Given a set of depth vs. clock time data we can calculate average rates of depth change between every pair of successive clock times. If we hypothesize a quadratic model we can choose three points on our approximate 'best-fit' curve to use to create a model function y = a t^2 + b t + c. Substituting the y and t coordinates of our three points we obtain linear three equations in a, b and c, which can be solved simultaneously for these parameters. This will give us our model.
We solve the simultaneous equations by the process of elimination.
We can use the model to predict depth at a given clock time or to find clock time at which a given depth occurs.
We can find the average rate of depth change between any two clock times, given a depth vs. clock time function. We evaluate the function at the two clock times to determine the depths corresponding to these clock times, then we calculate the change in depth and the difference in the clock times `dy / `dt. We use these differences to calculate the average rate.
For a specific quadratic function we can symbolically calculate the average rate between clock times t and t + `dt; imagining that `dt approaches zero we obtain the actual rate-above-change function dy / dt, or y'(t).
We can find the average rate of depth change between any two clock times, given a depth vs. clock time function. We evaluate the function at the two clock times to determine the depths corresponding to these clock times, then we calculate the change in depth and the difference in the clock times `dy / `dt. We use these differences to calculate the average rate. For a specific quadratic function we can symbolically calculate the average rate between clock times t and t + `dt; imagining that `dt approaches zero we obtain the actual rate-above-change function dy / dt, or y'(t).
We can generalize the above process to the general quadratic function y = a t^2 + b t + c, obtaining the general rate-of-change function y'(t) = dy / dt = ` a t + b.
From the rate function y'(t) of an unknown quadratic function we can determine the constants a and b for the function y(t) = a t^2 + b t + c. Using this knowledge we easily find the difference in the depths between two given clock times. The only thing we cannot find from the rate function is the constant c, which we need to determine the actual depth at a given time. However, without c we can still find the depth change between two given clock times.
The quadratic depth function y = a t^2 + b t + c implies a linear rate-of -depth-change function y ' = 2 a t + b. A linear rate-of-depth-change function y ' = m t + d implies a quadratic depth function y = 1/2 m t^2 + d t + c, where c is an arbitrary constant number while m and d are known if y ' is known. Thus the rate-of-depth-change function allows us to determine the change in depth between any two clock times; however to find the absolute depth at a clock time we must evaluate arbitrary constant c, which we can do if we know the depth at a given clock time.
The process of obtaining a rate function from a quantity function is called differentiation, and the rate function is called the derivative of the quantity function. The process of obtain the change-of-quantity function is called integration, and the quantity function is called the antiderivative or integral of the rate function.
From the function giving the rate at which a radioactive substance decays we estimate the number of decays over a substantial time interval. The process is depicted using a trapezoidal approximation graph.
#05:An exponential function is characterized by a growth rate r, a growth factor (1+r) and a quantity function Q(t) = Q0 (1+r)^t. Alternative forms of the exponential function include Q(t) = Q0 b^t and Q(t) = Q0 e^(kt).
Any function of this form has a doubling time tD such that for any t, Q(t+tD) = 2 Q(t), which we demonstrate algebraically and depict graphically.
Given a velocity vs. clock time function we can construct a trapezoid between any two graph points, with vertical altitudes running from the horizontal axis to the respective graph points. These altitudes represent the initial and final velocities over the corresponding time interval. The area of this trapezoid represents the product of the average of the initial and final velocities and the duration of the time interval, and therefore the distance that an object would move during the time interval at this average velocity. If the velocity function is not linear during time interval, it is very unlikely that the actual average velocity will equal the average of the initial and final velocities, and the distance so calculated will be an approximation rather than a precise value. The slope of the line segment between the graph points will represent the average rate at which the velocity changes (change in velocity divided by change in clock time).
In general if the graph represents the rate at which some quantity changes vs. clock time, a trapezoid can be constructed to approximate the change in the quantity between to given clock times, with the change in the quantity represented by the area of the trapezoid. More accurate approximations can be obtained by subdividing the trapezoid into a series of 'thinner' trapezoids, on which the line segments between graph points more nearly approximate the actual function.
The area under the graph between two clock time therefore represents the integral of the rate function between the two clock times. This integral represents the change in the quantity between these two clock time.
If the graph represents some quantity vs. clock time, then a similar trapezoid or series of trapezoids will have line segments between graph points which represent average slope between graph points, and which therefore represent average rates of change between the corresponding clock times. These average rates of change represent the approximate derivatives of the function depicted by the graph.
