Publications by Souleymane Doumbia, Group Member: Fomba Kassoh
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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Document
Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if (a) he bets 1 dollar each time (timid s...
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Document
Probability that the grandson of a man from Harvard went to Harvard # Transition matrix T <- matrix(c(1, 0, 0, 0.3, 0.4, 0.3, 0.2, 0.1, 0.7), byrow = TRUE, nrow = 3) # Calculate the 2-step transition matrix T_2 <- T %*% T # Probability that the grandson of a man from Harvard went to Harvard # Harvard is the first state...
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CLT and Generating Function
1. Exercise 11, page 363 We have been given: mu <- 0 sigma <- sqrt(1/4) Y1 <- 100 days <- 365 According to the Central Limit Theorem (CLT), the sum of a large number of independent, identically distributed random variables, each with finite mean and variance, will approximate a normal distribution. For the...
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