝐶 1 𝑚𝑜𝑙 𝐶 1 𝑚𝑜𝑙 𝐶𝑂2 44𝑔 𝐶𝑂2 𝑔𝐶𝑂2 = 24 𝑔 * 12 𝑔 * 1 𝑚𝑜𝑙 𝐶 * 1 𝑚𝑜𝑙 𝐶𝑂2 𝑔𝐶𝑂2 = 88𝑔 Decomposition Reaction - The inverse of a synthesis reaction wherein a compound is decomposed into two simpler substances ○ General Format - 𝐴𝐵 → 𝐴 + 𝐵 ○ Decomposition of Sodium chloride (Table Salt) into Sodium metal and Chlorine gas + 2𝑁𝑎𝐶𝑙(𝑠) → 2𝑁𝑎 (𝑠) + 𝐶𝑙2(𝑔) Gen Chem Reviewer by Ramon Caluag | 1  Single Displacement - A compound reacts with another substance to displace one of the components for the compounds (Metal displaces metal, non-metal displaces non-metal) ○ General Format - 𝐴𝐵 + 𝐶 → 𝐴𝐶 + 𝐵 ○ Liquid water reacts with sulfur in a single displacement reaction 2𝐻2𝑂(𝑙) + 2𝑆(𝑠) → 2𝐻2𝑆 + 𝑂2(𝑔) Combustion - A form of single displacement reaction where a compound containing carbon, hydrogen (and sometimes oxygen) reacting with oxygen gas to yield Carbon dioxide gas and water vapor and producing a great amount of heat (flame) ∆ ○ General Format - 𝐴𝐵 + 𝑂2(𝑔)→ 𝐻2𝑂(𝑔) ↑ + 𝐶𝑂2(𝑔) ↑ ○ Combustion of Methane 𝐶𝐻4 + 2𝑂2 → 𝐶𝑂2 + 2𝐻2𝑂 Double Displacement - Displacement wherein two compounds have a swapping or displacement of substances within where Metal displaces metal and non-metal displaces non-metal ○ General Format - 𝐴𝐵 + 𝐶𝐷 → 𝐴𝐷 + 𝐶𝐵 ○ Silver nitrate reacts with table salt in a double displacement reaction 𝐴𝑔𝑁𝑂3 + 𝑁𝑎𝐶𝑙 → 𝐴𝑔𝐶𝑙 + 𝑁𝑎𝑁𝑂3 Chemical Equations Describes reactions and summarizes reactions and its substances, reactants, products, and their respective amounts and phase Symbols in Chemical Equations Between products Heat as catalyst or "+" and reactants "∆" condition “Yields” divides Evolution of gas (gas as "→" product + reactant "↑" a product) State of matter Formation of precipitate "(𝑠)" "↓" Aqeuous solution or Catalyst Indicates use of a "(𝑎𝑞)" dissolved in water → catalyst Gen Chem Reviewer by Ramon Caluag | 2  Di-Atomic Atoms - “Element” + Gas, elements which on their own typically produce a di-atomic gas ○ 𝐼2 - Iodine Gas ○ 𝐻2 - Hydrogen Gas ○ 𝑁2 - Nitrogen Gas ○ 𝐶𝑙2 - Chlorine Gas ○ 𝐹2 - Fluorine Gas ○ 𝑂2 - Oxygen Gas ○ 𝐵𝑟2 - Bromine Gas ○ This can be memorized using the mnemonic “I Have No CLose Friends Other than my BRother” Examples: ○ What is the balanced chemical equation of the combustion reaction of Ethane Gas (𝐶2𝐻6) Unbalanced Equation: 𝐶2𝐻6(𝑔) + 𝑂2(𝑔) → 𝐻2𝑂(𝑔) + 𝐶𝑂2(𝑔) Balanced Equation: 2𝐶2𝐻6(𝑔) + 7𝑂2(𝑔) → 6𝐻2𝑂(𝑔) + 4𝐶𝑂2(𝑔) ∆ Add Symbols: 2𝐶2𝐻6(𝑔) + 7𝑂2(𝑔)→ 6𝐻2𝑂(𝑔) ↑ + 4𝐶𝑂2(𝑔) ↑ ○ What is the balanced chemical equation of the reaction between Cupric oxide (𝐶𝑢𝑂) and Ammonia (𝑁𝐻3) yielding Copper (II) ions, Water vapor, and Nitrogen gas Unbalanced Equation: +2 𝐶𝑢𝑂(𝑠) + 𝑁𝐻3(𝑔) → 𝐶𝑢 (𝑠) + 𝐻2𝑂(𝑔) + 𝑁2(𝑔) Balanced Equation: +2 3𝐶𝑢𝑂(𝑠) + 2𝑁𝐻3(𝑔) → 3𝐶𝑢 (𝑠) + 3𝐻2𝑂(𝑔) + 1𝑁2(𝑔) Add Symbols: +2 3𝐶𝑢𝑂(𝑠) + 2𝑁𝐻3(𝑔) → 3𝐶𝑢 (𝑠) ↓ + 3𝐻2𝑂(𝑔) ↑ + 1𝑁2(𝑔) ↑ Gen Chem Reviewer by Ramon Caluag | 3  Relations of Mass in Chemical Reactions Using a balanced Chemical equation, the yielded product of a reaction can be calculated given the mass of the reactants using the formula: 1 𝑚𝑜𝑙𝐴 𝑚𝑜𝑙𝐵 𝑚𝑜𝑙𝑎𝑟𝑚𝑎𝑠𝑠 𝐵 ○ 𝑚𝑎𝑠𝑠𝐴 × 𝑚𝑜𝑙𝑎𝑟𝑚𝑎𝑠𝑠𝐴 × 𝑚𝑜𝑙𝐴 × 1𝑚𝑜𝑙 𝐵 = 𝑚𝑎𝑠𝑠𝐵 ○ Mass A to Mol A: