quantities and co-variation of quantities
Module 2 : Investigation 2
MAT 170 | Precalculus
August 24, 2016
question 1 (a bit of review)
(e) Sam goes for a ride on a Ferris wheel. Suppose we want to use h
to denote the varying quantity of Sam’s height (in feet) above the
ground. What is wrong with each of the following attempts to define
the variable h ?
(i) Let h represent height.
(ii) Let h represent the number of feet.
(iii) Let h represent the height (in feet) above the ground.
(iv) Let h represent Sam’s height (in feet).
(v) Let h be Sam’s height above the ground.
2
question 1 (a bit of review) - possible answers
(e) Sam goes for a ride on a Ferris wheel. Suppose we want to use h
to denote the varying quantity of Sam’s height (in feet) above the
ground. What is wrong with each of the following attempts to define
the variable h ?
(i) Let h represent height.
Need to state what height is being measured, where it’s
being measured from, and what the units are.
(ii) Let h represent the number of feet.
Need to state what the “number of feet” is a measure of.
(iii) Let h represent the height (in feet) above the ground.
Need to state what is this the height of.
3
question 1 (a bit of review) - possible answers
(e) Sam goes for a ride on a Ferris wheel. Suppose we want to use h
to denote the varying quantity of Sam’s height (in feet) above the
ground. What is wrong with each of the following attempts to define
the variable h ?
(iv) Let h represent Sam’s height (in feet).
Need to state where the height is being measured from.
(v) Let h be Sam’s height above the ground.
Need to state the units.
4
expressions and formulas
what is an expression ?
Definition
An algebraic expression of a quantity is a way of expressing
that quantity using only addition, subtraction, multiplication,
division and exponentiation.
Example : If t denotes the amount of time (in minutes) since class
began, then
50 − t
is an algebraic expression for the remaining amount of time (in
minutes) until class ends.
Example : 5, 3 + 2 and
√
25 are all algebraic expressions of 5.
6
what is an expression ?
Often we will see the phrase :
“Express the varying quantity A in terms of the varying quantity
B.”
This means that we want to come up with an algebraic expression
for quantity A in which the only varying quantity involved in the
expression is the quantity B.
Example : If t denotes the amount of time (in minutes) since class
began, then
50 − t
an algebraic expression for the remaining amount of time (in minutes) until class ends in terms of the quantity t.
7
question 2
(b) (i.) Savanna and Porter spot each other on the beach when they
are 54 feet apart. Savanna does not move, but Porter starts walking
towards Savanna at a constant rate of 6 feet per second.
Let t represent the number of seconds elapsed since Porter started
walking toward Savanna. Write an expression to represent the
varying number of feet between Savanna and Porter in terms of t.
(c) Halle and Laura are twins. Let x represent Halle’s varying height
(in inches), and let y represent Laura’s varying height (in inches).
(i) Write an expression for the number of inches that Laura is
taller than Halle.
(ii) As x and y vary over the first 18 years of Halle and Laura’s lives,
what would it mean if the expression from (i) was negative ?
8
question 2 - possible answers
(b) (i.) Savanna and Porter spot each other on the beach when they
are 54 feet apart. Savanna does not move, but Porter starts walking
towards Savanna at a constant rate of 6 feet per second.
Let t represent the number of seconds elapsed since Porter started
walking toward Savanna. Write an expression to represent the
varying number of feet between Savanna and Porter in terms of t.
54 − 6t
9
question 2 - possible answers
(c) Halle and Laura are twins. Let x represent Halle’s varying height
(in inches), and let y represent Laura’s varying height (in inches).
(i) Write an expression for the number of inches that Laura is
taller than Halle.
y−x
(ii) As x and y vary over the first 18 years of Halle and Laura’s lives,
what would it mean if the expression from (i) was negative ?
That Halle was taller than Laura.
10
what is a formula ?
Definition
A formula defines how one varying quantity changes in terms
of one or more another varying quantities.
Example : Recall that if t denotes the amount of time (in minutes)
since class began, then 50 − t is an algebraic expression for the
remaining amount of time (in minutes) until class ends.
Let y represent the remaining amount of time (in minutes) until
class ends. Then
y = 50 − t
is a formula. This formula tells us how y varies in terms of t.
11
what is a formula ?
Definition
Suppose m and n are variables representing two varying quantities.
If we write a formula that expresses m in terms of n, that is,
m = “some algebraic expression involving n,”
we refer to m as the dependent variable and n as the independent variable.
12
questions 3 & 5
(3a) How is a formula different than an expression ?
(3b) The formula n = 13 − x represents the length of a melting icicle
in terms of x, the varying number of inches that have melted from
the 13 inch icicle. What does the “n =” convey ?
(5a) Suppose x is 2 more than y. Write a formula that defines x in
terms of y.
