VStoyanova_Assign13

Homework 13

1. Use integration by substitution to solve the integral below ??? ??????4e-^???7xdx

U=???7x dU=???7dx dx=dU/???7

4???e^{UdU/???7 4/???7???e}UdU 4/???7e^{U+C 4/???7e}???7x+C

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.

dN/dt=3150/t^{4???220 dN=(3150/t}4???220)dt N=???3150/t^{4dt??????220dt N=N0???3150/3t}3???220t N=N0???31503/3t^{3???220t N(1)=N0???1050/1}3???220(1) N0=6530+1050+220 N0=7800 N=7800???1050/t^3???220t

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

f(x) = 2x-9.

A1=(5.5???4.5)???1=1 A2=(6.5???5.5)???3=3 A3=(7.5???6.5)???5=5 A4=(8.5???7.5)???7=7 Area=1+3+5+7=16

Area=???8.54.5(2x???9)dx
Area=(2???x^2/2???9x)|8.5 4.5 Area=(8.5^{2???9???(8.5))???(4.5}2???9???(4.5)) Area=(72.25???76.5???20.25+40.5) Area=16

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

A=???4???1x+2dx??????4???1x^{2???2x???2dx A=1/2x}2|4???1+2x|4???1???[1/3x^3???x2???2x]|4???1 A=???[1/3x3???3/2x2???4x]|4???1 A=[3/2x^2+4x???1/3x3]|4???1

((3/2)*4^2 +4*4 -(1/3)*4^3) - ((3/2)*(-1)^2 +4*(-1) -(1/3)*(-1)^3)

## [1] 20.83333

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.

Let C be cost, r be the number of orders per year, and x be the number of at irons in an order. rx=110 so x=110/r, assume half an order is in storage at on average. Such that,

C=8.25r+3.75x/2 C=8.25r+206.25/r C’=8.25???206.2/5r^2 C’=0 r=sqrt(206.25/8.25) r=5 orders per year

6.Use integration by parts to solve the integral below.

ln(9x). x^{6dx ???ln(9x)???x}6dx U=ln(9x) dU=1/xdx dV=x^{6dx V=1/7x}7 ???UdV=UV??????VdU 1/7ln(9x)x^{7???1/7???x6dx 1/7x}7[ln(9x)???1/7] ???

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.

f (x) = 1/ 6x

F(x)=???e^{61f(x)dx=1 f(x)=1/6x F(x)=???e}61 1/6xdx F(x)=1/6 ???e^6 1 1/xdx F(x)=1/6[ln(x)|e^6 1 F(x)=1/6[ln(e6)???ln(1)] F(x)=1/6[6???0]=1 The sum of the definite integral for the function f(x)=1/6x on the interval [0,e^6] is 1. Therefore, we can say that the function f(x) is a probability density function.