If the clock times on a series of trapezoids are uniformly spaced, then if the slopes represent rates of change, then at any graph point the change in slope at that point divided by the uniform time interval between graph points will represent the approximate rate at which the slope changes at the graph point. Since the slope represents the rate at which the function changes, this rate of slope change will represent the rate at which the rate changes. This quantity is and approximate second derivative of the function.
By interpreting the altitude and width of a trapezoid, we can interpret what the product of average altitude and width represents, and we can interpret what is represented by the change in altitude divided by the width.
We work a review problem involving a quadratic depth function.
Testing the proportionality y = a x^3 for sandpile volume y vs. diameter x, we obtain questionable results. Looking at the proportionality for different sandpiles, and comparing with a DERIVE best fit for y vs. x data, we conclude that for the data considered y = .002 x^3 is a good, if not perfect, model.
Using the definition of the derivative we do a little algebra and determine that the derivative of y = a x^3 is y ' = 3 a x^2.
For a given diameter x we easily determine the rate y' = dy / dx= 3 a x^2 at which the volume y of a sandpile is changing, with respect to changes in x, at that diameter. Using this rate we can estimate the volume change for a given small change `dx in x. The volume change will simply be the product of the rate y' and change `dx in x: `dy = y' * `dx, or `dy = dy / dx * `dx.
The essence of the concept of the differential is that the change `dy in y corresponding to a change `dx in x is `dy = y'(x) * `dx.
Using the idea of the differential we approximate the volume of a sandpile with volume function y = .0031 x^3 at diameter x = 30.1, given its volume at x = 30. We interpret this result in terms of the tangent line to the graph of the y vs. x function at the x = 30 point.
We use the same function as before to obtain an equation for the tangent line at the x = 30 point. We use this equation to approximate the original function in the vicinity of the x = 30 point.
Testing a table of velocity of a falling object vs. distance fallen to see if velocity is in fact proportional to distance fallen, we see that the proportionality v = k x gives different values of k for different (x, v) points, so that the proportionality fails. It turns out that the proportionality v = k `sqrt(x) works, as can be checked in the same way.
Given a table of y vs. x data in which the x values are evenly spaced, in order to determine whether the set is exponential or not we need only look at the ratios of successive y values. If the ratio is constant, then the data indicates an exponential function. We see why this is so by looking at the form y = A b^x of an exponential function.
Given a proportionality y = k x^2 and values of y and x, we determine k. From the resulting y = k x^2 relationship we can determine y for any given x or x from any given y.
When an object cools in a constant-temperature room, the rate at which its temperature changes is proportional to the difference T -Troom between its temperature T and that of the room: rate = k (T - Troom). From a given rate and a given temperature we can evaluate k. The rate of temperature change is denoted dT / dt, so we have the proportionality dT / dt = k (T -Troom). This sort of equation, in which a derivative is treated as a variable, is called a differential equation.
A proportionality statement involving the rate of change of a quantity y, the quantity y itself and the independent variable x can be interpreted as a differential equation. We look at some examples of such situations.
To find the value of an inverse function g^-1(x) at a given value of x we can look at the graph of g(x). Locating the specified x on the y axis, we project over to the graph of g(x) and the up or down to the x axis; the value we obtain on this axis is the value of g^-1(x). Alternatively we can evaluate g(x) at different values of the independent variable until our result is sufficiently close to the specified x.
We construct the graph of the natural log function y = ln(x) by first constructing the graph of the exponential function y = e^x, then reflecting through the y = x line to get the graph of the inverse function, which is y = ln(x).
The forms y = A b^t and y = A e^(kt) are equivalent, with b = e^k or equivalently with k = ln(b). For exponential growth k is positive and b is greater than 1; for exponential decay k is negative and b is less than 1.
Most of the functions we use in calculus and in modeling the real world are composite functions of the form f(g(x)), with f and g usually being the simple power functions, exponential functions, logarithmic functions, polynomial functions, etc.. The ability to decompose a given composite function f(g(x)) into its constituent functions f and g is essential in later applications. It is to learn to do this now so that the skill is available when it is needed later. Don't wait to develop the skill until you need to apply to learn something else.
Given a differential equation of the form dT / dt = k (T - Troom), and given a value of T at a clock time t, we can determine the approximate value of T at clock time t + `dt by using the fact that `dT = dT / dt * `dt. If `dt is small enough that dT / dt doesn't change by much between t and t + `dt, the approximation will be a good one.
The process can be continued for successive intervals to determine approximations that t + 2 `dt, t + 3 `dt, etc.. The accuracy of the approximation decreases more and more rapidly with succeeding intervals.