Subject

Minimize the Surface Area of a Square‐Based Prism Minimizing surface area for a given volume is important when designing packages and containers to save on materials and reduce heat loss. Example 1: The Acme Box Company designs and manufactures boxes, cans, and other containers. It has been hired to design a box to hold 64…

Vector introduction Vectors are geometric objects a that have both a magnitude and direction. They are often used to represent quantities that vary with both time and space. A vector is a directed line. It can be used to show direction, velocity and acceleration. A vector has two components: magnitude and direction. Magnitude is the…

Cartesian Plane – Point-Slope Formula for Lines Lines on the plane: This is the imaginative title, lines on the plane part one. The point of this is to demystify formulas for equations of line. The first one, which we’re going to talk about in part one, is what we call the point slope formula for…

IGCSE Economics: Externalities – Costs and Benefits Calculations Social Costs = Private + External Costs Social costs and benefits are therefore the costs and benefits incurred by the entirety of society (producer, consumer and third party) as a result of the production and/or consumption of a good or service. Private costs and benefits are the…

CFA Level 1 – Quantitative Methods, Session 2 – Reading 11 (Notes, Practice Questions, Sample Questions) 1. Which of the following statements about testing a hypothesis using a Z-test is least accurate? A)If the calculated Z-statistic lies outside the critical Z-statistic range, the null hypothesis can be rejected.B)The calculated Z-statistic determines the appropriate significance level…

Tangent Lines – Slope of a Graph at a Poin We begin our tangent lines series. The key idea here, for the moment, is to imagine that we have a graph of a function y = f(x), and it’s shown in green. Now suppose we take a particular point x = a. So right here…

CFA Level 2 – Financial Reporting and Analysis, Session 6 – Reading 23 (Practice Questions, Sample Questions) 1. The financial statements of Pace Industries issued over the past five years show a progressively increasing net difference between the value of its pension fund and the projected future pension liability on the balance sheet. Pace most…

Practice in higher dimensions Here are three functions with a point and direction for each of them. Your task is to compute the gradients of the functions, then evaluate their gradients at the point given. Next, compute the directional derivative of the function at that point in the direction given by its vector component along…

Working with Quantifiers 2 All prime numbers are odd. We know this is false, because we know that two is a prime number and two is not odd. But let’s examine the logic that underlies that conclusion. Because in more complicated situations where we’re not familiar with the domain or with the result, all we’ll…

Introduction to Mathematical Thinking. Proofs What is a proof? Fermat’s Last Theorem If n≥3 the equation x^n + y^n = z^n has no solutions for positive whole numbers x, y, z. Goldbach Conjecture Every even number greater than 2 can be expressed as a sum of teo primes. In mathematics, a proof is a logically…