LCA Mathematical Applications flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Scientific notation form

A number written as a × 10^n where 1 ≤ |a| < 10 and n is an integer. It simplifies multiplying and dividing very large or very small quantities by operating on coefficients and exponents separately.

Negative exponent rule

For any nonzero base a, a^(−n) equals 1/a^n. A negative exponent indicates a reciprocal, not a negative result, so 2^(−3) = 1/8 rather than −8.

Rational exponent meaning

The expression a^(m/n) equals the n-th root of a^m, equivalently the n-th root of a raised to the m power. When n is even, the base must be nonnegative for a real result.

Percent change formula

Percent change equals (new value − old value) divided by old value, times 100. A positive result is an increase and a negative result is a decrease relative to the starting amount.

Absolute value equation solutions

For |x| = k with k > 0, the solutions are x = k and x = −k. If k = 0 the only solution is x = 0, and if k < 0 there is no solution.

Dimensional analysis conversion

Multiply a quantity by conversion ratios equal to one so unwanted units cancel diagonally. Choosing ratios so the target unit remains yields the converted value without memorizing formulas.

Slope formula

The slope through points (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). It measures vertical change per unit horizontal change; a vertical line has undefined slope.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so m₁ · m₂ = −1. A horizontal line and a vertical line are also perpendicular.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)]/(2a). It produces all real or complex solutions and follows directly from completing the square.

Discriminant interpretation

The value b² − 4ac in a quadratic determines the number of real roots: positive gives two, zero gives exactly one repeated root, and negative gives two complex conjugate roots.

Parabola vertex x-coordinate

For y = ax² + bx + c, the vertex lies at x = −b/(2a). Substituting that value gives the maximum when a < 0 and the minimum when a > 0.

Exponential growth model

The form y = a(1 + r)^t models growth where a is the initial amount and r is the per-period rate as a decimal. Decay uses y = a(1 − r)^t with 0 < r < 1.

Logarithm product and quotient rules

The log of a product equals the sum of logs, and the log of a quotient equals the difference of logs, for the same base. These rules convert multiplication into addition.

Inverse function procedure

To find an inverse, replace f(x) with y, swap x and y, then solve for y. The result satisfies f(f⁻¹(x)) = x, and its graph reflects the original across y = x.

Circle circumference and area

A circle of radius r has circumference 2πr and area πr². Arc length and sector area equal the corresponding fraction θ/360 of these full-circle values.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². It also underlies the distance formula between two points in the coordinate plane.

Volume of a cone

A cone with base area B and height h has volume V = ⅓Bh, or V = ⅓πr²h for a circular base. A pyramid follows the same one-third pattern.

Similar figure scaling

When two similar figures have a length scale factor k, their areas scale by k² and their volumes by k³. Doubling every length multiplies volume by eight.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. It is the Pythagorean theorem applied to horizontal and vertical differences.

Degree-radian conversion

A straight angle equals 180 degrees or π radians. Convert degrees to radians by multiplying by π/180, and radians to degrees by multiplying by 180/π.

Triangle area formula

A triangle with base b and height h has area ½bh. Heron's formula computes area from sides a, b, c using the semiperimeter s in √[s(s−a)(s−b)(s−c)].

Mean versus median

The mean is the sum divided by the count, while the median is the middle value of ordered data. Extreme outliers shift the mean far more than the median.

Standard deviation meaning

Standard deviation measures the typical distance of data values from their mean. In a normal distribution about 68 percent of values fall within one standard deviation.

Complement probability rule

The probability that an event does not occur equals 1 minus the probability that it does. This rule simplifies problems asking for at least one occurrence.

General addition rule

For any two events, P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap to avoid double counting outcomes in both events.

Conditional probability formula

The probability of A given B equals the probability of both events divided by the probability of B, provided P(B) > 0. It updates belief using known information.

Combination versus permutation

Permutations count ordered arrangements using n!/(n−r)!, while combinations count unordered selections using n!/[r!(n−r)!]. Use combinations when order does not matter.

Expected value formula

Expected value is the sum of each outcome value multiplied by its probability. It represents the long-run average result over many repetitions of the experiment.

Correlation coefficient range

The correlation coefficient r lies between −1 and 1, where signs show direction and magnitude shows linear strength. A value near zero indicates no linear relationship.

Combined work rate

If two workers complete a job alone in a and b hours, together they finish in 1/(1/a + 1/b) hours. Individual rates, expressed as jobs per hour, add.

Break-even point

Break-even occurs where total revenue equals total cost, found by solving R(x) = C(x). Above that production level the operation generates profit.

Percent error formula

Percent error equals the absolute difference between measured and accepted value, divided by the accepted value, times 100. It quantifies measurement accuracy.

Linear versus exponential model

A linear model fits data changing by a constant amount per unit, while an exponential model fits data changing by a constant percent per unit. Plot or compare successive ratios to decide.