The first step in calculating the mass of a yielded product is to convert the given massA (A signifying the reactant) to moles using mole concept or 1 𝑚𝑜𝑙 𝑚𝑜𝑙𝐴 = 𝑚𝑎𝑠𝑠𝐴 × 𝑚𝑜𝑙𝑎𝑟𝑚𝑎𝑠𝑠 𝐴 where mol is the numerator as we are looking for Mol ○ Mol A to Mol B: Using the Chemical Equation we can get our Mole Ratio or essentially how many moles of substance A it takes to produce x amount of moles of substance B Ex: 2𝐻2 + 1𝑂2 → 2𝐻2𝑂 where it takes one mole of Oxygen Gas to produce 2 moles of Water and 1 𝑚𝑜𝑙 𝑂2 thus the ratio is 1 𝑂2: 2 𝐻2𝑂 or 2 𝑚𝑜𝑙 𝐻2𝑂 Using the Mole ratio we can calculate Mol A to Mol B 𝑚𝑜𝑙 𝐵 using the formula 𝑚𝑜𝑙𝐵 = 𝑚𝑜𝑙 𝐴 × 𝑚𝑜𝑙 𝐴 ○ Mol B to Mass B The last step in calculating the mass of a yielded product is to convert the calculated Mol B into the Mass B or the total weight of the yielded product 𝑚𝑜𝑙𝑎𝑟𝑚𝑎𝑠𝑠𝐵 using the formula 𝑚𝑎𝑠𝑠𝐵 = 𝑚𝑜𝑙𝐵 × 1 𝑚𝑜𝑙 𝐵 where molar mass is the numerator as we are looking for the mass of B ○ Tip: In stoichiometric calculations and conversions more often that not the unit you are looking for is in the numerator and the given unit is in the denominator Gen Chem Reviewer by Ramon Caluag | 4  Examples: ○ 24 grams of Solid carbon reacts with oxygen gas to form how much Carbon dioxide gas 1𝐶(𝑠) + 1𝑂2(𝑔) → 1𝐶𝑂2(𝑔) ↑ Molar Mass of Solid Carbon: 12 g/mol Molar Mass of Oxygen Gas: 32 g/mol Molar mass of Carbon Dioxide: 44 g/mol 1 𝑚𝑜𝑙 𝐶 1 𝑚𝑜𝑙 𝐶𝑂2 44𝑔 𝐶𝑂2 𝑔𝐶𝑂2 = 24𝑔 * 12 𝑔 * 1 𝑚𝑜𝑙 𝐶 * 1 𝑚𝑜𝑙 𝐶𝑂2 𝑔𝐶𝑂2 = 88𝑔 ○ A neutralization reaction between Hydrochloric acid and Calcium hydroxide occurs and 0.75 grams of Calcium hydroxide is reacted to yield water and Calcium chloride. What is the balanced chemical reaction? How many moles of Hydrochloric acid would be needed to react with 0.75g of Calcium hydroxide? How much Calcium chloride will be yielded by 0.75g of Calcium hydroxide? Molar Mass of Calcium chloride: 110 g/mol Molar Mass of Calcium hydroxide: 74 g/mol Molar mass of Hydrochloric acid: 36 g/mol Unbalanced Equation: 𝐻𝐶𝑙(𝑎𝑞) + 𝐶𝑎(𝑂𝐻)2(𝑠) → 𝐻2𝑂(𝑙) + 𝐶𝑎𝐶𝑙2(𝑎𝑞) Balanced Equation: 2𝐻𝐶𝑙(𝑎𝑞) + 1𝐶𝑎(𝑂𝐻)2(𝑠) → 2𝐻2𝑂(𝑙) + 𝐶𝑎𝐶𝑙2(𝑎𝑞) Moles of HCl: 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 𝑚𝑜𝑙𝐻𝐶𝑙 𝑚𝑎𝑠𝑠𝐶𝑎(𝑂𝐻)2 × 𝑚𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 × 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 = 𝑚𝑜𝑙𝐻𝐶𝑙 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 2 𝑚𝑜𝑙𝐻𝐶𝑙 0. 75𝑔𝐶𝑎(𝑂𝐻)2 × 74𝑔𝐶𝑎(𝑂𝐻)2 × 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 = 𝑚𝑜𝑙𝐻𝐶𝑙 Answer: 0.02 mol HCl Mass of CaCl2: 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 𝑚𝑜𝑙𝐶𝑎𝐶𝑙2 𝑚𝑚𝐶𝑎𝐶𝑙2 𝑚𝑎𝑠𝑠𝐶𝑎(𝑂𝐻)2 × 𝑚𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 × 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 × 1 𝑚𝑜𝑙 𝐶𝑎𝐶𝑙2 = 𝑔𝐶𝑎𝐶𝑙2 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 1 𝑚𝑜𝑙𝐶𝑎𝐶𝑙2 110𝑔𝐶𝑎𝐶𝑙2 0. 