Section A: Algebra and Functions

Solving Equations and Analyzing Functions

This section reviews essential algebraic techniques and function analysis, foundational for problem-solving in physics.

• Solving Quadratic Equations: A quadratic equation is an equation of the form . Solutions can be found using factoring, completing the square, or the quadratic formula:

• Solving Rational Equations: Rational equations involve fractions with polynomials in the numerator and denominator. To solve, find a common denominator and solve for the variable, being careful to exclude values that make the denominator zero.

• Functions and Average Rate of Change: The average rate of change of a function from to is:

• Example: For , find and the average rate of change from to .

Section B: Geometry and Trigonometry

Vectors, Triangles, and the Unit Circle

This section covers vector decomposition, the Pythagorean theorem, and trigonometric identities, all crucial for analyzing physical systems.

• Vectors: A vector has both magnitude and direction. It can be decomposed into horizontal (x) and vertical (y) components using trigonometric functions:

• Pythagorean Theorem: In a right triangle with sides , , and hypotenuse :

• Trigonometric Functions in Quadrants: The values of and depend on the angle's quadrant. For example, in Quadrant I, both are positive.

• Trigonometric Identity: The fundamental identity:

• Unit Circle: The unit circle is a circle of radius 1 centered at the origin. The coordinates of a point at angle are .

• Example: For , the coordinates are .

Section C: Calculus

Derivatives, Integrals, and Motion

This section introduces calculus concepts applied to motion, including differentiation and integration of position, velocity, and acceleration functions.

• Velocity and Acceleration: For a position function :

  • Velocity:

  • Acceleration:

• Integration: The process of finding the original function from its derivative. For acceleration :

  • First integral gives velocity:

  • Second integral gives position:

• Vector Functions: For a position vector :

  • Velocity:

  • Acceleration:

• Displacement from Velocity: The displacement vector over a time interval is:

• Example: For , find and .

Summary Table: Key Concepts