questions 050921
So when you ask for average rate of change you're asking for slope, and slope is
the same thing as acceleration, correct?
<h3>The average rate of change of y with respect to x between two values of x is
the slope of the line segment between the corresponding points of the y vs. x
graph.
The slope of a velocity vs. clock time graph is acceleration. The slope of a
position vs. clock time graph is velocity.</h3>
When sketching the graph of the power function family y=A(x-h)^p+c for p=-2 A=1
h=0 c=-3 to 3 when you sovle the equation useing whole numbers between -3 and 3
is your graph just move one whole number or should you use certain fractions
between -3 and 3?
<h3>For a basic power function you are guided by the behavior at
x = -1,
x = 0,
x = 1/2,
x = 1 and
x = 2.
If h = 0 then there is no shift in the x direction and you simply use these
points.
If for example h = 2 then the graph shifts 2 units to the right and you would
use
x = -1 + 2 = 1,
x = 0 + 2 = 2,
x = 1/2 + 2 = 5/2
x = 1 + 2 = 3 and
x = 2 + 2 = 4.</h3>
A set of pendulum
frequency vs. length data may be modeled by the function f = A L^-.5,
where f is frequency and L is length. The number A is understood to be a
constant number, and is called a parameter of the model. If we have the
frequency at a known length we can substitute for f and L and determine the
value of the parameter A. Substituting this value of A into the form f = A L
^-.5, we obtain a mathematical model of frequency vs. length. To the
extent that we have accurate data our model will work for a pendulum of any
desired length. Since f is a multiple of the -.5 power of L we say that f is a
power function of L.
If we
observe the frequency of a pendulum at two lengths we can find the
two parameters A and p for the model f = A L^p. This equation has two
parameters, A and L. When we substitute the length and corresponding
frequency into this form we get an equation with A and p as unknowns. If we
substitute the length and frequency for our two observations we will therefore
obtain two equations in the two parameters A and p, which we can then solve to
obtain values of these parameters.
Substituting the values we
obtain for A and p into the form f = A L ^-p, we obtain a mathematical model
of frequency vs. length. Since f is a multiple of the p power of L we say
that f is a power function of L.
If we
have the form of a function, as with the form f = A L^-5 or the form f =
A L^p above, then if we substitute a number of data points equal to the number
of parameters in the model we will obtain a set of simultaneous equations
whose number is equal to the number of parameters. We may or may not be able to
obtain completely accurate solutions to these equations, but we often can solve
them precisely to obtain values of the parameters. When we cannot solve the
equations precisely we can almost always obtain approximate solutions.
For
example, the equations used to solve for the parameters of the model f = A L^p
can be solved exactly for A and p. However the precise solution
for p requires the use of logarithms, which may not be familiar to all students
at the beginning of this course (and even most students who are famliar with
logarithms will not remember the precise technique required). So most students
will be unable to solve the equations exactly at this point of the
course. However any student can solve the equations approximately using
trial and error. and will be able to master the use of logarithms at a
later point.
A set of depth
vs. clock time data for water flowing from a uniform cylinder through a
hole at the bottom of the cylinder can be very closely modeled by a
quadratic function of the form y = a t^2 + b t + c.
- One way to create
the model is to choose three well-spaced data points to
represent the data set.
- The t and y
coordinates of these data points are substituted into the form y = a
t^2 + b t + c, and the resulting equations are solved for
the parameters a, b and c.
- These parameters are
substituted back into the original form to obtain a
quadratic function y(t).
- The resulting
function y(t) will exactly fit the three selected data points,
in the sense that if the t coordinate of one of the selected points
is substituted for t, the resulting y value will be the y coordinate of that
same point.
A graph of the depth vs.
clock time data set vs. the quadratic function model shows that the model stays
very close to the data set.
- An analysis of the
residuals will show whether the model deviates in a systematic way from the
data set.
- The average
magnitude of the residuals is one measure of how well the model fits the
data.
- The pattern of the
residuals is another.
- If the
pattern of the residuals is random and the average magnitude of the
residuals is small compared to the depth changes observed, the model is
probably a good one.
- ** The standard
measure of the closeness of the model to the data is the 'standard
deviation', which can be pretty well understood to be the square root of an
average of the squared residuals. **
A quadratic
function y(t) has zeros for t values given by the quadratic formula.
- Depending on the
value of discriminant there are no real zeros (negative discriminant), one
real zero (discriminant zero) or two real zeros (discriminant positive).
The graph of a
quadratic function is a parabola; if the function has two distinct
zeros the vertex of the parabola lies on the vertical line which is halfway
between the two zeros.
- The location of the
vertical line on which the vertex lies is easily found
using the quadratic formula, whether the function has no zeros, one zero or
two zeros.