75𝑔𝐶𝑎(𝑂𝐻)2 × 74𝑔𝐶𝑎(𝑂𝐻)2 × 1 𝑚𝑜𝑙𝐶𝑎(𝑂𝐻)2 × 1 𝑚𝑜𝑙 𝐶𝑎𝐶𝑙2 = 𝑔𝐶𝑎𝐶𝑙2 Answer: 1.11 grams CaCl2 Gen Chem Reviewer by Ramon Caluag | 5  General Chemistry 1st Quarter Reviewer —————————————————— Electron Configuration - Lesson #2 Shell - Energy level where electrons are found Subshell - Shape of the orbital denoting S, P, D, F ○ S-block - Spherical shape, 2 electrons (1 orbital) ○ P-block - Dumbell shape, 6 electrons (3 orbital) ○ D-block - Cloverleaf shape, 10 electrons (5 orbital) ○ F-block - Complex shape, 14 electrons (7 orbital) Orbital - A location or space in the atom where you are likely to find electrons Energy Levels - Quantum mechanical model predicts the principal energy level Examples: 11 ○ 𝑁𝑎 - 1s2 2s2 2p6 3s1 16 ○ 𝑆 - 1s2 2s2 2p6 3s2 3p4 38 ○ 𝑆𝑟 - 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 Gen Chem Reviewer by Ramon Caluag | 6  Nobel Gases Group 18 Elements that are already at full valency (unreactive) 6 2 and end in 𝑛𝑃 (except for Helium ending in 1𝑠 ) Noble gases can therefore be used to show a shorter Noble Gas Configuration such as Boron from 1s2 2s2 2p1 turning into 2 1 [𝐻𝑒] 2𝑠 2𝑝 using the nearest noble gas as a reference Examples: 11 ○ 𝑁𝑎 - [Ne] 3s1 16 ○ 𝑆 - [Ne] 3s2 3p4 38 ○ 𝑆𝑟 - [Kr] 5s2 118 ○ 𝑂𝑔 - [Rn] 7s2 5f14 6d10 7p6 Drawing Orbital Diagram Visualization of the electron configuration through its orbitals Occupation of Orbitals - Minimizes energy in the atom following the electron configuration order in the image on the previous page Pauli Exclusion Principle - A single orbital can hold two electrons which must have opposite spins Hund’s Rule of Multiplicity - Must fill ↑ energy before ↓ energy to avoid electron repulsion Gen Chem Reviewer by Ramon Caluag | 7  Aufbau Principle - From Aufbauen meaning “to build”, orbitals are built up from increasing atomic number Examples: 1 ○ 𝐻 - 7 ○ 𝑁 - 10 ○ 𝑁𝑒 - Paramagnetism - Elements slightly attracted a magnet due to unpaired electrons ○ Ex: ↑↓ ↑ ↑ Two orbitals are unpaired and thus exhibits paramagnetism Diamagnetism - Non-attraction or slight repulsion to a magnet, filled in orbitals ○ Ex: ↑↓ ↑↓ ↑↓ All orbitals are paired and thus exhibits diamagnetism Gen Chem Reviewer by Ramon Caluag | 8  General Chemistry 1st Quarter Reviewer —————————————————— Quantum Numbers - Lesson #3 Describe the location, energy, shape, and orientation of each electron Each electron has unique set of quantum number Principal Quantum Numbers Represents the energy of an electron Values: 𝑛 = 1 𝑡𝑜 7 Predicted Values: Based on Period Actual Values: Based on Energy Level 6 ○ Example: 5𝑝 𝑛 = 5 Azimuthal Quantum Numbers (Angular Momentum) Represents the shape of an orbital Values: 𝑙 = 0 𝑡𝑜 3 Predicted Values: 𝑙 = 𝑛 − 1 Actual Values: Based on Subshell 2 ○ 1𝑠 = 0 (Spherical) 6 ○ 2𝑝 = 1 (Dumbell) 10 ○ 3𝑑 = 2 (Cloverleaf) 14 ○ 4𝑓 = 3 (Too complex) Gen Chem Reviewer by Ramon Caluag | 9  Magnetic Quantum Numbers Orientation of orbital in space Values: − 3 𝑡𝑜 3 Predicted Values: 𝑚𝑙 = (2𝑙) + 1 Actual Values: Based on last filled-in electron ○ -1 0 1 ↑↓ ↑ ↑ Ex above (2p4): 𝑚𝑙 =− 1 as the last filled electron is located on the − 1 orbital ○ -2 -1 0 1 2 ↑ ↑ ↑ ↑ Ex above (3d4): 𝑚𝑙 = 1 as the last filled electron is located on the 1 orbital Spin Quantum Numbers Describes spin of the electron Values: 𝑚𝑠 =+ 1/2 𝑜𝑟 − 1/2 Actual Values: Based on the spin of the last filled-in electron ↑↓ ↑ ↑ ○ Ex above (2p4): 𝑚𝑠 =− 1/2 as the