Project 1. (3 Points) Recognizing and Correcting Common Algebra Mistakes

Start Friday, September 7, Due Wednesday, September 19

As a tutor for an algebra class, you find that your students have made some errors, so you try to understand what they did and show them the correct way to do their problems.

1. __1 + x__ =
__1 + x__ =

x + x^{2} ~~x~~ + x^{2} x^{2}

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

______

2. Ö 3^{2}
+ 4^{2} = 3 + 4 = 7

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

3. 4(11 – 2)^{1/2}
= (44 – 8)^{1/2} = 36^{1/2} = 6

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

4. (x + 5)^{2}
= x^{2} + 25

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

5. x^{2} –
7x + 12 = (x – 2) (x – 6)

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

6. Let f (x) = x^{2},
then 3f (2) = f (6) = (6)^{2} = 36^{}

Write a sentence to describe the mistake: ______________________________________________________

Work out correctly and explain briefly:

Project 2. (3 Points) World Population Growth and Exponential Growth

Start Wednesday, October 24, Due Monday, November 5

Please follow the directions carefully and answer all questions. Label and show the scale for each axis of the graphs in a) and c), using even, linear scales.

World
Population Relative Rate of
Growth **a**) Graph
population by year

Year (millions) (percent/year) for 1650 to1999. Use a

1650 550 uniform (an even) scale for years.

__ .28__ Start
the population scale

1750 725 (also uniform) at 0.

__ __ **b**)
If P = P_{o}e^{kt} models

1850 1175 population growth between

__ __ between
two successive years

1900 1600 (exponential growth), then

__ __ k
= (1/t) ln (P/P_{o}) where

1950 2556 P_{o}
= population in the earlier year

__ __ P
= population in the later year

1980 4458 k = relative rate of growth of world

__ __ population per year (k constant)

1999 5996 t = number of years between the two

successive dates

(Source: World Almanac 2000) Use the formula to fill in the missing

values of k (converted to % per year),

the relative exponential rate of growth between each pair of years. The first value is shown. If growth were exponential for the whole period, the values of k would be about the same. Are they?

**c**) Make a separate graph of the relative rate of
population growth in percent per year for 1650 to 1999. Use a uniform (an even)
scale for years. Start the population relative growth rate scale (also uniform)
at 0. Plot each growth rate halfway between the years to which it applies, that
is, the first point on your graph should be (1700, .28).

**d**) What would the world population have been in 1980
if it had continued to grow exponentially at the relative rate of .28% per year
from 1650 to 1980? (Convert .28% to a decimal to use it in a formula.)

**e**) Between 1650 and 1980, did the relative rate of
population growth increase, decrease, or was it constant? (Look at your graph for
c)).

**f**) During that period, from 1650 and 1980, did
population grow exponentially, faster than exponentially, or slower than
exponentially? (For example, was world population in 1980 the same, more or
less than you calculated in part d)?)

**g**) What is the doubling time for world population for
the relative rate of growth of .28% per year?

**h**) What is the doubling time for world population for
the relative rate of growth you found for 1980-1999? With this doubling time,
how many times would population double during the lifetime of a person born in
1980 who lives for 90 years?

Project 3 Models with Sine and Cosine Functions (3 Points)

Start Friday, November 9, Due Monday, November 19.

Please follow the directions carefully and answer all questions.

Label and show the scale for each axis of the graphs.

(Source: Larson, Hostetler, Edwards, *Brief Calculus with
Applications*)

For each model given, **a)** graph the function for **3**
cycles, **b)** give the maximum and the minimum value, period, frequency,
amplitude, and center value (average of maximum and minimum) for each function.
Give word descriptions of the variables and the **units of measurement**, if
specified, in the labels of your graphs and in your answers for b).

- Respiratory Cycle: The velocity v in liters per second of air flow into and out of the lungs is

v = .8 sin (2πt/5), where t is time in seconds, v > 0 when air is being inhaled, v < 0 when air is being exhaled. In addition to giving the frequency in terms of cycles per second, also give the frequency in cycles per minute.