- The vertical
coordinate of the vertex is easily found by substituting the
horizontal coordinate into the function.
The graph points of the
parabola y = a t^2 + b t + c whose horizontal coordinates lie 1 unit to
the right and 1 unit to the left of the vertex have vertical
coordinates a units above the vertex.
- Thus to get to these
graph points from the vertex we think of moving over 1 unit and up a units.
- If a is negative,
then an upward displacement of a units is actually a downward displacement.
In order to find the
depth at a given clock time we simply substitute the
given clock time for y in the function y(t).
In order to find the
clock time at which the depth in the depth vs. clock time model
is equal to a given value, we recall that y represents the
depth.
- We therefore
substitute the desired depth for y and solve for t.
- For a quadratic
depth function y(t) we will use the quadratic formula to solve for
the desired clock time t.
- Often the quadratic
formula will give us two real solutions, while a depth
function will only pass a given depth at one clock time. In this case we
must choose the value of the clock time which is in the domain of
the actual function.
- If the depth
function is not quadratic, we will have to use another appropriate means to
solve the equation. For a linear depth function the solution is very easy.
Some of the methods used for other types of functions will be developed
later in the course.
The average rate
at which a depth function y(t) changes during a time interval is equal
to the change in depth divided by the duration of the time interval.
- This average rate
is, to a first approximation, associated with the midpoint of the
time interval.
- If the depths at the
beginning and at the end of the time interval are represented by points on a
graph of depth vs. clock time, then
- the rise of the
line segment from the first to the second point represents the change in
depth and
- the run of the
segment represents the duration of the time interval so that
- the
slope represents the average rate at which depth changes during the time
interval.
In general if a function
y(t) represents some quantity that changes with clock time, then the
average rate at which the quantity changes between two clock times is equal to
the change in the quantity divided by the change in the clock time.
- the change in
the quantity is represented by y(t2) - y(t1),
- the change in
clock time is t2 - t1 and
- the
average rate is ( y(t2) - y(t1) ) / (t2 - t1).
- This average rate is
represented by the slope of the line segment between the
graph points ( t1, y(t1) ) and ( t2, y(t2) ).
The graph of any
quadratic function can be thought of as a uniformly stretched and shifted
version of the basic quadratic function y = x^2.
- The graph of this
basic parabola can be 'fattened' or 'thinned' by stretching it in the
vertical direction by the appropriate factor a , which means that each point
on the parabola is moved a times as far from the x axis.
- If we stretch by a
factor a with | a | > 1, each point will move further from the x axis and
the resulting parabola will appear thinner.
- If we stretch by a
factor a with | a | < 1, each point will move closer to the x axis and the
resulting parabola will appear fatter.
- If a is positive,
the parabola will continue to open upward, whereas if a is negative, the
resulting parabola will open downward.
- After the parabola
is stretched to the right shape, the horizontal and vertical coordinates of
every point on the parabola are shifted horizontally and vertically, all by
the same amount, so that the vertex ends up in the right place.
If the basic quadratic
function y = x^2 is stretched by factor a , then shifted horizontally through
displacement h and vertically through displacement y, the resulting function
will be y(t) = a ( t - h ) ^ 2 + k.
- This relationship is
often expressed in the equivalent form y = k = a ( t - h ) ^ 2.
- By expanding the
square we can put the function into the form y(t) = a t^2 + b t + c.
- ** By a process
known as 'completing the square', we can also put y(t) = a t^2 + b t + c
into the form y(t) = a ( t - h ) ^ 2 + k. ** We do not use this process at
this stage of the course. We will use it later in relation to conic
sections. The technique is also important in calculus. **
We understand quadratic
functions and their uses better if we look at various sub-families of
the family of quadratic functions.
- Examples
of sub-families include:
- The set of
quadratic functions with h and k both zero, which consists of all
quadratic functions with vertex at the origin.
- The set of
quadratic functions with a = 1 and h equal to some fixed value, which
consists of all quadratic functions with vertex at x = h, opening
upward, and which are congruent to the basic parabola y = x ^ 2.
- The set of
quadratic functions with a = -1 and k equal to some fixed value, which
consists of all quadratic functions with vertex at y = k, opening
downward, and congruent to the basic parabola y = x ^ 2.
Quadratic
functions represent quantities whose rates change at a uniform rate.
- The rate at which
the depth of water in the flow experiment changes is changing at a constant
rate.
- The most typical
example of a situation which can be modeled by a quadratic function is that
of an object, for example an automobile, coasting down a uniform incline.
- The rate at which
the position of the automobile changes is its velocity.
- The velocity of the
automobile changes at a constant rate, as can be observed from the steady
motion of the speedometer needle.
