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Mathematics: Gradient of Parallel and Perpendicular Lines “Parallel lines have the same gradient” Remember the form y=ax+b, where “a” is the gradient? The gradient here is the same; it’s just the value of “b”, the y intercept that is different. The rule to learn here is that for two functions y=ax+b and y1 = a1x1…

IGCSE Business Studies: Ownership and Internal Organisation Ownership and Internal Organization The Legal forms of business organization This chapter introduces the main types of lega forms of business in the UK. These businesses can be classified as: sole traders, partnerships, limited liability companies, franchises, and multinationals. Each business has its own merits and drawbacks, which…

Introduction to Scatter Plots and Relationships Scatter Plot: A graph in the Cartesian plane which displays the joint distribution of two variables in which each point represents a pair of variable values. Carnings E=10.25h+50Independent variable • a variable that affects the value of another variable • Always placed on the horizontal axis• can take any…

IB Physics: Work, Energy and Power Work, Energy and Power Outline what is meant by work. Work is defined as the force multiplied by the distance moved in the direction of the force. Determine the work done by a non‑constant force by interpreting a force–displacement graph. If we plot the graph of a graph with…

The Pythagorean Theorem What is the Pythagorean theorem? The area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. Area1=Area2+Area3 AB2=BC2+AC2 C2=a2+b2 The square of the length of the hypotenuse is equal to the sum of the squares of the other two…

Relations & Functions A relationship can be shown between numbers in several ways: table of values, equation, graph, words, mappings. The first variable in a relation is called the independent variable. The second variable is the dependent variable. Domain – the set of values for which the independent variable is defined. Range – the set…

Optimization intuition: Worked example Here’s an example: we’ll consider the region R, which iS a triangle. The vertice of this triangle is (2, 2). The function f(x, y) = 2x – x2 – y is then applied to each point in the region. Our goal is to find the maximum of this function on the…

Magnitude The magnitude of a vector is the length of the vector. The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components. The magnitude of the vector v is denoted by V!. The magnitude of a vector is the length of the vector…

Putting it together The graph below shows the function x squared plus y squared. [Graph] We have seen before that the gradient of a function is always perpendicular to its level curves. And we see that it can be both big and small in some places. The value of the function is greater in areas…

Robot Arm Matrix Example 2 So let’s do one more. We will do the math and create a picture. We take 1 minus 1 zero one. We pick a vector 0, 0.1. Let’s subtract 0.1 from 0. What comes next? O plus 0.1, and 0 minus 0.1. We box that in yellow. Imagine in your…