Parametric Equations and Vectors

Parametric Equations

• Parametric equations represent a curve where the x and y coordinates are defined in terms of a third variable, often denoted as 't', called a parameter.
• The parameter 't' often represents time, but can be any variable.
• To find the rectangular equation (or Cartesian equation), solve for 't' in one equation and substitute into the other.
• Parametric equations are useful for describing curves that are not easily described by a single equation, like cycloids or spirals.
• The derivatives of x and y with respect to 't' can be used to determine the slope of the tangent line to the curve.
• The arc length of a curve defined parametrically is given by [ \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt ]
• Parametric equations can often be used for modeling motion.

Vectors

• A vector is a quantity with both magnitude and direction.
• Vectors can be represented graphically by arrows.
• Vectors can be added and subtracted using the parallelogram law or tip-to-tail method.
• Scalar multiplication of a vector multiplies both its magnitude and direction.
• Components of a vector can be determined using the coordinates on a coordinate system.
• Vectors can be used to represent displacements, forces, velocities, and other physical quantities.
• Magnitude/length of a vector: [ ||\bold{a}|| = \sqrt{a_x^2 + a_y^2 + a_z^2} ]
• Unit vector: a vector with a magnitude of 1.

Vector-Valued Functions

• A vector-valued function is a function that maps a scalar input to a vector output. This creates a curve in space.
• They are often written as (\bold{r}(t) = <x(t), y(t), z(t)>) where 't' is a parameter.
• Derivative of a vector-valued function represents the tangent vector to the curve.
• The derivative at a point represents the instantaneous velocity.
• Integrals of vector-valued function represent the accumulation of position of the vector from a specific time.

Polar Functions

• Polar coordinates represent a point in the plane using a distance from the origin (r) and an angle from the positive x-axis.
• The relationship between polar coordinates and Cartesian coordinates is:
• x = r cos(θ)
• y = r sin(θ)
• Polar equations are frequently used to define specific curves, such as circles, roses, limaçons, and spirals.
• Conversion between polar and rectangular equations is useful for graphing polar curves.
• Solving polar equations might involve trigonometry or algebraic manipulation.
• Polar graphs are inherently different than graphs in the rectangular plane.

Integration Techniques

• Substitution: Replacing part of the integral with a new variable.
• Integration by parts: Using the product rule in reverse. The formula is (\int u dv = uv-\int vdu). Choose (u) and (dv) based on the integral to reduce complexity.
• Partial Fraction Decomposition: Decomposing a rational function into simpler fractions before integrating. This technique is helpful for integrating rational functions that have repeating factors or denominators.
• Trigonometric Integrals: Using trigonometric identities to simplify integrals involving trigonometric functions.
• Trigonometric Substitution: Substituting trigonometric functions for algebraic expressions.
• Table of integrals: Using a table of common integrals to look for patterns.
• Recognizing basic integral patterns: Memorizing basic or common integrals for immediate substitution and streamlining the solution process.
• Recognizing and applying specific properties of definite integrals, like the order, limits, or intervals of integration, is essential.