Mathematical formula

Mathematical formula about algebra 

Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols. Here are some important algebraic formulas:

Quadratic formula: The quadratic formula is used to find the solutions of a quadratic equation, which is an equation in the form of ax^2 + bx + c = 0. The formula is: x = (-b ± sqrt(b^2 - 4ac)) / 2a

Slope formula: The slope formula is used to calculate the slope of a line given two points on the line. The formula is: m = (y2 - y1) / (x2 - x1) where m is the slope, (x1, y1) and (x2, y2) are the coordinates of the two points.

Distance formula: The distance formula is used to find the distance between two points in a coordinate plane. The formula is: d = sqrt((x2 - x1)^2 + (y2 - y1)^2) where d is the distance, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Factoring formulas: Factoring formulas are used to factorize algebraic expressions. Some of the most commonly used factoring formulas include:

• Difference of squares: a^2 - b^2 = (a + b)(a - b)
• Perfect square trinomial: a^2 + 2ab + b^2 = (a + b)^2
• Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Exponential and logarithmic formulas:

• a) Exponential growth: A(t) = A0(1 + r)^t, where A0 is the initial amount, r is the growth rate, and t is the time.
• b) Exponential decay: A(t) = A0(1 - r)^t, where A0 is the initial amount, r is the decay rate, and t is the time.
• c) Logarithm laws: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = nlog(a), where a and b are positive real numbers and n is any real number.

Trigonometric formulas:

• a) Pythagorean identity: sin^2(theta) + cos^2(theta) = 1
• b) Addition formulas: sin(alpha + beta) = sin(alpha)cos(beta) + cos(alpha)sin(beta), cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta)
• c) Double angle formulas: sin(2theta) = 2sin(theta)cos(theta), cos(2theta) = cos^2(theta) - sin^2(theta)

Matrix formulas:

• a) Matrix addition and subtraction: Given matrices A and B of the same size, A + B = B + A and A - B = -(B - A)
• b) Matrix multiplication: If A is an n x m matrix and B is an m x p matrix, then the product AB is an n x p matrix, and (AB)C = A(BC) for any compatible matrices A, B, and C.
• c) Determinant formula: The determinant of a 2x2 matrix [a b; c d] is ad - bc.

Mathematical formula about geometry

Mathematical formulas about geometry:

1. Perimeter and area of a rectangle: P = 2(l + w), A = lw, where l and w are the length and width of the rectangle.
2. Perimeter and area of a square: P = 4s, A = s^2, where s is the length of a side of the square.
3. Perimeter and area of a triangle: P = a + b + c, A = (1/2)bh, where a, b, and c are the lengths of the sides of the triangle and h is the height.
4. Pythagorean Theorem: In a right triangle with legs of length a and b and hypotenuse of length c, a^2 + b^2 = c^2.
5. Circumference and area of a circle: C = 2πr, A = πr^2, where r is the radius of the circle.
6. Volume of a rectangular prism: V = lwh, where l, w, and h are the length, width, and height of the prism.
7. Volume of a cube: V = s^3, where s is the length of a side of the cube.
8. Volume of a cylinder: V = πr^2h, where r is the radius of the base of the cylinder and h is the height of the cylinder.
9. Volume of a cone: V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
10. Volume of a sphere: V = (4/3)πr^3, where r is the radius of the sphere.
11. Surface area of a rectangular prism: SA = 2lw + 2lh + 2wh, where l, w, and h are the length, width, and height of the prism.
12. Surface area of a cube: SA = 6s^2, where s is the length of a side of the cube.
13. Surface area of a cylinder: SA = 2πr^2 + 2πrh, where r is the radius of the base of the cylinder and h is the height of the cylinder.
14. Surface area of a cone: SA = πr^2 + πrs, where r is the radius of the base of the cone and s is the slant height of the cone.
15. Surface area of a sphere: SA = 4πr^2, where r is the radius of the sphere.
16. Law of sines: (a/sin A) = (b/sin B) = (c/sin C), where a, b, and c are the lengths of the sides of a triangle and A, B, and C are the angles opposite those sides.
17. Law of cosines: a^2 = b^2 + c^2 - 2bc cos A, where a, b, and c are the lengths of the sides of a triangle and A is the angle opposite side a.
18. Interior angles of a polygon: The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
19. Area of a trapezoid: A = (1/2)(b1 + b2)h, where b1 and b2 are the lengths of the parallel sides of the trapezoid and h is the height of the trapezoid.

