College Algebra
Math 111
College Math 111 covers several key concepts and formulas that are essential for building a strong foundation in mathematics. Here are some of the main topics and rules you'll encounter in this course:
1. Algebraic Operations: You'll learn about basic operations like addition, subtraction, multiplication, and division of numbers, as well as the properties of these operations, such as the commutative, associative, and distributive properties.
Here are the rules for the commutative, associative, and distributive properties in mathematics:
Commutative Property:
The commutative property applies to both addition and multiplication. It states that the order of the numbers or terms does not affect the result.
- Addition: a + b = b + a
- Multiplication: a × b = b × a
Associative Property:
The associative property also applies to both addition and multiplication. It states that the grouping of numbers or terms does not affect the result.
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a × b) × c = a × (b × c)
Distributive Property:
The distributive property relates multiplication and addition/subtraction. It states how to expand the expression when multiplying a number by a sum or difference.
- Multiplication over addition: a × (b + c) = (a × b) + (a × c)
- Multiplication over subtraction: a × (b - c) = (a × b) - (a × c)
These properties are fundamental in algebraic manipulations and allow us to simplify and rearrange expressions while maintaining equality.
2. Equations and Inequalities: You'll work with linear equations and inequalities, solving for unknown variables and understanding concepts like slope and intercepts. You'll also encounter quadratic equations and systems of equations.
3. Functions: Functions play a central role in mathematics. You'll explore different types of functions, including linear, quadratic, exponential, and logarithmic functions. Understanding their properties, graphs, and transformations is crucial.
4. Trigonometry: Trigonometry focuses on angles and relationships within triangles. You'll study trigonometric functions like sine, cosine, and tangent, as well as the unit circle, trigonometric identities, and solving trigonometric equations.
5. Exponents and Logarithms: Exponents involve raising numbers to powers, while logarithms are the inverse operation of exponentiation. You'll learn about their properties, rules, and applications in solving equations and exponential growth/decay problems.
6. Graphing: Graphing allows you to visually represent mathematical functions. You'll learn how to plot points, sketch graphs of functions, identify key features such as intercepts and asymptotes, and analyze the behavior of functions.
7. Calculus Basics: Calculus is an advanced mathematical discipline, and College Math 111 introduces some fundamental concepts. You'll encounter limits, derivatives (slope of a function), and basic integration.
Remember to practice regularly and work through exercises to reinforce your understanding of these concepts. Math builds upon itself, so mastering the foundations covered in College Math 111 will set you up for success in more advanced math courses.
Here are some important ones to keep in mind:
1. Quadratic Formula: The quadratic formula is used to find the roots (solutions) of a quadratic equation. For a quadratic equation in the form ax^2 + bx + c = 0, the formula is x = (-b ± √(b^2 - 4ac)) / (2a).
2. Pythagorean Theorem: This theorem relates the lengths of the sides of a right triangle. For a right triangle with legs of lengths a and b and hypotenuse of length c, the theorem states that a^2 + b^2 = c^2.
3. Laws of Exponents: These rules govern the manipulation and simplification of expressions with exponents. Some key rules include:
- Product Rule: a^m * a^n = a^(m+n)
- Quotient Rule: a^m / a^n = a^(m-n)
- Power Rule: (a^m)^n = a^(m*n)
- Zero Exponent Rule: a^0 = 1
- Negative Exponent Rule: a^(-n) = 1 / a^n
4. Logarithmic Properties: Logarithms have several important properties. Some notable ones include:
- Product Rule: log(base b)(xy) = log(base b)(x) + log(base b)(y)
- Quotient Rule: log(base b)(x/y) = log(base b)(x) - log(base b)(y)
- Power Rule: log(base b)(x^p) = p * log(base b)(x)
5. Trigonometric Identities: Trigonometry involves various identities that relate the trigonometric functions. Some commonly used identities include:
- Pythagorean Identities: sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), 1 + cot^2(x) = csc^2(x)
- Double Angle Identities: sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x)
- Sum and Difference Identities: sin(x ± y) = sin(x)cos(y) ± cos(x)sin(y), cos(x ± y) = cos(x)cos(y) ∓ sin(x)sin(y)
6. Rules of Differentiation: Differentiation is a fundamental concept in calculus. Some key rules include:
- Power Rule: d/dx (x^n) = nx^(n-1)
- Product Rule: d/dx (uv) = u(dv/dx) + v(du/dx)
- Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)
These are just a few examples of formulas and rules you'll encounter in College Math 111. It's important to consult your textbook and lecture materials for a comprehensive list and to ensure you have a solid understanding of each concept.
Check back soon for important formulas
Trigonometry
Math 112
Trigonometry at the university level is a branch of mathematics that deals with the relationships between angles and sides of triangles. It's a fascinating subject that has numerous practical applications in fields like physics, engineering, and even computer graphics. Trigonometry plays a significant role in physics, especially in fields like optics, acoustics, and astronomy. It helps in understanding the behavior of light, sound, and celestial bodies, and contributes to advancements in navigation systems, GPS technology, and mapping applications.
