Untitled Flashcards Set

Math Exam Comprehensive Note Sheet

Number and Quantity (9 Questions)

Real Numbers:

• Properties of Operations:

  • Commutative Property: For addition ( a + b = b + a) and multiplication ( a × b = b × a), the order of the numbers does not affect the sum or product.

  • Associative Property: For addition ( (a + b) + c = a + (b + c)) and multiplication ( (a × b) × c = a × (b × c)), the way in which numbers are grouped does not change the sum or product.

  • Distributive Property: a(b + c) = ab + ac, relates to multiplying a number by a sum.

• Arithmetic Operations: Addition, subtraction, multiplication, and division involving integers, rational numbers, and irrational numbers are fundamental skills in arithmetic.

• Order of Operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to solve expressions correctly.

• Absolute Value: The distance from zero on the number line:

  • if a > 0, |a| = a;

  • if a < 0, |a| = -a.

Properties of Real Numbers:

• Closure Property: The sum or product of two real numbers is always a real number, indicating that real numbers are closed under addition and multiplication.

• Identity Property: For addition (a + 0 = a) and multiplication (a × 1 = a), there exist specific identity elements that do not change the number.

• Inverse Property: For addition, each number a has an additive inverse (-a) such that (a + -a = 0); for multiplication, a has a multiplicative inverse (1/a) such that (a × 1/a = 1) for a ≠ 0.

• Density Property: Between any two distinct real numbers, there exists another real number, demonstrating that there are infinitely many real numbers.

Complex Numbers:

• Definition: Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined as i² = -1.

• Addition/Subtraction: Combine like terms, such that (a + bi) + (c + di) = (a + c) + (b + d)i.

• Multiplication: Use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.

• Division: To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator: (a + bi) / (c + di) = [(a + bi)(c - di)] / [c² + d²].

• Polar Form: Represent complex numbers as r(cos θ + i sin θ), where r = √(a² + b²) and θ = arctan(b/a).

• Euler’s Formula: e^(iθ) = cos θ + i sin θ, a foundational concept in complex analysis.

Matrix Operations:

• Addition/Subtraction: Performed element-wise: A + B = (a₁ + b₁, a₂ + b₂) if A = (a₁, a₂) and B = (b₁, b₂).

• Scalar Multiplication: Multiply each element of the matrix by a scalar value: kA = (ka₁, ka₂).

• Matrix Multiplication: Adhere to the row-by-column rule: (AB)ij = ∑(Aik × Bkj) for suitable dimensions of A and B.

• Identity Matrix: A diagonal matrix with ones on the diagonal and zeros elsewhere, denoted I:I = [[1, 0, ...], [0, 1, ...], ...]

• Determinants: For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. For larger matrices, use cofactor expansion.

• Inverses: A matrix A is invertible if there exists another matrix B such that AB = I. Calculate the inverse using the formula: A⁻¹ = (1/det(A))adj(A).

• Applications: Solve systems of linear equations by (if A is invertible): x = A⁻¹b.

Vector Operations:

• Vector Addition/Subtraction: Vectors can be added or subtracted component-wise: A + B = (a₁ + b₁, a₂ + b₂).

• Scalar Multiplication: Multiply each component of a vector by a scalar: kA = (ka₁, ka₂).

• Dot Product: A · B = a₁b₁ + a₂b₂, used to determine angle relations between vectors.

• Applications: Used to calculate the angle between vectors via the formula cos(θ) = (A · B) / (|A||B|), and projection of one vector onto another.

• Cross Product (3D): A × B results in a vector perpendicular to both A and B, with magnitude |A||B|sin(θ).

• Magnitude and Direction:

  • Magnitude: The length of vector A = √(a₁² + a₂²).

  • Direction: Unit vector u = A / |A| gives the direction of the vector.

Quantities:

• Ratios and Proportions: Simplifying ratios and solving for unknowns in proportional equations; involves cross-multiplication.

• Units of Measurement: Converting between different units (e.g., metric to imperial) and understanding unit equivalences.

• Dimensional Analysis: Utilizing units to verify consistency in equations and understand relationships between different measurements.

• Scientific Notation: Express large or small numbers in the form a × 10^n, where 1 ≤ a < 10 and n is an integer. Performing operations with exponents requires knowledge of the laws of exponents, such as a^m × a^n = a^(m+n).

Algebra (23 Questions)

Expressions and Equations:

• Techniques for simplifying expressions, including combining like terms, factoring polynomials, and using identities.

• Solve linear equations of the form ax + b = 0 and quadratic equations in the standard quadratic form ax² + bx + c = 0 using several methods (factoring, completing the square, quadratic formula).

Systems of Equations:

• Solve systems using methods like substitution (solving one equation for a variable and substituting it into the other), elimination (adding or subtracting equations), and matrix methods (using arrays of coefficients).

Inequalities and Absolute Value Equations:

• Solve and graph inequalities to represent solutions on a number line. Understand compound inequalities and the role of absolute values in creating boundaries.

