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>

Introductory Flow Model, Functions, Function Families

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.

• In function notation

• 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 basic exponential function is y = 2 ^ t.

If p is an integer, then the function is defined for both positive and negative values of x.