If I refer back to the course I took, you can expect to learn a variety of concepts and techniques. The main goal is to develop your understanding of trigonometric functions, their properties, and how they relate to angles and sides of triangles. Some common topics covered in the course include:
1. Trigonometric functions: You'll dive into the main trigonometric functions like sine, cosine, and tangent, and learn how to evaluate them for different angles. These functions help you relate angles to the lengths of sides in a triangle.
2. Trigonometric identities: You'll explore various identities, such as the Pythagorean identities and double-angle formulas. These identities allow you to simplify and manipulate trigonometric expressions.
3. Trigonometric equations: You'll learn how to solve equations involving trigonometric functions. This involves applying techniques like factoring, substitution, and trigonometric identities to find solutions.
4. Graphs of trigonometric functions: You'll examine the graphs of functions like sine and cosine, understanding their periodic nature and how to identify key features such as amplitude, period, and phase shift.
5. Trigonometric applications: You'll see how trigonometry is used to solve real-world problems. This could include topics like navigation, physics problems involving angles and forces, or even wave phenomena.
By the end of the course, you should have a solid understanding of trigonometry and be able to apply it to various situations. You'll be able to solve trigonometric equations, manipulate trigonometric expressions, and interpret the graphs of trigonometric functions.
It's important to note that while formulas are certainly an integral part of trigonometry, the emphasis in a university course is more on understanding the concepts and their applications rather than simply memorizing formulas. However, you will become familiar with key formulas such as the Pythagorean theorem, the law of sines, and the law of cosines, which are fundamental to trigonometry!
Check back for a link to my Trig Cheat Sheet.
Calculus 1 & 2
Math 251 &252
Calculus 1 and Calculus 2 are foundational courses in college-level mathematics that introduce students to the concepts, techniques, and applications of calculus. Let's delve into each course in more detail:
Calculus 1, also known as Differential Calculus, focuses on the study of rates of change and the concept of limits. The main topics covered include:
1. Limits: You'll explore the concept of limits, which is fundamental to calculus. Limits allow us to understand the behavior of functions as they approach certain values. Key concepts include the limit definition, evaluating limits algebraically, and using limit properties.
2. Derivatives: Derivatives are used to determine the rate of change of a function at a specific point. You'll learn how to calculate derivatives using different techniques like the power rule, product rule, quotient rule, and chain rule. The derivative is denoted by "dy/dx" or "f'(x)." It has numerous applications, such as finding slopes of tangent lines, determining maximum and minimum values, and analyzing the behavior of functions.
3. Applications of derivatives: You'll explore various applications of derivatives, including optimization problems, related rates, and curve sketching. These applications involve using derivatives to solve real-world problems and understand the behavior of functions.
Here are more topics covered in Calculus 1:
- Limits and Continuity
- Differentiation
- Applications of Differentiation
- Implicit Differentiation
- Exponential and Logarithmic Functions
- Trigonometric Functions
- The Mean Value Theorem
- Anti-differentiation
- The Fundamental Theorem of Calculus
- Area under a Curve
- Applications of Integration
- Techniques of Integration
These topics form the foundation of Calculus 1 and provide a solid understanding of differentiation, integration, and their applications.
Calculus 2, also known as Integral Calculus, focuses on the concept of integration and its applications. The main topics covered include:
1. Integration: Integration involves finding the antiderivative of a function and calculating definite and indefinite integrals. You'll learn integration techniques like substitution, integration by parts, trigonometric substitution, and partial fractions.
2. Applications of integrals: Integrals have a wide range of applications, including finding areas under curves, calculating volumes of solids of revolution, and solving problems related to work, fluid pressure, and population growth.
3. Sequences and series: You'll study sequences and series, exploring their convergence and divergence. Topics include arithmetic and geometric sequences, Taylor series, and Maclaurin series. These concepts are used to approximate functions and solve problems in various fields.
The course outcomes for Calculus 1 and Calculus 2 are:
1. Understanding the fundamental concepts of calculus, such as limits, derivatives, and integrals.
2. Developing proficiency in calculating derivatives using differentiation rules and applying them to various problems.
3. Learning integration techniques and applying them to find definite and indefinite integrals.
4. Gaining the ability to solve real-world problems using calculus concepts and techniques.
5. Developing critical thinking skills, mathematical reasoning, and the ability to communicate mathematical ideas effectively.
Here's a list of topics commonly covered in Calculus 2:
- Integration Techniques: Further exploration of integration techniques, including integration by parts, trigonometric substitution, partial fractions, and other algebraic manipulations.
- Applications of Integration: More advanced applications of integration, such as finding the area between curves, volumes of solids of revolution, arc length, surface area, work, and fluid pressure.
- Improper Integrals: Understanding and evaluating improper integrals, which involve infinite limits of integration or integrands that are not defined over the entire interval.