Exponential and Logarithmic Functions:

• Properties: Key properties of exponents (a^m × a^n = a^(m+n), (a^m)^n = a^(mn)) and logarithms (log_a(xy) = log_a(x) + log_a(y)).

• Solving equations involving both exponential and logarithmic forms, understanding how one relates to the other.

• Applications: Analyze real-world phenomena like growth/decay processes and compound interest calculations.

• Modeling: Create algebraic equations/inequalities to model scenarios like budgeting, population growth, or decay, and write functions to map them.

Sequences and Series:

• Arithmetic Sequences: Defined by a constant difference between terms (d), with a formula for the nth term given by a_n = a₁ + (n - 1)d. The sum of the first n terms is S_n = n/2(a₁ + a_n).

• Geometric Sequences: Defined by a constant ratio between terms (r), with a formula for the nth term a_n = a₁ × r^(n-1). The sum of the first n terms (finite) is S_n = a₁(1 - r^n) / (1 - r) for r ≠ 1.

Functions (19 Questions)

Definition and Notation:

• Domain and range are crucial in defining functions: the set of all possible input values (domain) and the set of all possible output values (range). Function notation: f(x) denotes the function value at x.

Types of Functions:

• Linear: Graphs as straight lines, characterized by the equation y = mx + b, where m is the slope and b is the y-intercept.

• Quadratic: Parabolas described by y = ax² + bx + c, showcasing vertex and axis of symmetry.

• Polynomial: Higher-degree functions with varying coefficients and exponents.

• Rational: Functions expressed as the ratio of two polynomials, having vertical and horizontal asymptotes.

• Exponential and Logarithmic: Exponential functions grow rapidly, while logarithmic functions are their inverses and grow slowly, both being crucial in modeling.

• Trigonometric Functions: Relationship between angles and sides in triangles, including sine, cosine, and tangent.

Transformations:

• Operations performed on functions to shift (translations), flip (reflections), and change size (stretches/contractions) of their graphs.

Inverse Functions:

• Inverse functions are those that reverse the action of a function, denoted as f⁻¹(x), requiring the input-output relationship to be switched.

Modeling Relationships:

• Construct mathematical models using functions to describe real-world situations; interpret and analyze graphs to extract information about trends or data behaviors.

Composite Functions:

• Functions within functions, denoted as (f∘g)(x) = f(g(x)), essential for complex problem-solving and function manipulation.

Calculus (11 Questions)

Limits:

• Understanding and calculating limits, including one-sided limits and limits at infinity, are foundational concepts in calculus as they inform about function behavior near specific points.

L'Hôpital's Rule:

• Applied to evaluate limits that yield indeterminate forms (0/0 or ∞/∞) by differentiating the numerator and denominator until a determinate form is reached: [ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}. ]

Derivatives:

• Defined as the rate of change of a function, represented as f'(x) or dy/dx. Key rules include the power rule (d/dx [x^n] = nx^(n-1)), product rule, quotient rule, and chain rule, each facilitating differentiation under various conditions.

• Applications: Used in real-world problems like finding tangent lines, optimizing functions (maximum or minimum), and solving related rates concerns.

Integrals:

• Defined as the area under a curve, with definite integrals calculating the exact area between specific bounds and indefinite integrals providing general forms (antiderivatives).

• Fundamental Theorem of Calculus: Ties derivatives and integrals together, stating that if F is an antiderivative of f on [a, b], [ \int_a^b f(x)dx = F(b) - F(a). ]

Geometry and Measurement (17 Questions)

• Geometric Relationships: Parallel lines, triangles, circles, polygons, and 3D objects.

• Properties of Angles: Understanding congruence and similarity.

• Pythagorean Theorem: Fundamental for right triangles; a² + b² = c².

• Trigonometry: Relationships between sides and angles in triangles.

• Coordinate Geometry:

  • Equations of Lines: Slope-intercept form (y = mx + b).

  • Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²].

  • Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

  • Slope: m = (y₂ - y₁) / (x₂ - x₁).

• Conic Sections: Study of parabolas, ellipses, and hyperbolas.

• Measurement:

  • Surface Area and Volume: Formulas for geometric solids like cylinders, spheres, and pyramids.

• Units and Conversions: Understanding different measurement units and how to convert between them.

• Transformations: Techniques including reflections, rotations, translations, and dilations.

Statistics and Probability (11 Questions)

• Descriptive Statistics: Mean, median, mode, range, standard deviation, and variance.

• Probability:

  • Basic rules, conditional probability, independent events, and various probability distributions.

• Combinations and Permutations: Methods for counting arrangements and selections.

• Inferential Statistics:

  • Hypothesis testing, confidence intervals, and margin of error.

• Data Representation:

  • Utilizing histograms, box plots, scatter plots, and line graphs for visual data interpretation.

• Understanding Variability: Recognizing variability in data and interpreting trends.

Pedagogical Content Knowledge (1 Question)

• Instructional Design: Creating lesson plans that integrate multiple math domains.

• Differentiating Instruction: Strategies for catering to diverse learners.

• Analyzing Student Understanding: Using assessment to gauge comprehension.