Calculating the break-even point in economics

Determine the point at which total costs and total revenue are equal, resulting in no net loss or gain.

Applying the concept of linear independence in vector spaces

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.

Applying the Pythagorean theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Applying Pascal's triangle to binomial expansion

Use the nth row of Pascal's triangle to find the coefficients in the expansion of (a + b)^n.

Calculating the centroid of a geometric shape

Integrate the product of the shape's density function and its distance from a reference point, then divide by the shape's total mass or area.

Applying Euler's formula in complex analysis

Use e^(ix) = cos(x) + i*sin(x) to relate complex exponentials to trigonometric functions.

Applying the principle of conservation of energy in physics

The total energy of an isolated system remains constant over time, being conserved between its potential and kinetic forms.

Calculating the area of a trapezoid

Use the formula (a + b) / 2 * h, where a and b are the lengths of the parallel sides and h is the height.

Calculating the arc length of a function

Integrate the square root of 1 plus the square of the derivative of the function, with respect to the independent variable, over the interval.

Applying the concept of geometric series in summation

Sum a series where each term is a constant multiple of the previous term, using the formula S = a / (1 - r), for |r| < 1.

Calculating compound interest

Use the formula A = P(1 + r/n)^(nt), where A is the amount, P the principal, r the annual interest rate, n the number of times interest is compounded per year, and t the time in years.

Applying the concept of absolute value

Use absolute value to determine the distance of a number from zero on the number line, denoted as |a|.

Calculating the buoyant force on an object submerged in a fluid

Use Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid.

Calculating the cross product of two vectors

Use the determinant of a matrix constructed from unit vectors and the vectors' components to find a vector perpendicular to both.

Calculating the coefficient of variation

Divide the standard deviation by the mean and multiply by 100 to express it as a percentage, measuring relative variability.

Calculating the derivative of a function

Apply the power rule, sum rule, product rule, quotient rule, or chain rule depending on the function's form to find its instantaneous rate of change.

Applying the remainder theorem in polynomial division

When a polynomial f(x) is divided by (x - k), the remainder is f(k).

Applying the concept of conditional probability

Calculate the probability of an event given that another event has occurred, using P(A|B) = P(A and B) / P(B).

Applying the laws of logarithms to solve equations

Use properties of logarithms, such as log(a) + log(b) = log(ab), to combine or break down logarithmic expressions for solving.

Calculating the area of a parallelogram

Use the formula A = base * height, where the height is perpendicular to the base.

Applying the fundamental theorem of algebra

Every non-constant polynomial equation of degree n has exactly n roots in the complex number system.

Applying the midpoint rule for numerical integration

Estimate the area under a curve by averaging the heights of the function at the midpoints of intervals, then multiplying by the interval width.

Calculating the acceleration of an object

Use the formula a = (v - u) / t, where v is the final velocity, u is the initial velocity, and t is the time taken.

Applying Stoke's theorem in vector calculus

Relate a surface integral over a surface S to a line integral around the boundary curve of S.

Calculating the area of a circle

Use the formula A = pi * r^2, where r is the radius of the circle.