Assignment due 04/01
yeny.r
math_assignment.pdf
U-"
I . Find the derivative with respect to x if f(x) : u - r6x.
The derivative with respect to x if f(x) : n - i 6x is f '(x) :
2. Differenriate.
t(x): 15 e x
f'(*):f,
3. Differentiate.
21 q f1x.;: _ u'"ll
f '(x) -f Dif't'erentiate.
f(x):'o*5e7*
f'(*):f Diffbrentiate the following function.
f(x) : 'e "7x r'(*):I Dilferentiate.
-tr e
F(x): *x"
r'(*):I Find f '(x).
f{*) : e,'l;r8
4.
5.
6.
7.
f'(*):I
Page I
Assignment: HW 3.1
G lrffi
# Cr-HrrEC {-
Assignment: HW 3.1
dy
-:t
I
dxll
Page2
%.{flTIFE 8. Diff-erentiate.
Graph the function. Then determine critical values, inflection points, interval(s) over which the fixrction is increasing or decreasing, and the concavity.
f(x) : e a'
Choose the correct sketch of the given function.
l}n. Ce tc.
List the cntical values of f(x). Select the correct choice below and fill in any answer boxes within your choice.
CA I (t*" an ordered pair. Use a comtna to separate answers as needed.) There are no critical points.
List the inflection points of t(x). Select the correct choice beiow and iill in any answer boxes within your choice.
SA I (T*" an ordered pair. Use a comma to separate answers as needed-) i:.) B There are no inflection points.
Over what interval(s) is the firnction increasing? Select the correct choice below and fill in any answer boxes within your choice.
r'LA. ffi'- Bdls (Type your answer in interval notation" Use a comma to separate answers as needed")
r,_'rB. The function is never increasing.
Over what interva(s) is the function decreasing? Select the correct choice below and fill in any answer boxes within your choice.
Page 3
ifG. Assignment: HW 3.1
9.
9.
(cont.)
raA.
(Type your answer in interval notation. Use a comma to separate answers as needed.)
t.-i B. The function is never decreasing.
Over what interval(s) is the function concave up? Select the correct choice below and fiil in any answer boxes within your choice.
i--.:A. ffi (Type your answer in intewal notation. IJse a comma to separate aflswers as needed")
1--.)8. The fimction is never coucave up.
Over what interval(s) is the function concave down? Select Xhe correct choice below and fill in any ansqrer boxes within your choice.
r:A. ffi (Type your answer in interval notation. Use a comma to separate answers as needed.)
i _.i B. The function is never concave down"
Page4
ffi Assignment: HW 3.1
-F
rc,
Graph the function. Then determine critical values, inflection points, interval{s) over which the fi.urction is
increasing or decreasing, and the concavity'
1 I
tix): - e -x9 Choose the correct sketch of the given function
Cn. f,s.
List the critical values of f(x). Select the correct choice below and fill ia any answer boxes within your choice.
CA. I G*" an ordered pair. Use a coruna to separate answers as needed.) There are no critical Points.
List the inflection points of f(x), Select the correct choice below and fill in any answer boxes within your choice.
OA. I O*" an orderedpair. Use a cofiuna to separate answers as needed.)
i._.lB There are no inflection points.
Over what interva(s) is the function increasing? Select t}re correct choice below and fill in any answer boxes within your choice.
i.:jA ffi (Type your answer in interval notation. Use a comma to separate answers as needed")
i-.i n. The function is never increasing.
Over what intewal(s) is the function decreasing? Seiect the correct choice below and fill in any answer boxes within your choice.
Page 5
G Assignment: HW 3,1 -- -rr;;;-
10.
(cont.) cA.
(Type your answer in interv-al notation. tJse a comma to separate allswers as needed.)
i.-:rB The function is never decreasing.
Over what interval(s) is the function concave up? Select the correct choice below and fill in any an$wer boxes within your choice.
,-A. ffi ffi (Type your answer in interval notation. Use a comma to separate answers as needed')
{.-.r8. The function is never concave up.
Over what interva($ is the function conaavo down? Select the correct choice below and fill in any ailswer boxes within your choice.
i-A. ffi ffiir (Type your answer in interval notation. Lise a comma to separate answers as needed.)
i.-.i8. The function is never concave down.
