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Questions and Answers
What are the extremes in a proportion represented as $a/b = c/d$?
What are the extremes in a proportion represented as $a/b = c/d$?
If 30 mL represents 1/6 of a prescription, how would you calculate the total volume of the prescription?
If 30 mL represents 1/6 of a prescription, how would you calculate the total volume of the prescription?
What is the standard form of 1321000000 km?
What is the standard form of 1321000000 km?
Which of the following represents the correct evaluation of $6.4 × 10^8 - 5.2 × 10^7$?
Which of the following represents the correct evaluation of $6.4 × 10^8 - 5.2 × 10^7$?
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What is the result of expressing $log_{b} 36$ in terms of $x = log 2$ and $y = log 3$?
What is the result of expressing $log_{b} 36$ in terms of $x = log 2$ and $y = log 3$?
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What is the simplified form of $log 12 + log 8 - 2 log 6$?
What is the simplified form of $log 12 + log 8 - 2 log 6$?
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What is the standard form of 0.000000516 g?
What is the standard form of 0.000000516 g?
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When dividing $2.4 × 10^5$ by $6 × 10^8$, what is the resulting standard form?
When dividing $2.4 × 10^5$ by $6 × 10^8$, what is the resulting standard form?
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What is the expression for log 5 in terms of p and q if log 6 = p and log 3 = q?
What is the expression for log 5 in terms of p and q if log 6 = p and log 3 = q?
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If log 4 + 2 log p = 2, what is the value of p?
If log 4 + 2 log p = 2, what is the value of p?
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Express the logarithmic expression log 1000 + log 20 – 3 log 5 - 1 as a single logarithm.
Express the logarithmic expression log 1000 + log 20 – 3 log 5 - 1 as a single logarithm.
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What is the simplified form of 5log 2 − log 4?
What is the simplified form of 5log 2 − log 4?
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If you evaluate 4.23 × 10^17 ÷ 8.63 × 10^16, what is the result in standard form?
If you evaluate 4.23 × 10^17 ÷ 8.63 × 10^16, what is the result in standard form?
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What is the result of log 2 + 2log 18 − log 36 expressed as a single logarithm?
What is the result of log 2 + 2log 18 − log 36 expressed as a single logarithm?
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How should 0.2 × 10^3 be expressed in the form of 2^5?
How should 0.2 × 10^3 be expressed in the form of 2^5?
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What is the simplified form of 2ln(a) − 3ln(b) + 2ln(c) expressed as a single logarithm?
What is the simplified form of 2ln(a) − 3ln(b) + 2ln(c) expressed as a single logarithm?
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What is one method for solving a quadratic equation?
What is one method for solving a quadratic equation?
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When determining the domain of the function $f(x) = \frac{1}{x - 4}$, which statement is true?
When determining the domain of the function $f(x) = \frac{1}{x - 4}$, which statement is true?
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What is the purpose of the rational zero theorem in polynomial functions?
What is the purpose of the rational zero theorem in polynomial functions?
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Which of the following is NOT a characteristic of a quadratic function?
Which of the following is NOT a characteristic of a quadratic function?
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Identify the range of the function $f(x) = -2x^2 + 4$.
Identify the range of the function $f(x) = -2x^2 + 4$.
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What does the Remainder Theorem allow you to determine when dividing a polynomial?
What does the Remainder Theorem allow you to determine when dividing a polynomial?
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When using the Factor Theorem, what does it imply if a polynomial $f(x)$ has a factor $(x - a)$?
When using the Factor Theorem, what does it imply if a polynomial $f(x)$ has a factor $(x - a)$?
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What would be the first step in using polynomial division to find the quotient and remainder of $3x^2 + 6x - 7$ divided by $x + 4$?
What would be the first step in using polynomial division to find the quotient and remainder of $3x^2 + 6x - 7$ divided by $x + 4$?
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In the context of polynomials, how would one use the Rational Zero Theorem?
In the context of polynomials, how would one use the Rational Zero Theorem?
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Which method is typically used to find the vertex of a quadratic function?
