Notes, summaries, assignments, exams, and problems for Mathematics

Personal Finance Math and Interest Calculations

Interest and Yield Calculations

1. Simple Interest: Katie invests $2,300 in an account that pays 7% simple interest annually. Find the future value of the account after 9 years. Round your answer to the nearest cent.
2. Monthly Compounding: Suppose you invest $1,900 at a fixed rate of 5% per year, compounded monthly. Find the future value of the account after 6 years. Round your answer to the nearest cent.
3. Continuous Compounding: Suppose you instead invest your $1,900 in an account that earns 6% interest compounded continuously. What is the total amount of your investment after 7 years? Round your answer to the nearest cent.
4. Effective Annual Yield: Find the effective annual yield to the nearest hundredth

Chapter 1: Foundations of Statistics

Data: Information derived from observations, counts, measurements, or responses.

Statistics: The science of collecting, organizing, analyzing, and interpreting data to make informed decisions.

Population: The collection of all outcomes, responses, measurements, or counts of interest.

Sample: A subset or part of a population.

Parameter: A numerical description of a population characteristic.

Statistic: A numerical description of a sample characteristic.

Descriptive Statistics: Methods to organize, display, and summarize data (e.g., mean, range, graphs, tables).

Inferential Statistics: Using sample data to draw conclusions about a population.

Qualitative Data: Attributes, labels, or non-numerical entries.

Quantitative

Introduction: A Data Science Journey

My name is Amit Kadam, and I currently reside in Mumbai. I completed my Bachelor of Engineering (B.E.) degree in 2021. After graduation, the pandemic limited job opportunities, and my family faced financial challenges, so I took my first opportunity at Sterling as a Senior Associate, where I worked for 2.5 years.

Initially, I was responsible for document verification, but I was soon promoted to manage drug health screening processes. In this role, I handled candidate health reports, prepared data for analysis, and developed strong attention to detail and data-handling skills.

During this time, a friend who successfully transitioned into data science encouraged me to explore the field. I started by learning... Continue reading "Data Science Career Transition & Predictive Modeling" »

Question Bank #1 – Net Value Functions

L03 – Engineering Economics & Net Value Applications

Review Questions

Recall the nanoRIMS example discussed in lecture. If the net value of buying the nanoparticles is $0 (the reference), determine the net value per week of having a grad student make the nanoparticles based on the following information:

• Benefit = $896/week
• Cost:
  • Cost of consumable supplies per week: Ingredients & electricity to make one batch as accurately as a grad student does is $5/100 mL * 200 mL/week = $10/week
  • Cost of time: Grad student time is $15/hr * 9 hours/100 mL * 200 mL/week = $270/week
  • Cost of space: Occupying a whole fume hood space for 16 hours during working time is $12.50/hr * 16 hrs/week = $200/week
  • Cost of any device:

What is a Scatter Diagram?

Definition

A scatter diagram (or scatter plot) is a graphical representation of two variables where each point represents an observation consisting of paired values from two datasets. The horizontal axis (X-axis) represents one variable, and the vertical axis (Y-axis) represents the other.

Construction

Each point (x_i, y_i) is plotted on the graph for the corresponding values of the two variables.

Utility in Correlation Analysis

Scatter diagrams are essential for:

• Visualizing relationships: Helps identify if a linear or non-linear relationship exists.
• Direction of correlation:
  • Positive correlation: As X increases, Y increases (points slope upwards).
  • Negative correlation: As X increases, Y decreases (points slope downwards).

14. Obtain the regression equation of Y onX and correlation coefficient for the following: X; 4 6 8 10 12 , f ;7 9 8 12 15 1. Calculate the necessary sums: X Y X² XY 2 10 4 20 3 9 9 27 7 11 49 77 8 8 64 64 10 12 100 120 ΣX = 30 ΣY = 50 ΣX² = 226 ΣXY = 308 Export to Sheets 2. Calculate the slope (b): b = (nΣXY - ΣXΣY) / (nΣX² - (ΣX)²) where n is the number of data points (n = 5 in this case) b = (5 * 308 - 30 * 50) / (5 * 226 - 30²) b = (1540 - 1500) / (1130 - 900) b = 40 / 230 b ≈ 0.174 3. Calculate the y-intercept (a): a = (ΣY - bΣX) / n a = (50 - 0.174 * 30) / 5 a = (50 - 5.22) / 5 a ≈ 8.956 4. The fitted line: Substitute the values of a and b into the equation Y = a + bX: Y = 8.956 + 0.174X Therefore, the fitted straight... Continue reading "Regression Equation and Probability Addition Theorem Solutions" »

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Write an interpretation of r^2 using the template in the Activity 2.1 Readings. We will do this one as a class.

Template: The proportion of the variation in the Y variable that is explained by the SLR model with the X variable is r^2.

For slope : Template: As x var increases by 1 unit, we predict y var  will increase/dec  by ____  y var units.

For y-intercept: When x var = 0 units, we predict that the y var  will be ____ units..

For SLR: Error = epsilon = y - yhat = y - (betahat0 + betahat1x)

SSE = residual1^2 + res. 2^2 +…+ res.  n^2

Standard error of regression = Root MSE (in SAS language)

The text lists six conditions for simple linear... Continue reading "Understanding Simple Linear Regression: R-squared, Slope, and Conditions" »

Properties of Sets

Important Set Formulas

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• If A and B are disjoint: n(A ∪ B) = n(A) + n(B)
• Number of subsets of a set with n elements = 2ⁿ
• Number of proper subsets = 2ⁿ – 1
• n(P(A)) = 2ⁿ

1. Identity Function

f(x) = x

• Graph: Straight line through origin
• Line: y = x
• Domain & Range: ℝ
• Passes through: (0, 0), (1, 1), (-1, -1)

2. Constant Function

f(... Continue reading "Essential Mathematical Formulas: Sets, Functions, and Trigonometry" »

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)

1. Integration of Polynomials

Let f(x) = -x² + 2x + 3. The integral is:

• a) ∫(-x² + 2x + 3) dx = -x³/3 + x² + 3x + C
• b) [-x³/3 + x² + 3x] from 1 to 2 = 11/3

2. Finding f(x) from f'(x)

Let f'(x) = 3x² - 3. Given f(2) = 1, find f(x):

f(x) = x³ - 3x + C. Substituting f(2) = 1: 8 - 6 + C = 1, so C = -1. Thus, f(x) = x³ - 3x - 1.

3. Solving for f(x) with Initial Conditions

Let f'(x) = 6x² + 2x - 1. Given f(2) = 5:

f(x) = 2x³ + x² - x + C. Substituting f(2) = 5: 2(8) + 4 - 2 + C = 5, so 18 + C = 5, C = -13. Thus, f(x) = 2x³ + x² - x - 13.

4. Polynomial Integration

Let f(x) = 3x² - 4x:

• a) ∫f(x) dx = x³ - 2x² + C
• b) Evaluation results in 32.

5. Definite Integral Properties

Consider ∫₁⁵ f(x) dx = 6:

• a) ∫₁⁵ 2f(x) dx = 2 ∫₁⁵ f(x) dx = 2(6) = 12
• b) ∫₁³ 3 dx