11" Find the slope of the tangent line to the curve {x): e * at { * 1.8,0.17)'
The slope of the tangent line is [. (Round to two decimal places as needed.)
Page 6
fl-lrt.t
Use a calculator to find the natural logarithm, base e.
ln 0.0704
In0.0704: I (Round to four decimal places as needed.)
Solve e 2t:800 for t.2.
a 3.
4.
5.
t * [ (Round to three decimal places as needed.)
Use natural logarithms to solve the equation.
e - o.7ttt _ 4
Differentiate.
f(x): 15 lnx
r'(r) : I Difl-erentiate.
_l v . *7ln x -- -x4"4
dy_ dx
Differentiate.
f(x):ln(llx)
r'(*) : I
The solution is t: []. (Simplify your answer. Type an integer or a decimal. Round to the nearest thousandth as needed.)
6.
Page 1
T
Differentiate.
lnx Y:--r1x'"
dy* dx
Differentiate.
y: In (2*2 - 5x + 1)
Y,:T
g. Differentiate.
f(x): ln(e8*+3)
8.
10.
11.
r'(*) : tr Differentiate. (Hint: Use the Extended Power Rule.)
e(x): (tnx)t
s'(*):il
Find the equation ofthe line tangent to the graph of f(x) : (ln x)5 at x : 8.
v:n (Type your answer in slope-intercept form- Do not round until the final answer. Then round to two
decimal places as needed.)
Page 2
Assignment: HW 3.2
lZ. A model for consumers' response to advertising is given by the equation N(a) : 2400 + 550In a
where N(a) is the number of units sold, a is the amount spent on advertising, in thousands of dollars,anda>1.
a) How many units were sold after spending $1,000 on advertising'?
units were sold after spending $1,000 on advertising.
b) Find N'{u).
N'(u) : I Find N'(10).
N'(10) : [] ""iu per $1,000 spent on advertising (Simpliff your answer.)
c) Find the maximum value, if it exists. Select the correct choice below and filIin any answer boxes within your choice.
i.- rA' The maximr.r:n value is ffi units. There is no maximum value.
Find the minimum value. if it exists. Select the correct choice below and frll in any answer boxes within your choice.
i.-l,a The minimum value is ffi "*n. There is no minimum value.
d) Find "lgN'(a).
Does it make sense to spend more and more dollars on advertising'l
,l_EN'{r) : 550, so it does make sense. The derivative is always positive as a increases, so
the sales, and consequently profit. will continually increase.
iimN'(a) : 0, so it does not make sense. Eventually, the prof,rt from the increase in sales a-+s will be less than the amount spent on advertising to achieve that increase.
Page 3
Assignment: lIW 3.3 - 3.4
Suppose that Ps is invested in a savings account in which interest is compounded continuously at 6.5Yo per year. That is. the balance P grows at the rate given by the following equation.
dP 0.065P(r)
dr
(a)Find the function P(t) that satisfies the equation. Write it in terms of P6 and 0.065. (b)Suppose that $ 1000 is invested. What is the balance after 1 years? (c)When will an investment of $1000 double itself.r
(a) Choose the correct answer below.
P(r):Po"o'065t
P(t; : 6.s.tno, ' Ps: P(t) e o'065t
P(t): P(t) ' o'oost
(b) The balance after I year is $ (Type an integer or decimal rounded to two decimal places as needed.)
(c) The doubtingtime is I ,.*. (Type an integer or decimal rounded to two decimal places as needed.)
Page I
Assignment: HW 3.3 - 3.4
A ship carryhg 1000 passeugers is wrecked on a snall island from which the passengers are never rescued. The natural resources of the island restrict the population to a limiting value of 5820, to which the population gets closer and clossrbut which it never reaches. The population of the island after time t, in years, is approximated by fte logistic equation given belour. Complete parts (a) through {c).
5820 P(t):
I +4.82n -o5t
2.
a) Find the population after 0 years.
(itourirl to the nearcsi lntcger as nceded. i
b) Find the rate of change, P'(t) .
(t+4.82e -o's\2
ru,A26.20e -o'5t
29,052.40 - i)'5I
(t +5820, -o's1z
14,A26.20€ -B'st
(l +4.82, -o's12
29,052.40 - 0'51
(t +4.g2 u -ojt12
c) Sketch a graph of &e fuuctiom. Choose the correct Saph below.