Other function
families which we will take as basic for this course include the families of
linear, exponential and power functions.
The basic linear
function is y = x.
- The graph of this function is a
straight line characterized by a slope of 1 and a y-intercept y = 0.
- If this basic function is
vertically stretched by factor m, the line remains straight and its
slope becomes m and y = m * x.
- If the function is then
vertically shifted through displacement b, every point is displaced
y units in the vertical direction. Its y intercept thus becomes y = b and
its formula becomes y(x) = m x + b.
- To graph this function we can graph
the point ( 0 , b ) on the y axis and the point 1 unit to
the right of this point; this second point is displaced 1 unit
horizontally and m units vertically from (0 , b).
- A linear function will
exactly fit any two given data points for which the horizontal
coordinates differ.
Typical situations involving linear
functions include
- Force vs. displacement of pendulum
- Horizontal range of stream vs. time
for flow from side of vertical uniform cylinder
- Income: Money earned vs. hours
worked
- Demand: Demand for a product vs.
selling price (simplified economic model)
- Straight-line approximation to any
continuously changing quantity over a short time interval
The basic
exponential function is y = 2 ^ t.
- The graph of this function is
asymptotic to the negative t axis, passes through (0,1) and grows more and
more quickly, without bound, as t becomes large.
- Every time t increases by 1 this
function doubles.
- The function is generalized
by a compression by factor k in the horizontal direction, a vertical stretch
A, and a vertical shift c.
- The resulting function is
y(t) = A * 2 ^ ( k t ) + c.
-
- We think of first bringing every
point of the y = 2 ^ t function k times closer to the y axis.
- If k is negative the function is
also reflected about the y axis so that it becomes asymptotic to the
positive t axis.
- We then stretch the function
vertically by factor A, which changes the y intercept from (0, 1) to (0, A).
- We finally shift the function c
units vertically, which changes the y intercept to (0, A + c) and the
asymptote to the horizontal line y = c.
- ** This function also can be
expressed in the form y = A e ^ (kt) + c, where A and c are the same as
before and k differs. **
- ** This function can be expressed in
the third form y = A b ^ t + c, where A and c are the same as before and b =
2 ^ k. **
Typical situations involving exponential
functions include
- Compound interest: Value of
investment vs. time
- Unrestricted population growth:
Population vs. time
- Temperature approach to room
temperature: Temperature vs. time
- Radioactive decay: Amount
remaining vs. time
The power function family is actually a
multiple family of functions characterized by basic functions of the form y = x
^ p.
- The power p can be any real number,
either positive or negative.
- All basic power functions are
defined at least for positive values of x, and all pass through the point
(1, 1).
-
- If p is positive the function also
passes through the point (0, 0).
- If p is negative the function has a
vertical asymptote at x = 0 and approaches the x axis as an asymptote.
- The values of the function at x =
1/2 and x = 2, in addition to the characteristics summarized above,
indicate the general shape of its graph.
If p is an
integer, then the function is defined for both positive and negative values of
x.
- For even integers p
the graph of the basic power function is symmetric about the y axis and has
its lowest value at x = 0.
- A function symmetric about the y
axis is also called and even function.
- For odd integers p
the graph of the basic power function is anti-symmetric about the y
axis (also described as symmetric through the origin)
and has a point of inflection (where the graph changes from
downward curvature to upward curvature).
If p is a rational number with
denominator, in lowest terms, being odd then the function is defined for both
positive and negative values of x.
- The symmetry of the function is then
determined by the whether the numerator is even or odd by the same rule as
if p is and integer.
If p is neither and integer nor a
rational number with odd denominator, then the basic power function is
not defined for negative values of x.
- For example a rational power with
even denominator would require us to take an even root of a negative number,
which we cannot do.
- We cannot define a real irrational
power of a negative number.
For each value of p there is a
family of power functions y = A (x - h ) ^ p + k.
- As before, A is the vertical stretch
applied to the basic function y = x ^ p, while h and k are the horizontal
and vertical shifts.
- The vertical stretch A is applied
first, moving every point A times further from the x axis. In particular,
the point (1, 1) becomes (1, A).
- For positive p, the extreme
point or the point of inflection will then be shifted from
the origin to the point (h, k).
Typical situations
involving power functions include
- Period of pendulum vs. length (power
p = .5)
- Frequency of pendulum vs. length
(power p = -.5)
- Surface area vs. scale for a family
of geometrically similar objects (power p = 2)
- Volume vs. scale for a family of
geometrically similar objects (power p = 3)
- Strength vs. weight for
geometrically and physiologically similar individuals (power p = 2/3)
- Illumination vs. distance from a
point source (power p = -2)
- Illumination vs. distance from a
line source (power p = -1)