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.

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.

A quadratic function y(t) has zeros for t values given by the quadratic formula.

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 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.

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.

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.

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 graph of any quadratic function can be thought of as a uniformly stretched and shifted version of the basic quadratic function y = x^2.

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.

We understand quadratic functions and their uses better if we look at various sub-families of the family of quadratic functions.

Quadratic functions represent quantities whose rates change at a uniform rate.

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.

Typical situations involving linear functions include

The basic exponential function is y = 2 ^ t.

Typical situations involving exponential functions include

The power function family is actually a multiple family of functions characterized by basic functions of the form y = x ^ p.

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

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.

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 each value of p there is a family of power functions y = A (x - h ) ^ p + k.

Typical situations involving power functions include