Which method is typically used to find the vertex of a quadratic function?
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To determine the conditions for the equation $2x^2 - 4x + a = 0$ to have real roots, which discriminant condition must be satisfied?
To determine the conditions for the equation $2x^2 - 4x + a = 0$ to have real roots, which discriminant condition must be satisfied?
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If two square flower beds have a combined area of $18.5 m^2$, what is the formula used to express the combined area in terms of the side length $s$?
If two square flower beds have a combined area of $18.5 m^2$, what is the formula used to express the combined area in terms of the side length $s$?
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What is the primary goal when determining the stationary point of a function by the method of completing the square?
What is the primary goal when determining the stationary point of a function by the method of completing the square?
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What method can be used to find the stationary point of the function $f(x) = 3x^2 + 6x + 14$?
What method can be used to find the stationary point of the function $f(x) = 3x^2 + 6x + 14$?
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For the equation $x^2 - 3x + b + 1 = 0$ to have a repeated root, which condition must be satisfied?
For the equation $x^2 - 3x + b + 1 = 0$ to have a repeated root, which condition must be satisfied?
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What are the intercepts of the function $f(x) = 2x^3 - 5x - 2$?
What are the intercepts of the function $f(x) = 2x^3 - 5x - 2$?
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When applying the remainder theorem to find the remainder of the polynomial $3x^3 - 8x^2 - c + 19$ divided by $x-2$, what is the variable $c$ related to?
When applying the remainder theorem to find the remainder of the polynomial $3x^3 - 8x^2 - c + 19$ divided by $x-2$, what is the variable $c$ related to?
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What indicates that the function $f(x) = 3x^2 - 6x + 5$ has a minimum point?
What indicates that the function $f(x) = 3x^2 - 6x + 5$ has a minimum point?
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In the equation $x^2 + 3x - 4 = 0$, what are the approximate roots when solved correctly to two decimal places?
In the equation $x^2 + 3x - 4 = 0$, what are the approximate roots when solved correctly to two decimal places?
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To sketch the graph of the function $f(x) = 2x^2 + 3x - 8$, which key features must be identified?
To sketch the graph of the function $f(x) = 2x^2 + 3x - 8$, which key features must be identified?
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When finding the zeros of the function $f(x) = 2x - 5 - 2$, what is a potential root?
When finding the zeros of the function $f(x) = 2x - 5 - 2$, what is a potential root?
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Which polynomial has a factor of x + 1?
Which polynomial has a factor of x + 1?
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When the polynomial 2x^3 - 3x^2 - 7x + b is divided by 2x - 1, what is the remainder?
When the polynomial 2x^3 - 3x^2 - 7x + b is divided by 2x - 1, what is the remainder?
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If the expression 3x^4 - 5x^2 + cx + 9 gives a remainder of 6 when divided by x - 3, which equation must hold true?
If the expression 3x^4 - 5x^2 + cx + 9 gives a remainder of 6 when divided by x - 3, which equation must hold true?
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Which of the following expressions is exactly divisible by 2x + b?
Which of the following expressions is exactly divisible by 2x + b?
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Study Notes
Basic Mathematics and Statistics (BPC2111)
- This course covers basic mathematical and statistical concepts.
- Subtopics include basic calculations, functions, domain, codomain, and range, quadratic and reciprocal functions, and equations.
- Also includes graphs of quadratic and reciprocal functions, polynomial functions, division algorithms, zeros of polynomials, and the rational zero theorem.
- Other subjects in the course are quadratic equations (factorization, completing the square and quadratic formula), reciprocal equations and their graphs.
Functions
- A function takes an input (x) and produces an output (f(x)).
- Every input has one and only one output.
- Functions can be described using algebraic expressions, tables, or graphs.
Domain, Codomain, and Range
- Domain: The set of all possible input values (x).
- Codomain: The set of all possible output values (y) a function can produce.
- Range: The set of all actual output values (y) a function does produce.
Quadratic Equations
- A quadratic equation is an equation of the form ax² + bx + c = 0.
