Table of Basic Calculus I Concepts
Weeks 1-7
y = a t^2 + b t + c; fit 3 points; solve system for parameters
compare to data: check deviations and residuals
y vs. x: y on vertical, x on horizontal axis; depth vs. clock time: depth on vertical, clock time on horizontal axis
choose appropriate scale to spread graph out and show its behavior (e.g., curvature, zeros, vertex, E12etc.)
y = a t^2 + b t + c
graph is parabola with vertex at t = -b / (2a)
a t^2 + bt + c = 0 when and only when t = [-b +- `sqrt(b^2 - 4 a c) ] / (2a)
may be 2 zeros, 1 or no (when quantity under `sqrt pos., 0 or neg.)
substitute t into y(t); project from t axis to graph then to y axis
just substitute
solve y = y(t) for t; project from y axis to graph then to t axis
solve equation if possible; otherwise approximate
rise = change in depth; run = change in clock time; slope = rise / run = [y(t + `dt) - y(t)] / `dt = average rate
change in depth per unit of time
rise = change in depth; run = change in clock time; slope = rise / run = [y(t + `dt) - y(t)] / `dt = average rate
increasing slope -> increasing flow rate; constant slope -> constant flow E16rate
each consecutive pair of depth vs. clock time points -> ave. rate
ave. rate occurs at approximately midpoint of time interval
ave. height = ave. rate; width = time interval; area = ave. height * width representing ave. rate * time interval = change in depth
over short interval region under graph is approximately trapezoidal so area is area of trapezoid = midpoint ht. * width
each consecutive pair of depth change rate vs. clock time points -> ave. rate, time interval -> depth change
total depth change = sum of individual depth changes
ave. rate over small interval close to any rate found in interval
if interval shrinks to a point, limiting value is precise (instantaneous) rate at point
limiting value of ave. rate over intervals shrinking to 0
difficult concept -- rise / run shrinks toward 0 / 0 and what is that? Picture helps make sense
the instantaneous rate of change at that point
applies to depth as function of clock time; generalizes to any fn. of clock time; then E12to any function of any vbl.
if y = a t^2 + b t + c then derivative = y' = 2 a t + b
limiting value of [ y(t + `dt) - y(t) ] / `dt
if depth = a t^2 + b t + c then rate = y' = 2 a t + b
interpretation of limiting value of [ y(t + `dt) - y(t) ] / `dt
if y' = mt + b then y = .5 m t^2 + b t + c
just use derivative formula backward; c arbitrary
ave. rate * time interval, represented by area under graph
equal to difference of antiderivative function at beginning and ending clock time
derivative: finding rate of change function dy / dt = lim[`dt -> 0] (`dy / `dt) integral: finding function y(t) whose derivative is given y'(t)
generally more difficult, in fact usually impossible to find the function y(t) to match the given rate y'(t)
derivative of symbolic integral F(t) of f(t) is f(t); integral of symbolic derivative f(t) of F(t) is F(t) + c
nearly inverse processes; each 'undoes' the other, except for the integration constant c
heights represent y quantity, widths represent change in t quantity
convenient tangible way to visualize integral and derivative of a function
area = ave. ht. * width = change in quantity when y is rate; slope = `dy / `dt = rate of change of y
interpretation follows from meanings of vertical and horizontal coordinates of graph
approximate function by series of trapezoids, label areas, accum areas, slopes, rates of slope change by conventions
shows the most important characteristics of a graph in approximate form
dy = f'(t) * dt `dy = f`(t) * `dt, approximately, for small `dt
applies to continuous functions whose derivatives are continuous; approximation of change in y is good near t tangent-line approximation for y = f(t), tan line at point (a, f(a)) is slope = slope line (y - a) / (t - f(a)) = f`(a). Can be solved for y to get slope-int. form. Good approximation to f(t) near t = a. equivalent to differential; good until second derivative changes significantly differential eqns. expressing proportionality rate prop. function of indep. vbl., or of dep. vbl, or both. e.g.: y'(t) = k t^2; y'(t) = k y^.5; y'(t) = k t^3 / y^2.
sometimes easy to solve, sometimes difficult, sometimes impossible numerical soln. of diff. eq. Use table, initial value, appropriate increment of indep. vbl. Evaluate rate of change, use with increment to find change of dep. vbl., get new values. Repeat (iterate) process. Smaller `dt -> greater accuracy of approx. Equivalent to series of approximations by differentials. Errors accumulate.
predictor-corrector soln. of diff. eq. Use previous method to predict new value, use this value to predict new rate, then use ave. of two rates to predict 'corrected' new value. Same accuracy as original method with approx. 1/10 the steps (`dt can be approx. 10 times as big to give same accuracy) derivative function for function f(t) f'(t) is fn. whose value at t is the instantaneous rate of change of f graph of f' is graph of slopes of graph of f. Large values when slope of f large, zero when slope of f zero, small when slope of f graph is small algebraic determination of derivative for function y = f(t), calculate lim(`dt -> 0) [ ( f(t + `dt) - f(t) ) / `dt ] limit might not exist (e.g., at asymptote or point of jump discontinuity, or where right and left-hand limits are different) interpretation of derivative Units are units of rise / units of run. Expresses rate of change of dep. vbl. with respect to indep. vbl. Change in dep. vbl. per unit of change of indep. vbl. second derivative of y = f(t) Derivative of first derivative f'(t). Rate at which rate changes. Positive if graph of f(t) concave up, negative if graph of f(t) concave down, zero if graph of f(t) is linear. derivative and 2d derivative estimates from tables Approx. derivative is average rate of change over interval, associated with midpoint of interval. Approx. 2d derivative is average rate of change of derivative between midpoints, associated with midpoint of midpoints.
