Dilla university info & all exams

1. Pythagorean theorem: a² + b² = c²

2. Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

3. Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)

4. Slope-intercept form of a line: y = mx + b

5. Point-slope form of a line: y - y₁ = m(x - x₁)

6. Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)

7. Law of sines: a/sin A = b/sin B = c/sin C

8. Law of cosines: c² = a² + b² - 2ab cos C

9. Sum of angles in a triangle: A + B + C = 180°

10. Area of a triangle: A = (1/2)bh

11. Volume of a sphere: V = (4/3)πr³

12. Volume of a cylinder: V = πr²h

13. Volume of a cone: V = (1/3)πr²h

14. Surface area of a sphere: A = 4πr²

15. Surface area of a cylinder: A = 2πr² + 2πrh

16. Surface area of a cone: A = πr² + πrs, where s is the slant height

17. Binomial theorem: (a + b)ⁿ = Σ(n choose k)a^(n-k)b^k, where Σ is the sum from k=0 to n, and (n choose k) is the binomial coefficient

18. Fundamental theorem of calculus: ∫a^b f(x) dx = F(b) - F(a), where F is the antiderivative of f

19. Derivative of a constant: d/dx(c) = 0

20. Power rule for derivatives: d/dx(xⁿ) = nx^(n-1)

21. Product rule for derivatives: d/dx(fg) = f'g + fg'

22. Quotient rule for derivatives: d/dx(f/g) = (f'g - fg')/g²

23. Chain rule for derivatives: d/dx(f(g(x))) = f'(g(x))g'(x)

24. Mean value theorem: if f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that f'(c) = (f(b) - f(a))/(b-a)

25. Intermediate value theorem: if f is continuous on [a,b], then for any y between f(a) and f(b), there exists c in [a,b] such that f(c) = y

26. Rolle's theorem: if f is continuous on [a,b] and differentiable on (a,b), and if f(a) = f(b), then there exists c in (a,b) such that f'(c) = 0

27. Integration by substitution: ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x)

28. Integration by parts: ∫u dv = uv - ∫v du

29. L'Hopital's rule: if lim(x → a) f(x)/g(x) = 0/0 or ∞/∞, then lim(x → a) f(x)/g(x) = lim(x → a) f'(x)/g'(x)

30. Taylor series: f(x) = Σ(n=0 to ∞) f^(n)(a)/n!(x-a)^n, where f^(n) is the nth derivative of f

31. Euler's formula: e^(ix) = cos(x) + i sin(x)

32. De Moivre's theorem: (cos x + i sin x)^n = cos(nx) + i sin(nx)

33. Fundamental trigonometric identities: sin² x + cos² x = 1, 1 + tan² x = sec² x, 1 + cot² x = csc² x

34. Double angle formulas: sin 2x = 2sin x cos x, cos 2x = cos² x - sin² x, tan 2x = (2tan x)/(1 - tan² x)

35. Half angle formulas: sin(x/2) = ±√((1 - cos x)/2), cos(x/2) = ±√((1 + cos x)/2), tan(x/2) = ±√((1 - cos x)/(1 + cos x))

36. Sum-to-product formulas: sin A + sin B = 2sin((A+B)/2)cos((A-B)/2), cos A + cos B = 2cos((A+B)/2)cos((A-B)/2), sin A - sin B = 2cos((A+B)/2)sin((A-B)/2), cos A - cos B = -2sin((A+B)/2)sin((A-B)/2)

37. Product-to-sum formulas: cos A cos B = (1/2)(cos(A-B) + cos(A+B)), sin A sin B = (1/2)(cos(A-B) - cos(A+B)), sin A cos B = (1/2)(sin(A+B) + sin(A-B)), cos A sin B = (1/2)(sin(A+B) - sin(A-B))

38. Hyperbolic functions: sinh x = (e^x - e^-x)/2, cosh x = (e^x + e^-x)/2, tanh x = sinh x/cosh x

39. Inverse trigonometric functions: arcsin x, arccos x, arctan x

40. Logarithmic identities: log(xy) = log x + log y, log(x/y) = log x - log y, log x^n = n log x

41. Exponential identities: e^x+y = e^x e^y, (e^x)^n = e^(nx), e^0 = 1

42. Binomial coefficients: (n choose k) = n!/(k!(n-k)!)

43. Pascal's triangle: each entry is the sum of the two entries above it

44. Fermat's little theorem: if p is a prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p)

45. Chinese remainder theorem: if m₁, m₂, ..., mₙ are pairwise coprime integers and a₁, a₂, ..., aₙ are any integers, then there exists an integer x that satisfies the system of congruences x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂), ..., x ≡ aₙ (mod mₙ)