(5b) For the formula you created in part (5a), which variable is the
dependent variable and which is the independent varaible.
13
questions 3 & 5 - possible answers
(3a) How is a formula different than an expression ?
An expression gives the value of a single quantity.
A formula gives a relationship between two or more varying quantities.
(3b) The formula n = 13 − x represents the length of a melting icicle
in terms of x, the varying number of inches that have melted from
the 13 inch icicle. What does the “n =” convey ?
The quantity that n represents (the remaining length of the icicles
in inches) is equal to the expression 13 − x.
14
questions 3 & 5 - possible answers
(5a) Suppose x is 2 more than y. Write a formula that defines x in
terms of y.
x = y+2
(5b) For the formula you created in part (5a), which variable is the
dependent variable and which is the independent variable.
x is dependent,
y is independent.
15
change in a quantity’s value
change in a quantity’s value
Understanding how varying quantities change was the motivating
question behind the invention (discovery ?) of Calculus.
In fact, working with changes of a given quantity is so common we
have special notation.
We will denote the change in a given variable x by ∆x.
Example : If x represents the number of people on ASU’s Tempe
campus, then ∆x = 1200 represents a positive change (increase)
in the number of people on campus by 1200.
On the other had, ∆x = −1200 represents a negative change (decrease) in the number of people on campus by 1200.
17
question 4
The original length of a candle before it is lit is 18 inches.
(a) If x represents the number of inches burned from a candle
and y represents the number of inches of the candle that
remain, what does 18 − x represent ?
(b) Write a formula that represents y in terms of x.
(c) As the value of x increases from 2 to 9, what is ∆y ?
(d) Could you have written the formula in (b) without using a
constant quantity ?
18
question 4
The original length of a candle before it is lit is 18 inches.
(a) If x represents the number of inches burned from a candle
and y represents the number of inches of the candle that
remain, what is 18 − x and what does it represent ?
18 − x is an expression for the number of inches remaining
on the candle. This is the same quantity that y represents.
(b) Write a formula that represents y in terms of x.
y = 18 − x
19
question 4
The original length of a candle before it is lit is 18 inches.
(c) As the value of x increases from 2 to 9, what is ∆y ?
When x = 2 we have
y = 18 − 2 = 16.
When x = 9 we have
y = 18 − 9 = 9.
So y starts with a value of 16 inches and ends with a value
of 9 inches. Therefore, the change in y as the value of x
increases from 2 to 9 is −7 inches, that is,
∆y = −7.
20
question 4
The original length of a candle before it is lit is 18 inches.
(d) Could you have written the formula in (b) without using a
constant quantity ?
No, with x and y defined as they are, we wouldn’t be able
to write the formula in (b) without the constant quantity
18.
21
using graphs to understand the change
in a quantity’s value
question 1
Sam boards a Ferris wheel from the bottom and rides around several
times before getting off. The following graph represents Sam’s height
above ground (in feet) with respect to the amount of time (in
seconds) since the Ferris wheel began moving for one complete
rotation.
(a) The point (2, 4.9) is plotted on
the graph. Explain what this
point means in the context of the
Ferris wheel situation.
(b) As the number of seconds
since the Ferris wheel began
moving increased from 2 to 8
seconds, how much did Sam’s
height in feet above the ground
change ?
23
question 1 - possible answers
Sam boards a Ferris wheel from the bottom and rides around several
times before getting off. The following graph represents Sam’s height
above ground (in feet) with respect to the amount of time (in
seconds) since the Ferris wheel began moving for one complete
rotation.
(a) The point (2, 4.9) is plotted on
the graph. Explain what this
point means in the context of the
Ferris wheel situation.
This point tells us that 2 seconds after the Ferris wheel
began moving, Sam’s height
above the ground was 4.9
feet.
24
question 1 - possible answers
Sam boards a Ferris wheel from the bottom and rides around several
times before getting off. The following graph represents Sam’s height
above ground (in feet) with respect to the amount of time (in
seconds) since the Ferris wheel began moving for one complete
rotation.
(b) Let h denote Sam’s height (in
feet) above the ground. As the
number of seconds since the
Ferris wheel began moving
increased from 2 to 8 seconds,
what is ∆h ?
At 2 second, Sam is 4.9 feet
above the ground. At 8 seconds he is 27 feet above
ground. So ∆h = 27 − 4.9 =
22.1 feet.
25
question 1
Sam boards a Ferris wheel from the bottom and rides around several
times before getting off. The following graph represents Sam’s height
above ground (in feet) with respect to the amount of time (in
seconds) since the Ferris wheel began moving for one complete
rotation.
(d) Estimate the time(s) during
the first 20 seconds since the
Ferris wheel began moving when
Sam’s height above ground was
24 feet. Represent your solutions
on the graph and explain how
you got them.
At approximately 7 seconds
and 13 seconds.
26
27