1. \[\int 4e^{-7x} dx\] Set u=−7x and du=−7dx.

\[\int 4e^{-7x} dx = \int\frac{-7 * 4}{-7}e^{-7x}dx \] \[= \int\frac{-4}{7}e^{u}du\] \[-\frac{4}{7}e^u + constant\] \[-\frac{4}{7}e^{-7x}\]

  1. \[\frac{dN}{dt} = -\frac{3150}{r^4}-220\] find a function $ N(t)$ to estimate the level of contaimination if the level after 1 day was 6530 bacteria per cubic centimeter. \(\frac{dN}{dt} = -\frac{3150}{r^4}-220\) \(dN = (-\frac{3150}{t^4}-220)dt\) \(N = \int - \frac{3150}{t^4}-220dt\) \(N = \int - \frac{3150}{t^4}dt - \int220dt\) \(N = -\frac {3150}{3t^3} - 220(1)+C = 6530\) \(C=7800\)

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

Since square has an area of 1 and each rectangle has a width of 1, we get: a1 = 11 = 1 a2 = 13 = 3 a3 = 1*5 = 5 a4 = 1.7 = 7 total area = 1 + 3 + 5 + 7 = 16

  1. Find the area of the region bounded by the graphs of the given equations: \[y = x^2 - 2x - 2, y = x + 2\]

plot the functions

f1 <- function(x) x^2-2*x -2
f2 <- function(x)x+2

curve(f1, -3, 5)
curve(f2, add = TRUE)

find the area that is bound by the 2 intersections:

  1. x is the number of flat irons to order

\[C = 8.25r + \frac{3.75x}{2}\] \[C = 8.25r + \frac{206.25}{n}\] \(c' = 8.25 - \frac{206.25}{n^2}\) \(c' = 0\) \(0 = 8.25 -\frac{206.25}{n^2}\) \(n = 5\)

  2. .Determine whether \(f(x)\) is a probability density function on the interval \([1,e6]\) . If not, determine the value of the definite integral. \(f ( x ) = \frac{1}{6x}\)

\(\int{e^6}{1}f(x)dx\) \(\int{e^6}{1}\frac{1}{6x}dx\) \(\frac{1}{6}\int{e^6}{1}\frac{1}{x}dx\) \(\frac{1}{6}ln(x)|\frac{e^6}{1}\) \(\frac{1}{6}ln(e^6) - \frac{1}{6}ln(1)\) \(\frac{1}{6} * 6 - \frac{1}{6} * 0\)