- The sine wave for a tuning fork, which sounds A below middle C, is given by

y = .001 sin (440 πt).

- Blood pressure in millimeters of mercury at time t in seconds is given by

P = 110 – 25 cos (11 πt/6). Give the frequency in terms of heartbeats per second and

also in heartbeats per minute. (Remember that taking the negative of a function

reflects it in the x axis, and adding a constant moves the graph upwards.)

- Predator-Prey Cycle: The population P of a predator species is given by

P = 5800 + 1000 sin (πt/12), where t is time in months. The population P of a prey

species is P = 9500 + 2800 cos (πt/12). Graph both models on the same axes. When

the predator population is highest, is the prey population increasing or decreasing?

When the prey population is highest, is the predator population increasing or

decreasing? (Again remember that adding a constant to a function moves the graph

upwards.) (Note the different trigonometric functions in the two population

functions.)

Project 4 Pythagorean Triples (3 Points)

Start Wednesday, November 21, Due Wednesday, December 5.

Please read the description and example, then follow the directions at the bottom of this page carefully and answer all questions.

If three positive numbers a,b,c satisfy the equation a^{2}
+ b^{2} = c^{2}, a triangle with sides of length a,b,c forms a
right triangle. Certain positive integers (whole numbers) satisfy the equation
a^{2} + b^{2} = c^{2}, for example a=3, b=4, c=5 and
a=20, b=21, c=29. A right triangle with sides whose lengths are integers is an
integer right triangle.

In an integer right triangle, the sines and cosines of the acute angles are positive rational numbers (fractions with positive integers). If both the sine and cosine of an angle are positive rational numbers, then there is an integer right triangle with that angle.

For example

Triangle 1

β 5 sin
α = 3/5 sin β = 4/5 α = sin^{-1}(3/5) ≈
36.9^{○}

3 cos
α = 4/5 cos β = 3/5 β = sin^{-1}(4/5) ≈
53.1^{○}

α

4

Triangle 2

t 29 sin
s = 20/29 sin t = 21/29 s = sin^{-1}(20/29) ≈ 43.6^{○}

20 cos s = 21/29 cos t = 20/29 t
= sin^{-1}(21/29) ≈ 46.4^{○}

s

21

If two angles from integer right triangles, which have rational number sines and cosines, are added or subtracted, the sine and cosine of the resulting angle (calculated from the formulas for the sine and for the cosine of a sum or difference of angles) are rational numbers. If the new sine and cosine are positive, a new integer triangle can be formed.

For example, add angle α from Triangle 1 and s from Triangle 2

sin(α + s) = sin α cos s + cos α sin s = (3/5)(21/29) + (4/5)(20/29) = 143/145

cos(α + s) = cos α cos s - sin α sin s = (4/5)(21/29) – (3/5)(20/29) = 24/145

This gives the new integer triangle below, where m = α + s

Triangle 3

n sin m = 143/145 sin n = 24/145 m = sin^{-1}(143/145) ≈ 80.5^{○}

143 145 cos m = 24/145 cos n = 143/145 n = sin^{-1}(24/145) ≈ 9.5^{○}

m

24

Project Instructions: Use the sum/difference of angles
formulas starting with the angles above (α,β,s,t,m,n) to generate
four different new integer right triangles (Triangles 4,5,6,7). Sketch each new
triangle as above, naming the acute angles, listing the sines and cosines, and
giving degree approximations for the angles. Write sines and cosines as **fractions,
not as decimals**.

(Note 1: An angle may be added to itself, or two angles from different triangles may be added or subtracted. To avoid negative new sines or cosines, the new angle should be between 0 and 90 degrees.)

(Note 2: The fractions obtained for sines and cosines must be reduced to lowest terms. Do not use a triangle that was obtained before.)

(Note 3: Fractions with integers that are too large to be calculated out as whole numbers should not be used.)