Functions/Equations/Calculus

A geologist you spoke with is concerned about the rate of land erosion
around the base of a dam. Another geologist is studying the magna
activity within the earth in an area of New Zealand known for its
volcanic activity. One of the shortcuts they apply when doing
calculations in the field is to use synthetic division. After they
explain the concept and process to you, you decide to try and solve some
problems.

Use synthetic division to determine if the first set of numbers are
zeros of the given polynomial

a. -3, 2. f(x) = 3x3 + 5x2 - 6x + 18.

a. -4, 2. f(x) = 3x3 + 11x2 - 2x + 8.

3 5 -6 18

↓ -9 +12 -18

3 -4 +6 0 = 0 – a root

3 5 -6 18

↓ +6 +22 +32

3 +11 +16 +50 = +100 – not a root

1b. f(x)=3x^3=11X^2-2x+8

3 11 -2 8

↓ -12 4 -8

3 -1 2 0 = 0 – a root

3 11 -2 8

↓ 6 34 64

3 17 32 72 = 144 – not a root

Given the polynomial f(x) = 2x 3 -5x2- 4x + 3, find all the solutions if
the function is completed.

a) f(x) =0

b) f(x+2)=0

d) f(2x) = 0

For this problem I have chosen to rational root test

They are the following

a.

f(x) = 0

2x^3-5x^2 – 4x+3=0

Possible solutions +1, + 1/2 , +3, + 3/2

Answer Ѕ, -1, 3)

f(1) = 2(1)^3-5(1)-4(1)=3 = 0?

= 2 -5-4+3 = -4 – thus not a solution

f(1) = 2(-1)^3) -5(-1)^2 -4(01)+3 = 0?

= 2-5+4+3=0

0=0 – thus a solution

f(1/2) = 2(1/2)^3) – 5(1/2)^2) -4(1/2) +3 = 0 ?

= 2/8 – 5/4 – 4/2 = 3 = 0 ?

= ј - 5/4 – 8/4 + 12/4 = 0

0 = 0 – thus a solution

f (-1/2) = 2 (-1/2/)^3) – 5 (-1/2/)^2) -4 (-1/2) + 3 = 0?

= -2/8 – 5/4 +2 + 3 = 0?

-1/4-5/4+8/4+12/4= 0?

= 14/4 - thus not a solution

f (3) = 2(3) ^3 – 5(3)^2 – 4(3) +3 = 0?

54 – 45-12+3 = 0?

0=0 – thus a solution

f(-3) = 2(-3)^3 -5(3)^2 – 4(-3) =3 = 0?

2(-27) -5(9) +12+3 =0?

-54-45+12+3+0?

-84 – thus not a solution

f(3/2) = 2 (3/2)^3 -5(3/2)^2-4(3/2)+3=0?

2(27/8)-5(9/4)-12/2+3=0?

27/4-45/4-24/4+12/4=0?

-30/4 – thus not a solution

f(-3/2) = 2 (-3/2)^3 – 5(3/2)^2-4(-3/2)+3 = 0?

2(-27/8) – 5(9/4)+12/2+3=0?

-27/4-45/4+24/4+12/4=0?

-36/4=-9 – thus not a solution

2b – f(x+2) = 2(x+2)^3 -5(x+2)^2 -4(x+2)+3

=2(x^3+6x^2+12x+8)-5(x^2+4x+4)

-4x-8+3=0

=2x^3+12x^2+24x+16=5x^2-20x-20

-4x-8+3=0

f(x+2)=2x^3+7x^2-9

Possible roots – Ѕ, 9/2, 3/2, 1

Answer 1, -3/2 , 3)

f(1/2) = 2 (1/2^3)+7(1/2^2-9

=2(1/8)+7(1/4)-9=-7 – thus not a solution

f (-1/2) =2 (-1/2)^3 +7 (-1/2)^2 – 9

=2 (-1/8) +7(1/4) -9

= -1/4 + 7/4 – 36/4 = -30/4 – thus not a solution

f (9/2) = 2 (9/2)^3 =8(92/) – 9 = 0?

= (729/8) +7(81/4) – 9

= 315 – thus not a solution

f(-9/2) = 2(-9/2)^3 =7(-9/2^2 – 9 = 0?

= 2(-729/8) = 7(81/4) – 9

-162/4 -9

-49.5 – thus not a solution

f(3/2) = 2(3/2)^3 = 7(3/2)^2 – 9=0?

2(27/8) +7(9/4) -9

27/4+63/4-9

=13.5 – thus not a solution

f(-3/2) = 2(-3/2)^3 = 7(-3/2)^2 – 9 = 0?

2(-27/8) +7(9/4) – 9

-54/8+63/4-9

-27/4+63/4-36/4

=0 – thus a solution

f(1) = 2(1)^3 +7(1)^2-0 = 0?

2+7-9

=0 – thus a solution

f(-1) =2 (-1)^3 +7(-1)^2 -9 =0?

-2+7-9

= -4 – thus not a solution

2c. f(2x) = 0

F(x) = 2x^3-5^2-4+3

F(2x)=2(2x)^3-5(2x)^2-4(2x)+3

=16x^3-20x^2-8x+3

Possible roots 3,3/16,3/2,3/8,3/4,1,1/16,1/2

Answer 3/2,-1/2,1/4

f(3) = 16(3)^3-20(3)^2-8(3)+3=0?