Mathematical formula about calculus

Mathematical formulas in calculus:

1. Derivative of a constant: d/dx(c) = 0, where c is a constant.
2. Power rule for derivatives: d/dx(x^n) = nx^(n-1), where n is a constant.
3. Product rule for derivatives: d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x), where f(x) and g(x) are functions of x and f'(x) and g'(x) are their derivatives.
4. Quotient rule for derivatives: d/dx(f(x)/g(x)) = [f'(x)g(x) - f(x)g'(x)]/g(x)^2, where f(x) and g(x) are functions of x and f'(x) and g'(x) are their derivatives.
5. Chain rule for derivatives: d/dx(f(g(x))) = f'(g(x))g'(x), where f(x) and g(x) are functions of x and f'(x) and g'(x) are their derivatives.
6. Derivative of e^x: d/dx(e^x) = e^x.
7. Derivative of ln(x): d/dx(ln(x)) = 1/x.
8. Derivative of sin(x): d/dx(sin(x)) = cos(x).
9. Derivative of cos(x): d/dx(cos(x)) = -sin(x).
10. Derivative of tan(x): d/dx(tan(x)) = sec^2(x).
11. Derivative of cot(x): d/dx(cot(x)) = -csc^2(x).
12. Derivative of sec(x): d/dx(sec(x)) = sec(x)tan(x).
13. Derivative of csc(x): d/dx(csc(x)) = -csc(x)cot(x).
14. Antiderivative of a constant: ∫ c dx = cx + C, where c is a constant and C is the constant of integration.
15. Power rule for antiderivatives: ∫ x^n dx = (1/(n+1))x^(n+1) + C, where n is a constant and C is the constant of integration.
16. Integration by substitution: If u = g(x) is a differentiable function, then ∫f(g(x))g'(x) dx = ∫f(u) du.
17. Integration by parts: ∫u dv = uv - ∫v du, where u and v are functions of x and dv/dx is the derivative of v with respect to x.
18. Fundamental theorem of calculus (part 1): If f(x) is continuous on the closed interval [a, b], then F(x) = ∫f(t) dt from a to x is an antiderivative of f(x), that is, F'(x) = f(x).
19. Fundamental theorem of calculus (part 2): If F(x) is an antiderivative of f(x), then ∫f(x) dx from a to b = F(b) - F(a).
20. Mean Value Theorem: If f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
21. Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that relates to the continuity of functions. The theorem states that if a function f(x) is continuous on a closed interval [a, b] and takes on values f(a) and f(b) at the endpoints of the interval, then for any value c between f(a) and f(b), there exists at least one value x in the interval [a, b] such that f(x) = c.

Mathematical formula about statistics

Mathematical formulas in statistics:

1. Mean (or average): μ = (x1 + x2 + ... + xn)/n, where x1, x2, ..., xn are the individual data values and n is the sample size.
2. Median: The middle value in a dataset after arranging the values in ascending or descending order.
3. Mode: The value that occurs most frequently in a dataset.
4. Range: The difference between the largest and smallest values in a dataset.
5. Variance: s^2 = ∑(xi - x̄)^2/(n-1), where xi is each individual data value, x̄ is the mean, and n is the sample size.
6. Standard deviation: s = √(s^2), where s^2 is the variance.
7. Coefficient of variation: CV = (s/x̄) x 100%, where s is the standard deviation and x̄ is the mean.
8. Chebyshev's theorem: For any dataset, at least (1 - 1/k^2) percent of the data falls within k standard deviations of the mean, where k is any positive integer greater than 1.
9. Empirical rule (68-95-99.7 rule): For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean.
10. Z-score: z = (x - μ)/s, where x is an individual data value, μ is the mean, and s is the standard deviation.
11. Central limit theorem: The sampling distribution of the sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
12. Sampling distribution of the mean: μx̄ = μ, where μx̄ is the mean of the sample means and μ is the population mean.
13. Standard error of the mean: SE = s/√n, where s is the standard deviation and n is the sample size.
14. Confidence interval: x̄ ± z*(SE), where x̄ is the sample mean, z is the critical value from the standard normal distribution, and SE is the standard error of the mean.
15. Hypothesis testing: A statistical method used to test a claim or hypothesis about a population parameter based on a sample statistic.
16. Null hypothesis: The hypothesis that there is no significant difference or relationship between two variables or populations.
17. Alternative hypothesis: The hypothesis that there is a significant difference or relationship between two variables or populations.
18. Level of significance (alpha): The probability of rejecting the null hypothesis when it is actually true.
19. Test statistic: A standardized value used to test the null hypothesis.
20. P-value: The probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.
21. Type I error: Rejecting the null hypothesis when it is actually true.
22. Type II error: Failing to reject the null hypothesis when it is actually false.
23. One-sample t-test: A hypothesis test used to compare the mean of a single sample to a known population mean.
24. Independent samples t-test: A hypothesis test used to compare the means of two independent samples.
25. Analysis of variance (ANOVA): A hypothesis test used to compare the means of three or more independent samples.

