Pre-Calculus BC Honors - Midterm Review

These flashcards cover key concepts from the Pre-Calculus BC Honors midterm review, including mathematical definitions and principles relevant to the exam.

Limit

The value that a function approaches as the input approaches a specified value.

Continuity

A function is continuous at a point if the limit exists at that point and equals the function's value.

Polar Curve

A curve expressed in polar coordinates, defined by the distance from a reference point and the angle from a reference direction.

Asymptote

A line that a graph approaches but never touches.

Removable Discontinuity

A point at which a function is not defined or does not have a limit, but can be 'removed' by redefining the function at that point.

Trigonometric Identity

An equation that establishes a relationship between trigonometric functions that is true for all values of the involved variables.

Synthetic Division

A simplified method for dividing a polynomial by a linear binomial.

End Behavior

The behavior of a function as the independent variable approaches positive or negative infinity.

Domain

The set of all possible input values (x-values) for a function.

Range

The set of all possible output values (y-values) for a function.

Polynomial Function

A function that can be expressed in the form of a polynomial, which is a sum of terms consisting of a variable raised to non-negative integer powers.

Logs

A mathematical function that represents logarithmic relationships, often used to solve for exponents in equations.

natural log

The logarithm to the base e, where e is an irrational constant approximately equal to 2.718. It is commonly used in calculus and exponential decay problems.

Conjugate

The conjugate of a binomial expression is formed by changing the sign of the second term. For example, the conjugate of (a + b) is (a - b).

Factoring

Limit

The value that a function approaches as the input approaches a specified value.

Continuity

A function is continuous at a point if the limit exists at that point and equals the function's value.

Polar Curve

A curve expressed in polar coordinates, defined by the distance from a reference point and the angle from a reference direction.

Asymptote

A line that a graph approaches but never touches.

Removable Discontinuity

A point at which a function is not defined or does not have a limit, but can be 'removed' by redefining the function at that point.

Trigonometric Identity

An equation that establishes a relationship between trigonometric functions that is true for all values of the involved variables.

Synthetic Division

A simplified method for dividing a polynomial by a linear binomial.

End Behavior

The behavior of a function as the independent variable approaches positive or negative infinity.

Domain

The set of all possible input values (x-values) for a function.

Range

The set of all possible output values (y-values) for a function.

Polynomial Function

A function that can be expressed in the form of a polynomial, which is a sum of terms consisting of a variable raised to non-negative integer powers.

Logs

A mathematical function that represents logarithmic relationships, often used to solve for exponents in equations.

natural log

The logarithm to the base e, where e is an irrational constant approximately equal to 2.718. It is commonly used in calculus and exponential decay problems.

Conjugate

The conjugate of a binomial expression is formed by changing the sign of the second term. For example, the conjugate of (a + b) is (a - b).

Factoring

How are logarithms related to exponential functions?

Logarithms are the inverse operations of exponential functions, meaning that logb(b^x) = x and b^{\logb(x)} = x.

What is the Product Rule for Logarithms?

The product rule states that the logarithm of a product is the sum of the logarithms: logb(MN) = logb(M) + log_b(N).

What is the Quotient Rule for Logarithms?

The quotient rule states that the logarithm of a quotient is the difference of the logarithms: logb(M/N) = logb(M) - log_b(N).

What is the Power Rule for Logarithms?

The power rule states that the logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number: logb(M^p) = p \cdot logb(M).

What is the Change of Base Formula for Logarithms?

The change of base formula allows you to convert a logarithm from one base to another: logb(x) = \frac{\logc(x)}{\log_c(b)}, commonly using base 10 or base e.

Limit

The value that a function approaches as the input approaches a specified value.

Continuity

A function is continuous at a point if the limit exists at that point and equals the function's value.

Polar Curve

A curve expressed in polar coordinates, defined by the distance from a reference point and the angle from a reference direction.

Asymptote

A line that a graph approaches but never touches.

Removable Discontinuity

A point at which a function is not defined or does not have a limit, but can be 'removed' by redefining the function at that point.

Trigonometric Identity

An equation that establishes a relationship between trigonometric functions that is true for all values of the involved variables.

Synthetic Division

A simplified method for dividing a polynomial by a linear binomial.

End Behavior

The behavior of a function as the independent variable approaches positive or negative infinity.

Domain

The set of all possible input values (x-values) for a function.

Range

The set of all possible output values (y-values) for a function.

Polynomial Function

A function that can be expressed in the form of a polynomial, which is a sum of terms consisting of a variable raised to non-negative integer powers.

Logs

A mathematical function that represents logarithmic relationships, often used to solve for exponents in equations.

natural log

The logarithm to the base e, where e is an irrational constant approximately equal to 2.718. It is commonly used in calculus and exponential decay problems.

Conjugate

The conjugate of a binomial expression is formed by changing the sign of the second term. For example, the conjugate of (a + b) is (a - b).

Factoring

The process of breaking down a polynomial or other mathematical expression into a product of simpler expressions or factors.

How are logarithms related to exponential functions?

