Elina Azrilyan

November 22th, 2019

Assignment 13

Problem 1.

Use integration by substitution to solve the integral below.

knitr::include_graphics("13-1.jpg")

Problem 2.

Biologists are treating a pond contaminated with bacteria. The level of contamination is changing at a rate of dN/dt = - (3150 / t^4) - 220 bacteria per cubic centimeter per day, where t is the number of days since treatment began. Find a function N(t) to estimate the level of contamination if the level after 1 day was 6530 bacteria per cubic centimeter.

knitr::include_graphics("13-2.jpg")

Problem 3.

Find the total area of the red rectangles in the figure below, where the equation of the line is f (x) = 2x - 9.

knitr::include_graphics("13-3.jpg")

Problem 4.

Find the area of the region bounded by the graphs of the given equations. y=x^2-2x-2; y=x+2;

library(ggplot2)
fx <- function(x){ (x^2 - 2*x - 2) }
gx <- function(x){ (x + 2) }

ggplot(data = data.frame(x = 1), mapping = aes(x = x)) + stat_function(fun = fx, mapping = aes(color = "fx") ) + stat_function(fun = gx, mapping = aes(color = "gx") ) + scale_x_continuous(limits = c(-2,5)) + scale_y_continuous(limits = c(-3.5, 8))
## Warning: Removed 10 rows containing missing values (geom_path).

From the graph above we can see that the intercept point values for x are -1 and 4.

fxint <- integrate(fx, lower = -1, upper = 4) 
gxint <- integrate(gx, lower = -1, upper = 4) 
round(gxint$value - fxint$value,1)
## [1] 20.8
Problem 5.

A beauty supply store expects to sell 110 flat irons during the next year. It costs $3.75 to store one flat iron for one year. There is a fixed cost of $8.25 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

knitr::include_graphics("QFormula.png")

Source: https://en.wikipedia.org/wiki/Economic_order_quantity

D<-110
K<-8.25
h<-3.75

#Optimal Lot Size
Q<-sqrt(2*D*K/h)
Q
## [1] 22
#Optimal number of orders per year
D/Q
## [1] 5
Problem 6.

Use integration by parts to solve the integral below.

knitr::include_graphics("13-6.jpg")

Problem 7.

Determine whether f ( x ) is a probability density function on the interval [1, e^6]. If not, determine the value of the definite integral.

knitr::include_graphics("13-7.jpg")