- Sequences and Series: Studying sequences (ordered lists of numbers) and series (sums of sequences), including convergence and divergence of series, tests for convergence (such as the geometric series test, p-series test, and comparison test), and operations on series (such as arithmetic operations and differentiation/integration of power series).
- Power Series: Investigating power series, which are infinite series in which the terms involve powers of x, including Taylor series and Maclaurin series expansions. Applications of power series include approximating functions, evaluating limits, and solving differential equations.
- Parametric Equations and Polar Coordinates: Working with parametric equations to describe curves and their derivatives, as well as understanding polar coordinates and their conversions to rectangular coordinates. Applications include graphing curves, calculating areas, and solving related rates problems.
- Differential Equations: Introducing basic concepts of differential equations, including first-order differential equations, separable equations, and solving initial value problems. Applications of differential equations can be found in various scientific and engineering fields.
- Vector Calculus (sometimes included): Introducing vector-valued functions, vector fields, line integrals, and Green's Theorem. This topic connects calculus with physics and engineering applications, such as work and flux integrals.
Please note that the specific topics covered in Calculus 2 may vary slightly depending on the institution and instructor. This list provides a general overview of the core topics typically included in the course.
Calculus is used extensively in many fields, including physics, engineering, economics, and computer science. Its applications range from modeling the motion of objects to analyzing complex systems. Calculus provides a powerful toolkit for understanding and quantifying change, rates, and accumulated quantities. The formulas and techniques you learn in Calculus 1 and Calculus 2 serve as a foundation for further study in advanced calculus and other mathematical disciplines.
Check back soon
Mnemonics can be helpful for remembering key formulas, concepts, or steps in calculus.
Here are a few mnemonics that can help:
1. SOH-CAH-TOA: This popular mnemonic is used in trigonometry but can also be useful in calculus when dealing with trigonometric functions.
- SOH: Sine = Opposite/Hypotenuse
- CAH: Cosine = Adjacent/Hypotenuse
- TOA: Tangent = Opposite/Adjacent
2. DERIVATIVE: To remember the steps for finding derivatives using the power rule, you can use the mnemonic "DERIVATIVE":
- D: Differentiate term by term
- E: Exponents come down
- R: Reduce the exponent by 1
- I: Include the constant if present
- V: Verify the chain rule and apply it if necessary
- A: Add up all the derivatives
- T: Take the derivative of each term
- I: Include the constant if present
-V: Verify the chain rule and apply if necessary
- E: Evaluate the derivative at specific points if needed
3. LIPET: This mnemonic is helpful when determining the order of functions to apply the chain rule in calculus.
- L: Logarithmic functions
- I: Inverse trigonometric functions
- P: Polynomial functions
- E: Exponential functions
- T: Trigonometric functions
4. CUBIC: When factoring a cubic polynomial, the mnemonic "CUBIC" can help you remember the different steps:
- C: Check for common factors
- U: Use the rational root theorem to find possible rational roots
- B: Begin factoring using a rational root
- I: Identify remaining factors using long division or synthetic division
- C: Check for any other factors
5. RHR: In calculus, when determining the limits of trigonometric functions, you can use the mnemonic "RHR" for the Right Hand Rule:
- R: Right-hand limit as x approaches a value from the right
- H: Horizontal asymptote, if applicable
- R: Right-hand limit as x approaches Certainly! Here are a few more mnemonics for calculus:
6. FOIL: This mnemonic is commonly used when multiplying two binomials. It stands for:
- F: First terms multiply
- O: Outer terms multiply
- I: Inner terms multiply
- L: Last terms multiply
7. PEMDAS: While not specific to calculus, this mnemonic is useful for remembering the order of operations in mathematical expressions. It stands for:
- P: Parentheses first
- E: Exponents (i.e., powers and square roots, etc.)
- M: Multiplication and division (from left to right)
- D: Division and multiplication (from left to right)
- A: Addition and subtraction (from left to right)
- S: Start at the left and work towards the right
8. ABCD Rule: When integrating by parts, the ABCD rule can help determine which function to differentiate and which function to integrate. It stands for:
- A: Algebraic functions (polynomials, logarithms, etc.)
- B: Trigonometric functions
- C: Exponential functions
- D: Inverse trigonometric functions
9. MATHC: This mnemonic can help you remember the common trigonometric identities:
- M: Memorize (sin^2θ + cos^2θ = 1)
- A: Add/Subtract (sin(A ± B) = sinAcosB ± cosAsinB)
- T: Tangent (tanθ = sinθ/cosθ)
- H: Half-Angle (sin(θ/2) = ±√[(1 - cosθ)/2])
- C: Co-Function (sin(π/2 - θ) = cosθ)
10. SHIP and CHAD: These mnemonics are used to remember the signs of trigonometric functions in each quadrant of the unit circle:
- SHIP: (S)ine is (H)ighest in the (I) first quadrant, (P)ositive for sine and cosine.
- CHAD: (C)osine is (H)ighest in the (A) second quadrant, (D)ecimal for tangent.