power functions
chapter 9.1


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PART I : power functions

1) A power function has the form: y = kxp.
The area , A(x) , is proportional to the square of its radius x. Find the repression of the function A(x) = ____________. k = ________ and p = _________
A(x) is a power function.
A power function f(x) is a function of the form f(x) = k xp, p and k are constants. 

2) A) The weight , w(x) , of an object is inversely proportional to the square of the object's distance, x, from the earth's center. Find w(x) = _________. p = ________
B) If of an object  the distance is 3959 miles from earth's center, its weight is 44 pounds. Find the constant k = _______

C) Fill the value table:
Use your TI. Enter the function in Y1. Set the TABLE to ASK. (TBLESET function). Then use the key TABLE

d, milesw = f (d), lbs
4000
5000
6000
7000
8000

Use your TI to graph the function: you need to set the right window.

Conclusion:
A quantity y is directly proportional to a power of x if    y = _____, k and n are constants
A quantity is inversely proportional to xn if y = _______, k and n are constants, n >0

3) which of the following functions are power functions ? For each power function, state the value of the constants k and p in the formula y = kxp
see the board.

4) The radius of a sphere is directly proportional to the cube root of its volume. If a sphere of radius 18.2 cm has a volume of 25, 253.4 cm3, what is the radius of a sphere whose volume is 30,000 cm3 ?

5) Consider the family of power functions: y = x2, y = x4, y=x6 ...
A) using your TI , trace y = x2, y = x4
Which one goes down quicker?  _______ Which one flattens more ? ________ As you increase the power, the function goes down even ________ and flattens even ________.
The "long-run"behavior (for x really big + or - ) is the same.
B) predict the graph of y = - x2 . check with your TI.
 
C) predict y = 2 x2   .  Then compare with x2 on the TI (Y1=x2 and Y2= 2x2) what happened ?

D) predict y = 0.5 x2   . Same question. What happened ?
conclusion?

6) Consider the family of function y = x3, y = x5 , y = x7 ...
A) Using your TI trace and compare y = x3, y = x5

B) predict the graph of y = - x3

C) check the graph of  y = - 2 x3

6) The radius of a sphere is directly proportional to the cube root of its volume. If a sphere of radius 18.2 cm has a volume of 25,252.4 cm3, what is the radius of a sphere whose volume is 30,000 cm3 ?

PART II Assignments (p. 381 p. 382)

1) do 1 to 6 (easy)

2) do 14, 15, 16, 17

3) 18, 19,

4) 28

5) 34 Let's do this one together.
The circulation time of a mammal - that is, the average time it takes for all the blood to circulate once and return to the heart - is governed by the equation
t = 17.5 m 1/4
where m is the body mass of the mammal in kilograms. and t is the circulation time in seconds.

A) Complete the table which shows typical body masses in kilograms for various mammals.
Use your TI. ENter the function in Y1 and use TABLE to find the y values.

animalbody mass (kg)circulation time(sec)
blue whale910000
African elephant5450
white rhinoceros3000
hippopotamus2520
black rhinoceros1170
horse700
lion180
human70

B) Compare the mass of an hippotamus and a human, The ratio is ___ : ___
THe mass of an hippo is _____  times the mass of an human.
Compare their circulation time, the ratio is only ___: ____
The circulation time of an hippo. is ___ times the circulation time of an human.
What is the relationship between these 2 factors ?

C) If the circulation time of one mammal is twice of another, what is the relationship between their body masses ?

6) 37
Two oil tankers crash in the Pacific ocean. The spreading oil slick has a circular shape, and the radius of the circle is increasing at 200 meters per hour.

A) Express the radius of the spill, r, as a power function of time, t, in hours since the crash.

B) Express the area of the spill,A, as a power function of time t.

C) Clean-up efforts begin 7 hours after the spill. How large an area is covered by oil at that time?



7) HW: 42 revisited
The force of gravity as expressed by Newton : F(R) = G m M /R2
A) This is the force of gravity  exerted, at a distance, by a planet of radius ____ and of mass ____ on an object of mass_______. G is a constant (9E9). THe force's unit is ___________ (why ?).
Suppose m and M stays constant. If m is the mass of a person, F(R) represent his/her _________.

B) Suppose a man weight's is 180lbs for a given radius of the planet. (his mass m and the mass M of the planet stay constant). How does he weigh on the surface of a planet whose radius is three times as large ? one-third as large ?

C) If his weight is four times as large, the radius has to be ..........................

D) What is the weigh is multiplied by 3 ? What is the relationship between the radius R1 and R2 ?



8) 43
One of Kepler's three laws of planetary motion states that the square of the period , P, of a body orbiting the Sun is proportional to the cube of its average distance , d, from the sun. The earth has a period of 365 days and its distance from the sun is approximately 93,000,000 miles.
A) Find P as a function of d
B) The planet Jupiter has an average distance from the sun of 483,000,000 miles. How long in Earth days is a Jupiter year ?

 

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