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GROUP WORK 1, SECTION 3 1

Doing a Lot with a Liftle

Section 3.1 introduces the Power Rule: 4rn : hx'-r,where n is anyreal number. The good news is thatdxthis rule, combined with the Constant Multiple and Sum Rules, allows us to take the derivative of even themost formidable polynomial with ease! To demonstrate this power, try problem 1:

1. A formidable polynomial :

"f (x) :x10 + t*' + i*t - 5x7 - 0.33x6 * rxs - Jl*a - +z

Its derivative:

f' (r) :

The ability to differentiate polynomials is only one of the things we've gained by establishing the power Rule.Using some basic definitions, and a touch of algebrii, there are all kinds of functions that can be differentiatedusing the Power Rule.

2. AII kinds offunctions:

.f (x): iF + i/2 g(x): + - ,L h @):xs -t'E +2x3 {7 " \-'l

"t,Their derivatives:

-f' (*) : g, (x) : ht (x) :

Unfortunately, there are some deceptive functions that look like they should be straightforward applicationsof the Power and constant Multiple Rules, but actually require a little thought.

3. Some deceptive functions :

f (x) : (2x)a s (x) : (x3)sTheir derivatives:

f'(*): g'(x):

t54

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!

Doing a Lot with a Little

The process you used to take the derivative of the functions in Problem 3 can be generalized. In the first case,

-f (x) : (2*)4, we had a function that was of the form (kx)", where k and,n were constants (k : 2 and

n : 4).In the second case, g (x) : ("3)s, we had a function of the form (xk)". Now we are going to find a

pattem, similar to the Power Rule, that will allow us to find the derivatives of these functions as well.

4. Show that your answers to Problem 3 can also be written in this form:

.f' (x) :4(2r)3 '2 g' (x) : s (rt\o .l*'-Y",t

And now it is time to generalize the Power Rule. Consider the two general functions, and try to find €xpros:

sions for the derivatives similar in form to those given in Problem 4. You may assume that n is an integer.

5. Two general functions'.

f (x): (kx)n g(x): (xk)'Their derivatives:

.f' (x) :g'(x) :

155

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I

GROUP WORK 1, SECTION 3.2

Back and Forth (Form A)

Compute the following derivatives. Write your answers at the bottom of this sheet, where indicated. When

finished, fold the top of the page backward along the dotted line and hand to your parhrer.

Do not simplify.

1.f(x):5x4+)x2-+

2. s @) :2\E - 4ffi

3. h (x) :2xd

xa-4x+34. ; (;; : ------;--.-=;-

e"+l

ANSWERS

f' (x) :

g' (x) :

h'(x):

j' (x) :

kt (x) :

161

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GROUP WORK 1, SECTION 3.?

ack and Fortft {form B}

Compute the foltrowing derivatives. Wrile your an$^/ers at the bgttom of this sheet, wbere indicated. When

finishe4 fold the top of the page backward along the dotted line and hand to your partner.

Do rrot simpli$.

1, f (r): -2x3 + $r2 - S

xd +6xzzs@): s

g. h (x): (r3 + xz + 2x) {bc2 - 2x4 + sx)

d4.j(x): #

5. &(x) : ^fr -22{

ANSWERS

f' (x):

d &):

h'{x}:

j'tx):

l{ (x\:

lm

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t

GROUP WORK 2, SECTION 3.2

Sparse Data

Assume that f(x) and g (x) aredifferentiable functions about which we know very little. In fact, assume that

all we know about these functions is the following table of data:

This isn't a lot of information. For example, we can't compute f' (3) with any degree of accuracy. But we

are still able to figure some things out, using the rules of differentiation.

1. Let h (x) : e' f (r). What is ht (0)?

2. Let j (x) : -a"f @) g (x). What is 7'(1)?

3. Let fr (l : !IP. What is k' (-2)?s (r)

5.Letm(-): *t

What ismt(t)?

4. Let t (*) : *3 g (x). If lt (2) : -48,what is g' (2)?

163

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x f (x) f' (x) g (x) g'(x)

-21

0I2

J

-95

J

-5

1

79

_J

J

-54928

8

1

-36

?

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GROUP WORK 1, SECTION 3.3

The Magnilicent Six

The derivative of f (x) : sinx was derived for you in clads. From this one piece of information, it is possible

to figure out formulas for the derivatives of the other five trigonomekic functions. Using the trigonometric

identities you know, compute the following derivatives. Simplify your answers as much as possible.