Logarithms are the inverse operations of exponential functions, meaning that logb(b^x) = x and b^{\logb(x)} = x.

What is the Product Rule for Logarithms?

The product rule states that the logarithm of a product is the sum of the logarithms: logb(MN) = logb(M) + log_b(N).

What is the Quotient Rule for Logarithms?

The quotient rule states that the logarithm of a quotient is the difference of the logarithms: logb(M/N) = logb(M) - log_b(N).

What is the Power Rule for Logarithms?

The power rule states that the logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number: logb(M^p) = p \cdot logb(M).

What is the Change of Base Formula for Logarithms?

The change of base formula allows you to convert a logarithm from one base to another: logb(x) = \frac{\logc(x)}{\log_c(b)}, commonly using base 10 or base e.

Greatest Common Factor (GCF) Factoring

A factoring method where the largest common monomial factor is extracted from each term in a polynomial.

Difference of Squares Factoring

A method to factor a binomial of the form (a^2 - b^2), which factors into (a - b)(a + b).

One-Sided Limit

The limit of a function as x approaches a value from either the left (values less than the limit point) or the right (values greater than the limit point). Denoted as \lim{x \to a^-} f(x) or \lim{x \to a^+} f(x).

Pythagorean Identity

A fundamental trigonometric identity that states for any angle \theta, sin^2(\theta) + cos^2(\theta) = 1.

Limit

The value that a function approaches as the input approaches a specified value.

Continuity

A function is continuous at a point if the limit exists at that point and equals the function's value.

Polar Curve

A curve expressed in polar coordinates, defined by the distance from a reference point and the angle from a reference direction.

Asymptote

A line that a graph approaches but never touches.

Removable Discontinuity

A point at which a function is not defined or does not have a limit, but can be 'removed' by redefining the function at that point.

Trigonometric Identity

An equation that establishes a relationship between trigonometric functions that is true for all values of the involved variables.

Synthetic Division

A simplified method for dividing a polynomial by a linear binomial.

End Behavior

The behavior of a function as the independent variable approaches positive or negative infinity.

Domain

The set of all possible input values (x-values) for a function.

Range

The set of all possible output values (y-values) for a function.

Polynomial Function

A function that can be expressed in the form of a polynomial, which is a sum of terms consisting of a variable raised to non-negative integer powers.

Logs

A mathematical function that represents logarithmic relationships, often used to solve for exponents in equations.

natural log

The logarithm to the base e, where e is an irrational constant approximately equal to 2.718. It is commonly used in calculus and exponential decay problems.

Conjugate

The conjugate of a binomial expression is formed by changing the sign of the second term. For example, the conjugate of (a + b) is (a - b).

Factoring

The process of breaking down a polynomial or other mathematical expression into a product of simpler expressions or factors.

How are logarithms related to exponential functions?

Logarithms are the inverse operations of exponential functions, meaning that logb(b^x) = x and b^{\logb(x)} = x.

What is the Product Rule for Logarithms?

The product rule states that the logarithm of a product is the sum of the logarithms: logb(MN) = logb(M) + log_b(N).

What is the Quotient Rule for Logarithms?

The quotient rule states that the logarithm of a quotient is the difference of the logarithms: logb(M/N) = logb(M) - log_b(N).

What is the Power Rule for Logarithms?

The power rule states that the logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number: logb(M^p) = p \cdot logb(M).

What is the Change of Base Formula for Logarithms?

The change of base formula allows you to convert a logarithm from one base to another: logb(x) = \frac{\logc(x)}{\log_c(b)}, commonly using base 10 or base e.

Greatest Common Factor (GCF) Factoring

A factoring method where the largest common monomial factor is extracted from each term in a polynomial.

Difference of Squares Factoring

A method to factor a binomial of the form (a^2 - b^2), which factors into (a - b)(a + b).

One-Sided Limit

The limit of a function as x approaches a value from either the left (values less than the limit point) or the right (values greater than the limit point). Denoted as \lim{x \to a^-} f(x) or \lim{x \to a^+} f(x).

Pythagorean Identity

A fundamental trigonometric identity that states for any angle \theta, sin^2(\theta) + cos^2(\theta) = 1.

Factoring by Grouping

A factoring technique used for polynomials with four or more terms, where terms are grouped to find common binomial factors.

Factoring Trinomials

The process of breaking down a trinomial (a polynomial with three terms) into a product of two binomials, often of the form (x + a)(x + b).

Two-Sided Limit

The limit of a function as x approaches a value 'a' from both the left and the right sides. For the two-sided limit to exist, the one-sided limits from the left and right must exist and be equal: \lim{x \to a} f(x) = L if and only if \lim{x \to a^-} f(x) = L and \lim*{x \to a^+} f(x) = L.

Logarithmic Equation

An equation that involves one or more logarithms of a variable, typically solved by using properties of logarithms or converting to exponential form.

Exponential Equation

An equation in which variable expressions occur as exponents, often solved by taking logarithms of both sides or writing both sides with the same base.