432-180-24+3

= 231 – thus not a solution

f(-3) = 16(3)^3+-20(3)^2-8(-3)+3=0?

-432-180+24+3

=585 – thus not a solution

f(3/16) = 16 (3/16/)^3-20(3/16/)^2-8(3/16)+3 =0?

16(27/4096)-20(9/256)-3/2+3

.1055-.7031-1.5+3

.9024 – thus not a solution

f(-3/16) = 16(-3/16)^3-20(3-3/16)^2-8(-3/16)+3=0?

16(-27/4096)-20(9/256)+3/2+3

-.1055-.7031+1.5+3

=3.69 – thus not a solution

f(3/2) =16(3/2)^3-20(3/2)^2-8(#/2)+3=0?

=54-45-12+3

=0 – thus solution – root

f(-3/2) = 16(-3/2)^3 -20(-3/2)^2-8(-3/2)+3=0?

16=(-27/8)-20(9/4)+12+3

-54-45+12+3

= -84 – thus not a solution

f(3/8) = 16(3/8)^3-20(3/8)^2-8(3/8)+3=0?

16=27/512-20(9/64)-3+3

.8438-2.8125-3+3

=-1.9687 – thus not a solution

f(-3/8) = 16(-3/8)^3-20(-3/8)^2-8(-3/8)+3

=-.8438-2.8125+3+3

=2.3437 – thus not a solution

f(3/4) = 16(3/4)^3-20(3/4)^2-8(3/4)+3=0?

6.75-11.25-6+3

=-7.5 – thus not a solution

f(-3/4) = 16(-3/4)^3 -20(-3/4)^2-8(-3/4)+3=0?

-6.75-11.25+6+3

=-9 – thus not a solution

f(1) = 16(1)^3-20(1)^2-8(1)+3=0?

=16-20-8+3

-4-8+3

=-9 – thus not a solution

f(-1) = 16(-1)^3-20(-1)^2-8(-1)+3=0?

-16-20+8+3

=-25 – thus not a solution

f(1/16) = 16(1/16)^3-20(1/16)^2-8(1/16)+3=0?

16/4096-20/256-1/2+3

.0039-.0781-1/2+3

=2.4258 – thus not a solution

f(1/2) = 16(1/2)^3-20(1/2)^2-8(1/2)+3=0?

16(1/8)-20(1/4)-4+3

2-5-4+3

-3-4+3

=-4 – thus not a solution

f(-(1/2) = 16(-1/2)^2-20(-1/2)^2-8(-1/2)+3=0?

-2-5+4+3

=0 – thus a solution

Today you visited the R & D (Research and Development) department of a
pharmaceutical manufacturer. The people in this lab have to calculate
the recommended dosage of the products they are developing. Medicine is
expressed as a function of the concentration in parts per million, with
the potency of the dosage declining over a period of time.

The function N(t) = (0.8t + 1000 ) / (5t+4), where t=>15 gives the body
concentration N(t), in parts per million, of a certain dosage of
medication after time t, in hours.

Find the horizontal asymptote of the graph and complete the following
statement: N(t) approaches ? as t approaches infinity.

Explain the meaning of the answer to a) in terms of the medical dosage
application.

Horizontal asymptote:

R(x) = p(x)/z(x)

N(t) - .8t+1000/5t+4

If degreeo f p(x) equals degree of q(x) then r(x) has horizontal
asymptote of y=an/bn=.8/5=.16



Thus as t increases the body concentration of the drug approaches .16.

A retailer you spoke with in New York City's fashion district imports
haute couture from European designers. One of the accommodations which
must be considered when importing fashion from other countries is the
difference in the size charts. A function that will convert dress sizes
in the United States to those in Italy is

g(x) = 2(x+12).

Find the dress sizes in Italy that correspond to sizes 6, 10, and 14 in
the United States

Y=2(x+12)

Y=2x+24

U.S. dress size Italy dress size

6 2(6)+24 = 36

10 2(10)+24 = 44

14 2(14)+24=52

Find a formula for the inverse of the function.

Y=2x+24

Then x=2y+24

x-24+24

x-24/2+y

f-1(x)=x-24/2

$

@

B

d

¤

Ё

є

ј

в

д

ж

и

&

(

N

P

R

T

B

¤

¦

Ё

x

є

 

¦

Ё

®

І

ґ

ё

є

Ь

Use the inverse function to find the dress sizes in the United States
that correspond to 36, 44, and 60 in Italy.

Italy dress size Italy dress size

36 36-24/2 = 6

44 44-24/2 = 10

60 60-24/2 = 18

Last week's profits from a dry cleaners was $2000. Suppose the $2000 is
invested at interest rate k, compounded continuously, and grows to
$2983.65 in 5 years.

What is the interest rate? 8%

A=pert

2983.65 = 2000 e^5t

2988.65/2000=e^5t

1.4918=e^5t

ln1.4918=ln e^5t

.4=5t

.08=t

8% interest rate

Find the exponential growth function. A=Pert

What will the balance be after 10 years? $4451.08

A=(2000) e^.08(10)

A=2000 e^.8

=2000(2.2255) = $4451.08

When will the $2,000 double? – 8.6643 yrs.

4000 = 2000 e^.08t

2= e^.08t

ln2=ln e^.08t

.6931=.08t

.6931/.08=t = 8.6643 yrs