1. (sin-rr)' :

2. (cosx)' :

3. (tanx)t :

4. (secx)/ :

5. (cotx)' :

167

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GROUP WORK 1, SECTION 3.4

Unbroken Ghain

For each of the following function, of r, write the equation for the derivative function. This will go a lot more

smoothly if you remember the Sum, Product, Quotient, and Chain Rules... especially the Chain Rule! Please

do us both a favor and don't simplify the answers

1. f (x): sin 3x .f' (x) :

2. S @): (sin 3-r)3 g' (x) :

3. h (x): (sin3x)3 + 5:r

4. j (x): [1sin:")3 + 5r]5

h' (x):

j' (x) :

1s.k(,v):x*l k'(x)=

6./(x):l;4 l' (x) :

174

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m' (x):[{sin:x)3 + sx]5x

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GROUP WORK 2, SECTION 3 4

Ghain Rule Without Formulas

Consider the function s f and.g given by the following graph:

Defineh:fog.1. Compute h'(l).

2. Compute h'(0).

3. Does ht (2) exist?

175

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1

4ii ]-- -l--"--li1:

ii

I :\:t, u; 1

t:-|- i -ri

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GROUP WORK 1, SECTION 3.6

Logarithmi c D iflerentiation

Let f (x): (x * 1)r. As of now, none of the rules that we've leamed seem to help us find ft (x).

1. Why can't you use the Power Rule to compute f' (*)?

2. Why can't you use a formula like ftt. :3' ln3 to compute -f' (")?

3. Start with the equation y : (x + 1)'. Take the logarithm of both sides to get lny : ln ((x + 1)"). Then

use a property of logarithms to help you find the derivative of ln ((x + 1)").

4. Now use implicit differentiation to find *r"r.

5. Since lny ' h ((x + 1)'), you can equate your answers to Parts 3 and 4. Do so, and then replace all y

terms by (.x * 1)'. Why can you do this substitution?

dv6. Perform some algebra to get f alone on the left hand side, and you are done!

191

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Losarithmic Differentiatiofl

7.[-nt g (x) : d ln(-r+l). What is the relationship between f (x) and,g (x)?

s. If g (x) : g'ls('+l), compute d (x). What techniipes are you using in this case?

9. Since you have done the same problem in two different wajs, show that your answers to problems 6 and8 are identical.

t0. Let y : f (r) be implicitly defiired by xshr : .yt*'. Compute y/ in terms of x and y. (HINT Canlogarithms help you?)

1V.

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GROUP WORK 1, SECTION 3 B

The Rule of 72

In this exercise, we attempt to answer the question asked by many investors: 'oHow long is it going to take for

me to double my money?'1

1. Consider an investment of $ 100 invested at 5o/o, compounded continuously. How long would it take forthe investor to have $200?

2. What would the doubling time be if the initial investment were $1,000? $i0,000? What effect does

changing the principal have on the doubling time, and why?

One of the first things that is taught in an economics class is the Rule of 72.It can be summarized thusly:

"The number of years it takes an investment to double

is equal to 72 divided by the annual percentage interest rate."

3. What would the Rule of 72 say the doubling time of a 5olo investment is? Is it a good estimate?

4. Repeat Problems 1 and 3 for investments of 3oh, 80 , l2oA and l8o/o. What can you say about the accuracy

of the Rule of 72?

5. Derive a precise formula for the time T to double an initial investment.

6' There is an integer that gives a more accurate answer for continuous or nearly continuous compounding

than the Rule of 72. What is this number? Check your answer by using it to estimate the doubling time ofa 50% investment.

7. It tums out that there is a reason that we use the number 72 in the Rule. It has to do with one of the

assumptions we made. Why do economists use the Rule of 72?

201

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GRoUP \|*HjEcTr0N 3 8

A patient is brought to a hospitai and grven a sedative to help him sleep. The doctor wants to operate, but

cannot safely do so until the concenhation of sedative in his body is'less than 0.03 milligrams/liter. ln this

exercise, we will determine how many hours the doctor nnrst wait until he can operate.

The following table of data was obtained by monitoring the levels of the sedative in the patient's blood.

Samples were taken every ten minutes, and the concentration of the drug was determined and reported in

milligrams per liter.

Time (minutes) Concentration (mdl)

0

10

20

30

4A

50

60

7A

80

90

10.0

6.47

3.68

2.23

1.35

0.820

0.498

43020.183

0.111

l. Is the rate of change of concentration with respect to time a constant?

2- Estimate the rate of change of concentration with respect to time at 30 minutes, 60 minutes, and

80 minutes.

, 3. Show that the rate of change is (roughly) proportional to the concentration C. Write this relationship as a

diferential equation fot dC/dt-

{. Find the constant of proportionality-

M.

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