last filled electron is spinning downwards Gen Chem Reviewer by Ramon Caluag | 10  General Biology 1 3rd Quarter Reviewer By Ramon Jacob L. Caluag 11-STEM 6 —————————————————— Cell Structure and Functions - Lesson #1 Biology - Study of life ○ Anatomy - Parts and Structure of Body ○ Physiology - Organ Functions ○ Botany - Study of Plants ○ Zoology - Study of Animals ○ Microbiology - Microorganisms ○ Parasitology - Parasites ○ Genetics - Heredity ○ Biochemistry - Chemical compositions and processes in organisms ○ Cytology - Study of Cells Cell Theory Robert Hooke - Coined the term “cell” from observing cork Anton van Leewenhoek - Observed microscopic objects from pond water (animalcules - “moving objects”) Matthias Schleiden - Plant cell (Garden) Theodor Schwann - Animal Cell (Swan) Rudolf Virchow - Cell division Classical Cell Theory ○ All organisms are made-up of cells ○ Basic unit of life is the cell ○ Cells come from other cells Modern Cell Theory ○ Chemical compositions of cells are mostly similar ○ DNA is passed during cell division ○ Energy flows within the cell General Biology 1 by Ramon Caluag | 1  Life Signs of Life ○ Growth & Development ○ Reproduction ○ Adaptation ○ Movement ○ Metabolisis ○ Organization ○ Respond to Stimuli Hierarchy of Life ○ Biosphere, Biome, Ecosystem, Community, Population ○ Organism, Organ Systems, Organs, Tissues ○ Cells, Organelle, Molecule, Atom Theories on the Origin of Life ○ Spontaneous Generation - Life came from non-living objects (Generally rejected) ○ Biogenesis - Life came from living organisms ○ Abiogenesis - Life came from atoms and molecules Biomolecules - Organic Molecules produced and used by organisms and are essential to biological processes ○ Carbohydrates ○ Lipid (Fats) ○ Protein ○ Nucleic Acid DNA - Deoxyribonucleic Acid made up of 4 different types of nitrogenous bases (A=T, G=C) ○ Adenine ○ Thymine ○ Guanine ○ Cytosine Parts of the Cell General Biology 1 by Ramon Caluag | 2  Protection ○ Cell Wall - Barrier and Structure, Peptoglycan for prokaryotes, Cellulose for plant cells Absent in animal cells ○ Plasma Membrane - Cell membrane, phospholid barrier, selectively permeable in animal cells ○ Cytoplasm - Cystosol (Jelly), Holds organelles, transportation medium Genetic Control ○ Nucleus - Brains of the cell, controls cellular processes, contains DNA Absent in Prokaryotic Cells Manufacture ○ Ribosomes - Red dots, synthesis of amino acids (20 types), monomers of proteins ○ Endoplasmic Reticulum - Interconnected network of different membranes Rough ER - Contains ribosomes creating secretory proteins Smooth ER - Does not contain ribosomes, generates lipids (fats as stored energy and internation insulation) Modification ○ Golgi Body - Flattened sheets of sacs and tubes, modifies, sorts, and packages different substances Looks like spotify logo Storage ○ Vacuole - Stores important substances, bigger in plant cells Absent in prokaryotes ○ Vesicles - Smaller vacuoles storing monomers Breakdown ○ Lysosome - Digestive sacs (Lysocine), breaks down molecules and defective cell parts General Biology 