Ca. Cc. *o*a.
Page 2
ffi
Assignment: IIW 3.3 - 3.4
Pharmaceutical firms invest significant money in testing any new medication. After the drug is
approved for use, it still rakes time for physicians to fully accept and start prescribing the
medication. The acceptance by physicians approaches a limiting value of 100%, or 1, after time t, in months" Suppose that the percentage P of physicians prescribing a new cancer medication is
approximated by the equation below. Complete parts (a) through (c).
P(0:100(l - n -o'36t1
a) What percentage of doctors are prescribing the medication after 16 months?
o//o
1I)o nor lound unlil thr ltnal ausu,er. Then rouncl to titr ttcaresi lLrnth as rl*edcd^ i
b) Find P'( l6), and interpret its meaning.
P'(16): (Tlo not ro*yiil untjl the linal answer. Thrn routtd t* th* nearest lenth ;:s rreeded.)
Choose the correct interpretation below.
At l6 months, 0.1% of doctors are prescribing the medication.
At t6 months, the percentage of doctors who are prescribing the medication is decreasing by 0.1% per month.
At 16 months, 0.l7o of doctors are not prescribing the medication.
At 16 months, the percentage of doctors who are prescribing the medication is increasing by 0.1% per month.
c) Sketch a graph of the function. Choose the correct graph below'
f,n *o.Ce LJ U.
Page 3
a J.
Assignment: HW 3.3 - 3.4
4. A chemical substance has a decay rute of 6.70/o per day. The rate of change of an amount N of the chemical is given by the equation
dN - 0.067 N
dt
where t is the number of days since decay began. Complete parts a) through c).
a) Let No represent the amount of the chemical substance at t: 0. Find the exponential function that models the situation.
N(t)=Nnil
tr) Suppose that 400 g of the chemical substance is present at t: 0. How much will remain after 6 days?
After 6 days, g will remain. (Round to the nearest wh*le ttun"lber as needcii" )
c) After how many days will half of the 400 g of the chemical substance remain?
After days, half of the chemical substance will remain. (iiound [r] one dccirnal pl*cc as neerled")
5. The decay rate, k, for a particular radioactive element is 2.8t/o, where time is measured in years. Find the half:life of the element.
The hall:-life is years. (RounC to oue clecimal;rlace a-r needed.i
6. The amount of carbon-I4 present in animal bones t years after the animal's death is given by P(t) : Po e - 0'000120e7t. How old is an ivory tusk that has lost 38% of its carbon-l4?
The ivory tusk is years old. (liound to the te arest intcger ;is ne e de d. )
7. Find the present value of $3000 payable at the end of 2 years, if money may be invested atTo/o with interest compounded continuously.
The present value of $3000 is $[. 6oond to the nearest cent as needed-)
Page 4
Assignment IIW 3.3 - 3.4
How much money must you invest now at 4.9o/o rnterest compounded continuously in order to have $ 10,000 at the end of 6 years?
You must invest $[. (Rounrl 1o the nearest ccnL as ncedcd.)
9 . The sales, S, of a product have declined in recent years. There were 203 million sold in i 984 and 1.2 million sold in 1994, Assume the sales are decreasing according to the exponential decay model, S(t):Soe -kt. a) Find the value k and write an exponential function that describes the number sold
after time, t, in years since 1984. b) Estirnate the sales of the product in the year 2002. c) In what year (theoretically) will only I of the producr be sold ?
a) Rounded to six decimal places, k:
b) To the nearest unit, the total is
c) Only one unit willbe sold in the year (Round lo lhe irearesl vcilr as irr:r":dr:{i.)
10. A coroner arrives at a murder scene at 9 p.m" She tinds the temperafure of the body to be 86.3"F. She waits one hour, takes the temperature again, and finds it to be 82.4"F. She notes that the room temperature is 72"F. Assuming that the temperature of the body was 98.6"F when the murder occured, when was the murder committed?
The murderwas commited at approximately approximatefy I la.m.
,
,r* : (Do not round until the final answer. Then round to the nearest hour.)
i--lil
Page 5