Mathematical formula about exponential

Mathematical formulas related to exponential functions:

1. Exponential function: f(x) = a^x, where a is a constant and x is the variable.
2. Exponential growth: y = a(1 + r)^t, where a is the initial value, r is the growth rate, and t is the time.
3. Exponential decay: y = a(1 - r)^t, where a is the initial value, r is the decay rate, and t is the time.
4. Compound interest: A = P(1 + r/n)^(nt), where A is the amount after time t, P is the principal, r is the annual interest rate, and n is the number of times interest is compounded per year.
5. Continuously compounded interest: A = Pe^(rt), where A is the amount after time t, P is the principal, r is the annual interest rate, and e is the constant 2.71828.
6. Exponential derivative: d/dx (a^x) = a^x * ln(a), where ln(a) is the natural logarithm of a.
7. Logarithmic function: y = loga(x) if and only if x = a^y, where a is a constant.
8. Logarithmic identity: loga(xy) = loga(x) + loga(y).
9. Change of base formula: loga(x) = logb(x)/logb(a), where a, b, and x are positive real numbers and a ≠ 1, b ≠ 1.
10. Exponential rules: a^m * a^n = a^(m+n) and (a^m)^n = a^(mn), where a is a constant and m and n are integers.
11. Half-life: t(1/2) = ln(2)/k, where t(1/2) is the half-life, ln(2) is the natural logarithm of 2, and k is the decay constant.
12. Doubling time: t(2) = ln(2)/r, where t(2) is the doubling time and r is the growth rate.
13. Logistic function: f(x) = L/(1 + e^(-k(x - x0))), where L is the carrying capacity, k is the growth rate, and x0 is the midpoint of the sigmoid curve.
14. Natural logarithm: ln(x) = loge(x), where e is the natural base (approximately 2.71828).
15. Derivative of natural logarithm: d/dx (ln(x)) = 1/x.
16. Derivative of exponential: d/dx (e^x) = e^x.
17. Product rule for derivatives: d/dx (fg) = f'g + fg', where f and g are functions and f' and g' are their respective derivatives.
18. Chain rule for derivatives: d/dx (f(g(x))) = f'(g(x))g'(x), where f and g are functions and f' and g' are their respective derivatives.
19. Power rule for derivatives: d/dx (x^n) = nx^(n-1), where n is a constant.
20. Second derivative: d^2/dx^2 (f(x)) is the derivative of the derivative of f(x).
21. Taylor series: f(x) = ∑(n=0 to ∞) (f^(n)(a)/n!)(x-a)^n, where f^(n)(a) is the nth derivative of f(x) evaluated at a.

Mathematical formula about Logarithmic function

1. Logarithmic function: y = loga(x) if and only if x = a^y, where a is a positive real number and x and y are positive real numbers.
2. Logarithmic identity: loga(xy) = loga(x) + loga(y).
3. Change of base formula: loga(x) = logb(x)/logb(a), where a, b, and x are positive real numbers and a ≠ 1, b ≠ 1.
4. Inverse function: a^loga(x) = x and loga(a^x) = x.
5. Natural logarithm: ln(x) = loge(x), where e is the natural base (approximately 2.71828).
6. Derivative of natural logarithm: d/dx (ln(x)) = 1/x.
7. Logarithmic differentiation: if y = u^v, then ln(y) = v ln(u) and dy/dx = y(ln(u) du/dx + v u'/u).
8. Logarithmic integration: ∫(dx/x) = ln|x| + C.
9. Logarithmic rules: loga(xy) = loga(x) + loga(y), loga(x/y) = loga(x) - loga(y), and loga(x^r) = r loga(x), where a, x, and y are positive real numbers and r is a real number.
10. Change of base formula for natural logarithms: loga(x) = ln(x)/ln(a).
11. Euler's number: e = lim (n→∞) (1 + 1/n)^n, where e is approximately 2.71828.
12. Logarithmic differentiation of exponential functions: if y = a^x, then ln(y) = x ln(a) and dy/dx = a^x ln(a).
13. Inverse properties of logarithms: loga(a^x) = x and a^loga(x) = x.
14. Exponential properties of logarithms: loga(a^x) = x loga(a) = x and loga(xy) = loga(x) + loga(y).
15. Natural logarithmic properties: ln(e^x) = x and e^ln(x) = x.
16. Product rule for logarithms: loga(xy) = loga(x) + loga(y).
17. Quotient rule for logarithms: loga(x/y) = loga(x) - loga(y).
18. Power rule for logarithms: loga(x^r) = r loga(x).
19. Derivative of logarithmic function: d/dx (loga(x)) = 1/(x ln(a)).
20. Common logarithm: log10(x) = log(x).
21. Exponential decay model: y = a e^(-kt), where y is the amount after time t, a is the initial amount, k is the decay constant, and e is the constant 2.71828.
22. Exponential growth model: y = a e^(kt), where y is the amount after time t, a is the initial amount, k is the growth constant, and e is the constant 2.71828.
23. Half-life formula: t(1/2) = ln(2)/k, where t(1/2) is the half-life, ln(2) is the natural logarithm of 2, and k is the decay constant.