formula sheet for math

Guri Bee

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21-10-2024

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Your Ultimate Math Formula Cheat Sheet: From Algebra to Calculus

Memorizing formulas can be a daunting task, especially when you're facing a mountain of mathematical concepts. But fret not! This article will equip you with a handy formula sheet, covering essential concepts from basic algebra to calculus. We'll break down each formula, provide examples, and offer insights to make your math journey smoother.

Note: This cheat sheet is meant to be a starting point. It's crucial to understand the concepts behind these formulas to apply them effectively.

1. Algebra

• Basic Operations:
  • Addition: a + b = c
  • Subtraction: a - b = c
  • Multiplication: a * b = c
  • Division: a / b = c (where b ≠ 0)
• Exponents:
  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n) (where a ≠ 0)
  • Power of a power: (a^m)n = a^(m*n)
• Linear Equations:
  • Slope-intercept form: y = mx + c (where m is the slope, and c is the y-intercept)
  • Point-slope form: y - y1 = m(x - x1) (where m is the slope, and (x1, y1) is a point on the line)
• Quadratic Equations:
  • Standard form: ax^2 + bx + c = 0
  • Quadratic formula: x = [-b ± √(b^2 - 4ac)] / 2a

Example: Solve the quadratic equation 2x^2 + 5x - 3 = 0 using the quadratic formula.

Here, a = 2, b = 5, and c = -3. Substituting these values into the quadratic formula:

x = [-5 ± √(5^2 - 4 * 2 * -3)] / (2 * 2) x = [-5 ± √(49)] / 4 x = (-5 ± 7) / 4

Therefore, x = 1/2 or x = -3.

2. Geometry

• Area:
  • Square: A = s^2 (where s is the side length)
  • Rectangle: A = l * w (where l is the length, and w is the width)
  • Triangle: A = (1/2) * b * h (where b is the base, and h is the height)
  • Circle: A = πr^2 (where r is the radius)
• Perimeter:
  • Square: P = 4s (where s is the side length)
  • Rectangle: P = 2l + 2w (where l is the length, and w is the width)
  • Triangle: P = a + b + c (where a, b, and c are the sides)
  • Circle: C = 2πr (where r is the radius)
• Volume:
  • Cube: V = s^3 (where s is the side length)
  • Rectangular prism: V = l * w * h (where l is the length, w is the width, and h is the height)
  • Sphere: V = (4/3)πr^3 (where r is the radius)

Example: Calculate the volume of a sphere with a radius of 5 cm.

V = (4/3) * π * (5 cm)^3 V = (4/3) * π * 125 cm^3 V ≈ 523.6 cm^3

3. Trigonometry

• SOH CAH TOA:
  • Sine (sin): Opposite / Hypotenuse
  • Cosine (cos): Adjacent / Hypotenuse
  • Tangent (tan): Opposite / Adjacent
• Pythagorean Theorem: a^2 + b^2 = c^2 (where a and b are the legs of a right triangle, and c is the hypotenuse)
• Unit Circle: This is a circle with a radius of 1 unit, used to visualize trigonometric functions.

Example: Find the sine of 30 degrees using the unit circle.

The sine of an angle in the unit circle is represented by the y-coordinate of the point where the angle intersects the circle. For 30 degrees, the y-coordinate is 1/2. Therefore, sin(30°) = 1/2.

4. Calculus

• Derivatives:
  • Power rule: d/dx (x^n) = nx^(n-1)
  • Product rule: d/dx (u * v) = u * dv/dx + v * du/dx
  • Quotient rule: d/dx (u/v) = (v * du/dx - u * dv/dx) / v^2
• Integrals:
  • Power rule: ∫ x^n dx = (x^(n+1)) / (n+1) + C (where C is the constant of integration)
  • Integration by parts: ∫ u dv = uv - ∫ v du

Example: Find the derivative of the function f(x) = x^3 + 2x.

Using the power rule:

f'(x) = 3x^(3-1) + 2 * 1x^(1-1) f'(x) = 3x^2 + 2

Additional Resources:

• Khan Academy: https://www.khanacademy.org/ - offers free, comprehensive math courses and resources.
• Paul's Online Math Notes: https://tutorial.math.lamar.edu/ - provides clear explanations and practice problems.
• Wolfram Alpha: https://www.wolframalpha.com/ - a powerful computational engine that can solve mathematical problems and provide step-by-step solutions.

Remember: This formula sheet is your companion, not your crutch. Embrace the power of understanding the concepts and applying them confidently. Happy learning!