1 by Ramon Caluag | 3  ○ Peroxisomes - Hydrogen peroxide (Disinfectant), Metabolizes lipids Energy Processing ○ Mitochondria - Generates ATP (currency for energy) or adenosine triphosphate through cellular respiration in animals Absent in Plant Cells and Prokaryotes ○ Plastids - Photosynthesis of glucose (C6H12O6) in plant cells Chloroplast - Contains chlorophyll (green pigment), absorbing sunlight, produces & stores glucose Chromoplast - Contains carotenoids (red, green, orange, carrot color), flowers ad fruit Leukoplast - No pigment, stores starch Support, Movement, & Communication ○ Cytoskeleton - Filamentous/Fiber-like, Support & Structure Microfilament - Actin Intermediate Filament - Protein Sub-units Microtubule - Tubulin ○ Centrioles - Made up of microtubles, produces spindle fibers during cell division ○ Cell Junction - Way of communication Plasmodesmata - Plants Animal Cells Tight Junctions - Allows biomolecules to move Anchoring Junctions - Connects cells like a rivet (Attached) Gap Junctions - Protein channel allowing exchange of ions and molecules Major Cell Parts Prokaryotic Eukaryotes Eukaryotic (plant) (Animal) Capsule ✔️ Cell Wall ✔️ ✔️ General Biology 1 by Ramon Caluag | 4 Plasma ✔️ ✔️ ✔️ Membrane Mitochondria ✔️ Plastids ✔️ Vacuole ✔️ ✔️ Plasmodesmata ✔️ Cytoplasm ✔️ ✔️ ✔️ Nucleus ✔️ ✔️ Nucleoid Region ✔️ —————————————————— General Biology 1 3rd Quarter Reviewer —————————————————— Tissues - Lesson #2 Animal Tissues Epithelial Tissues - Animals, coverings/linings of organs ○ Protects, absorbs, secretes, and excretes ○ Shapes - Squamous (Irregular), Cuboidal (Square), Columnar (Elongated) ○ Arrangement - Simple (One layer), Stratified (Many layers), Pseudostratified (Fake stratified, columnar) Connective Tissues - Functions for movement, structure, insulation, transportation, immunity, and clotting General Biology 1 by Ramon Caluag | 5  ○ Ligaments - Bone to Bone ○ Tendon - Muscle to Bone ○ Cartilages - Provides support to body ○ Osteocytes - Bone cells (Movement and structure) ○ Adiopose - Fats for insulation ○ Blood Cells Erythocytes - Transport nutrients, Red Blood Cell Leukocytes - Defense, Immune System, White Blood Cell Thrombocytes - Clotting and Wounds, Platelets Muscular Tissues ○ Skeletal Muscles - Striated (has lines) and Multinucleated, Voluntary Movements ○ Cardiac Muscles - Striated and uninucleated, blood circulation, involuntary movements ○ Smooth Muscles - Non-striated and multinucleated, bodily and organ functions Neurons - Reception and transmission of information and impulses Plant Tissues Protective Tissue ○ Epidermis - Outer covering of younger plants (green stem) ○ Peridermis - Outer covering of older plans (bark) Ground Tissues ○ Parenchyma - Photosynthesis, storage, gas exchange ○ Colenchyma - Primary cell wall and support ○ Sclerenchyma - Secondary cell wall and support Vascular Tissues ○ Xylem - Water-conducting (roots) ○ Phloem - Food-conducting (leaves) Meristimatic Tissues - Active cells undergoing cell division constantly in the roots General Biology 1 by Ramon Caluag | 6  ———————————————