- Methods to solve quadratic equations include factorization, completing the square, and using the quadratic formula.
Quadratic and Reciprocal Functions
- Quadratic functions are functions with a degree of 2 (x²). Reciprocal functions involve 1/x.
- Graphing involves analyzing properties like direction of opening, intercepts, and stationary points (maximum/minimum).
Reciprocal Function
- A reciprocal function is a function where the output is 1 divided by the input (e.g., 1/x).
Polynomial Functions
- Polynomial functions are mathematical expressions that involve non-negative integer powers of a variable (x).
- These often include the concepts of division and finding zeros (roots) of a polynomial.
Polynomial Functions: Division Algorithm
- The division algorithm states that when a polynomial P(x) is divided by a polynomial D(x), there exists a quotient Q(x) and a remainder R(x) with a degree of R(x) < degree of D(x).
Polynomial Functions: Factor Theorem
- If a polynomial f(x) has a root a, then x - a is a factor of f(x).
Polynomial Functions: Rational Zero Theorem
- If a polynomial has integer coefficients and a rational zero p/q, where p and q are integers with no common factors other than 1, then p is a factor of the constant term and q is a factor of the leading coefficient.
Domain, Codomain, and Range
- Domain: The set of all possible input values.
- Codomain: The set of all possible output values a function can produce.
- Range: The set of all actual output values a function does produce.
Real Numbers
- Integers: ... -3, -2, -1, 0, 1, 2, 3 ...
- Rational numbers: Numbers that can be expressed as a fraction p/q, with p and q as integers and q not equal to 0.
- Irrational numbers: Numbers that cannot be expressed as a fraction.
- Real numbers: The set of all rational and irrational numbers.
Significant Figure
- Significant figures are the digits in a number that carry meaning contributing to its precision.
- Rules for determining significant figures depend on the type of number (integers, decimal numbers, etc).
Percent
- Percent (%) means "per hundred".
- To convert a fraction to a percentage, divide the numerator by the denominator, and multiply the result by 100.
Variation
- Types of variation: Direct, inverse, and joint.
- Direct variation: A direct relationship between two quantities (e.g., y = kx, where k is a constant).
- Inverse variation: An inverse relationship between two quantities (e.g., y = k⁄x, where k is a constant).
- Joint variation: A relationship where a variable is dependent on several other variables.
Arithmetic Progression (AP)
- An arithmetic progression is a sequence of numbers where each term is obtained by adding a fixed value (common difference) to the previous term.
- Formulas for the nth term (tₙ) and sum of n terms (Sₙ) of an AP are given.
Geometric Progression (GP)
- A geometric progression is a sequence of numbers where each term is obtained by multiplying a fixed value (common ratio) to the previous term.
- Formulas for the nth term and the sum of n terms of a GP are given.
Standard Form
- Standard form or scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.
- Useful for very large or very small numbers.
Laws of Logarithm
- Logarithms are inverse operations of exponentiation.
- Laws exist to simplify and manipulate expressions involving logarithms.
Derivatives and Differentiation
- Procedures of finding a derivative or gradient of a function (f(x)) are explained.
- Differentiating functions from the first principles are explained
- Rules for differentiation, including the product rule, quotient rule, and chain rule, are provided.
Higher Derivatives and Stationary Points
- Second, or higher, derivatives reveal the nature of stationary points.
Exponential Growth and Decay
- Exponential growth and decay are characterized by functions of the form N = N₀e⁻λt, where N₀ is the initial amount.
Definite Integral
- Integration can be used to find the area under a curve.
- Definitions and formulas for finding definite integrals are given.
Area Under a Curve (AUC)
- The area under a curve can be determined by calculating the definite integral of the function.
Area Between Two Curves
- Finding the area between the curves of two different functions often involves calculating the definite integrals.
Asymptotes and Continuity
- Vertical asymptotes occur when a limit to infinity exists.
- Horizontal asymptotes occur at the limit at infinity.
- A function is described to be continuous if the limit at a point equals the value of the function at that point.
Limits
- Describing the behavior of a function as the input value approaches a certain point.
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