Publications by Souleymane Doumbia, Group Member: Fomba Kassoh
Functions of Multiple Variables
\(\underline{PROBLEM 1}\) Given points: (5.6, 8.8), (6.3, 12.4), (7, 14.8), (7.7, 18.2), (8.4, 20.8) We use the method of least squares to find the best-fitting line for the given data points. Linear Regression Equation The linear regression equation is given by: \[ y = mx + b \] where: - \(y\) is the dependent variable, - \(x\) is the independen...
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Domain and Range of Multi-variable Functions
Domain and Range of Multi-variable Functions Function 1: \(f(x, y) = x^2 + y^2 + 2\) Domain The function \(f(x, y) = x^2 + y^2 + 2\) involves square terms of \(x\) and \(y\), which are defined for all real numbers. Therefore, the domain of \(f\) is all ordered pairs of real numbers: \[ \text{Domain} = \mathbb{R}^2 \] Range Since \(x^2\) and \(y^...
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Taylor Series Expansions of Popular Functions
Taylor Series Expansions of Popular Functions Function \(f(x) = \frac{1}{1-x}\) The Taylor series expansion of \(f(x) = \frac{1}{1-x}\) around \(x = 0\) is: fx <- function(x) 1/(1 - x) taylor_series_1x <- taylor(fx, x0 = 0, n = 7) print(taylor_series_1x) ## [1] 1.007014 1.001710 1.000293 1.000029 1.000003 1.000000 1.000000 1.000000 Function \(f(x...
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Taylor Series Approximations
Finding the \(n^{th}\) Taylor Series for \(f(x) = e^{-x}\) at \(c = 0\) Formula Derivation To find the Taylor series for the function \(f(x) = e^{-x}\) centered at \(c = 0\), we first compute the derivatives of \(f(x)\) at \(x = 0\). The Taylor series expansion of a function \(f(x)\) around a point \(c\) is given by: \[ f(x) = \sum_{n=0}^{\infty} ...
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Univariate & Multivariate Calculus
Problem 1: Using integration by substitution to solve the integral: \[ \int 4e^{-7x} \, dx \] Let \(u = -7x\), therefore \(du = -7dx\) or \(dx = -\frac{1}{7}du\). Substituting into the integral, we get: \[ \int 4e^{u} \cdot \frac{du}{-7} = -\frac{4}{7} \int e^{u} \, du = -\frac{4}{7}e^{u} + C \] Now, since \(u = -7x\), we replace \(u\) back with \(...
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Calculating Area Enclosed by Two functions
Packages library(ggplot2) library(pracma) Problem Statement We want to calculate the total area enclosed by the following functions: \[f(x) = 2x^2 + 5x - 3\] \[g(x) = x^2 + 4x - 1\] Functions Definition f <- function(x) { 2 * x^2 + 5 * x - 3 } g <- function(x) { x^2 + 4 * x - 1 } Intersection Points To find the areas enclosed by the curves, we f...
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Regression Analysis
Packages library(readr) library(ggplot2) Loading Dataset dataset = read_csv('https://raw.githubusercontent.com/Doumgit/SP24-D605-F.-Computational-Mathematics/main/Dataset.csv') ## Rows: 190 Columns: 10 ## ── Column specification ──────────────────────────────────────�...
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605 Discussion 12
Loading the data # Loading the mtcars dataset data(mtcars) Creating the necessary variables within the mtcars dataset For our model, we can use mpg as the dependent variable \(Y\) and include hp (horsepower) as the continuous predictor \(X\), am as the dichotomous predictor \(D\), and we’ll create an interaction term between hp and am. We’ll a...
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Linear regression
Loading cars data set summary(cars) ## speed dist ## Min. : 4.0 Min. : 2.00 ## 1st Qu.:12.0 1st Qu.: 26.00 ## Median :15.0 Median : 36.00 ## Mean :15.4 Mean : 42.98 ## 3rd Qu.:19.0 3rd Qu.: 56.00 ## Max. :25.0 Max. :120.00 cars_data = cars head(cars_data) ## speed dist ## 1 4 ...
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Discussion Board 11
Loading Exam data set library(tidyverse ) ## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ── ## ✔ dplyr 1.1.3 ✔ readr 2.1.4 ## ✔ forcats 1.0.0 ✔ stringr 1.5.0 ## ✔ ggplot2 3.4.4 ✔ tibble 3.2.1 ## ✔ lubridate 1....
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