Notes, summaries, assignments, exams, and exercises for Mathematics for Secondary education

In computer graphics, output primitives are the basic geometric structures used to describe and construct images. The simplest primitives are Points and Lines. Before these primitives can be displayed on a screen, they must undergo scan conversion—the process of digitizing a continuous geometric shape into a discrete grid of screen pixels.

Points and Lines

Points

A point is a zero-dimensional geometric object. In a digital system, a point is represented by an ordered pair of coordinates (X, Y). To display a point, the graphics control system locates the corresponding row and column address in the frame buffer and sets its intensity or color value.

Lines

A line is a one-dimensional geometric object defined by two endpoints: (X₁, Y₁) and (X₂, Y₂)

Total Cost = w₁x₁ + w₂x₂ + … + w_nx_n

Isocost function:

x₂ = -w₁x₁/w₂ + tc/w₂ where the slope is: -w₁/w₂ ratio of relative factor prices with negative sign

Cost minimization mate: -w₁/w₂ = -MP₁/MP₂ (isoquant curve)

Cost minimization implies producing a certain amount of output (y > 0) with the lowest possible cost. As with the maximization of profits, the firm is getting the most out of the resources it is using.

What is the difference between cost minimization and profit maximization? Cost minimization is a necessary condition for the maximization of profits, but not the other way around. For example, if you bring costs down to zero then output and profits will be zero too, failing to maximize profits while minimizing costs. As before, when...

9.14 Nonconstant growth. Computech corporation is expanding Rapidly and currently needs to retain all of its earnings; hence, it does not Pay dividends. However, investors expect computech to begin paying dividends, Beginning with a dividend of $0.50 coming 3 years from today. The dividend Should growth rapidly at a rate of 35% per year during years 4 and 5; but after Year 5, growth should be constant 7% per year. If the required return on computech Is 13%, what is the value of the stock today?

solution

Calculate the dividend cash flows and place them on a time Line.  Also, calculate the stock price at The end of the supernormal growth period, and include it, along with the Dividend to be paid at t = 5, as CF5.  Then, enter the cash flows as...

Assessment is one of the most important components of education as it provides evidence of students' learning outcomes and their growth over time. This process helps not only students but also teachers, families, and other stakeholders in many ways. For example, assessment is a tool to evaluate the effectiveness of teaching and teachers, and it can provide insight into program effectiveness for program developers. According to Brookhart and Nitko (2011), assessments are used to measure, evaluate, and document learning outcomes, proficiency, and educational needs.

Assessment can be formative or summative (Crooks, 2001). Formative assessments are used to monitor day-to-day learning of students, provide feedback to both teachers (i.e., about instructions)...

Cost of the Building: Land+Construction

When was the building bought? YTD amort.construction 2020 / construction P&L --> it has been amortized x years until 2020

At which price did the company sell the building? ((Land + Construction) - YTD amort.constr) + Profit from selling

useful life of the building? construction / construction P&L

Occupancy: 33%. VERY LOW !! ===> 70 rooms x 600 ADR x occupancy x 365 days = 5.059.000 €

F&B Menu: 7 € ==> VERY LOW & doesn't match with an 600 ADR !! [(80% of 100 people for lunch) + (80% of 100 people per Dinner) ] * Menu Price * 365 days = 409.000 €
F&B Breackfast: 18€/person. Reasonable with 600 ADR 80% of 1,5 clients x 33% occupancy * 70 rooms * 365 days * Breackfast Price...

Key Terms in Linear Programming & Optimization

Constraint

An equation or inequality that rules out certain combinations of decision variables as feasible solutions.

Problem Formulation

The process of translating a verbal statement of a problem into a mathematical statement called the mathematical model.

Mathematical Model

A representation of a problem where the objective and all constraint conditions are described by mathematical expressions.

Decision Variable

A controllable input for a linear programming model.

Objective Function

The expression that defines the quantity to be maximized or minimized in a linear programming model.

Nonnegativity Constraints

A set of constraints that requires all variables to be nonnegative.

Linear Program

A mathematical...

Selecting Samples

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

A random sample of size n from an infinite population is a sample selected such that the following conditions are satisfied.

Point Estimation

By making the preceding computations, we perform the statistical procedure called point estimation. We refer to the sample mean x as the point estimator of the population mean m, the sample standard deviation s as the point estimator of the population standard deviation s, and the sample proportion p as the point estimator of the population proportion p. The numerical value obtained for x, s, or p is called the point...

Bond Price Sensitivity

7.7 Interest rate sensitivity. An investor purchased the following 5 bonds. Each bond had a par value of $1,000 and an 8% yield to maturity on the purchase date. Immediately after the investor purchased them, interest rates fell, and each of them had a new YTM of 7%. What is the percentage change in price for each bond after the decline in interest rates? Fill in the following table:

N=10 Y=8 or 7 PV=? FV=1000 PTM=0 except coupon=100

Default Risk Premium

6.4 Default risk premium. A treasury bond that...

Business Documents

Commercial documents, also called "proof," constitute the evidence of any business operation conducted by a company. Thus, each proof tests a type of transaction. It is known that any operation involves at least two parts. For each of them, it will have a different meaning (as discussed later: buyer or seller, debtor or creditor, or collector payer, etc.).