Trapezoidal approx. graph: derivatives are slopes, 2d. derivatives are slope change rates. If intervals of indep. vbl. are uniform, then 2d derivatives will be associated with interior points of original partition.
Riemann sums Approx. areas of intervals of partition of interval from t = a to t = b. Left sum uses left altitude on interval, right sum uses right altitude. If function increasing these are lower and upper sums. Difference between left and right sums depends only on fn. values at a, b and on increment `dt; all areas except first area of left-hand sum and last of right-hand sum cancel. Integral of increasing (or decreasing) function is squeezed between left- and right-hand sums; difference -> 0.
Area of trapezoidal approx. similarly squeezed.
Difference of Riemann sums r.h. - l.h. = difference = f(b) * `dt - f(a) * `dt
= ( f(b) - f(a) ) * `dt
Intermediate terms of difference cancel. Total difference for incr. or decr. fn: 'stack' small rectangles representing differences on indiv. subintervals.
DERIVE syntax for integrals, Riemann sums LEFT_RIEMANN(function, vbl. left, right, number of subintervals); in MISC.MTH INT(function, vbl. left, right)
RIGHT_RIEMANN in instructor-modified MISC.MTH utility file.
Permits efficient calculation of Riemann sums for partitions involving many intervals, and permits observation of how integral is squeezed between left and right sums. exponential functions y = A (1+r) ^ t (init. value A, growth rate r) y = A b^t (init. value A, growth factor b)
y = A e^(kt) (init. value A, growth factor e^k)
Consistent half-life or doubling time by solving b^`dt = 2 (or 1/2) for `dt, so `dt = log(2) / log(b) or log(1/2) / log(b), or e^(k `dt) = 2 or 1/2, so `dt = ln(2) / k or ln(1/2) / k.
quadratic functions zeros and vertex by quad. formula; symmetric about vert. line thru vertex zeros [-b +- `sqrt(b^2 - 4 a c)] / (2a); vertex at x = -b / (2a). logarithmic function inverse of exponential fn y = b^x is log[base b] (x). base-10 log simply log(x); base-e log is ln(x)
reverse columns of imaginary 'complete' table; reflect exp. graph across y = x line graph asymptotic to neg. y axis, passes thru (1,0); concave downward but no horiz. asymptote.
laws of logarithms log(ab) = log a + log b; log(a^b) = b log a; log(1) = 0; etc. NOT log(a+b) = log(a) + log(b) (fallacy of linearity) NOT log(a) / log(b) = log(a/b). log(a/b) = log(a * b^-1) = log a + log(b^-1) = log a + (-1) log b = log a - log b
solving exponential eqn. b^x = c take logs of both sides log(b^x) = log(c) -> x log(b) = log(c) -> x = log(c) / log(b)
sine function y = sin(`theta) is y coord. of point on unit circle at angle 'theta from pos. x axis. if `theta = `omega * t (motion on unit circle at const. angular vel.) then `theta changes by 2 `pi when `omega * `dt = 2 `pi so `dt = 2 `pi / `omega is period and hence `omega = 2 `pi / period
period of y = sin(`omega * t) is 2 `pi / `omega
sin fn. with period T is y = sin(2 `pi / T * t)
inverse sine function sin^-1(x) is the angle `theta such that sin(`theta) = x table obtained by reversing columns of sin(t) vs. t graph graph by reflecting y = sin(`theta), for -`pi/2 < `theta < `pi/2, across y = x line
cosine function x projection of point on unit circle period as for sine fn.; phase is `pi/2 ahead of that of sine fn. tangent function y/x for point on circle centered at origin; equal to sin(`theta) / cos(`theta) vertical asymptotes at `pi/2, 3 `pi / 2, etc. (wherever cos(`theta) = 0) polynomial function y = A(x-x1)(x-x2) ... (x^2 + b1 x + c1) ... or y = an x^2 + a(n-1) x^(n-1) + ... + a1 x + a0
product of linear and irreducible quadratic factors
each linear factor gives a zero
max. # of lin. factors is n so max. # of zeros is n, max # of turning pts. is n-1 zeros may be single, so fn. passes through in linear fashion, or multiple, so fn. passes through in power-fn. fashion
far-right and far-left behavior determined by leading term an x^n
rational fn. polynomial / polynomial zero if numerator has zero, denominator doesn't
vert. asymptote if denominator has zero, numerator doesn't
approaches x axis as horiz. asymptote if degree of num. less than deg. of denom. approaches infinity at far right and left if degree of denom. less than deg. of num.
approaches nonzero horiz. asymptote if deg. of num. equal deg. of denom.
When we observe water depth vs. clock time for water flowing from the uniform cylinder through a hole near the bottom, we obtain a series of data points each specifying a clock time and a depth. If we select three of these points and substitute each into the form of a quadratic function y = a t^2 + b t + c, we obtain three simultaneous linear equations in the three unknowns a, b and c, which are the parameters of our quadratic model. Solving the equations we obtain the values of these parameters, which we substitute into the form y = a t^2 + b t + c to obtain a specific quadratic function. This function is a model, not necessarily the best model, of our real-world depth vs. clock time.
We compare this model with our data by substituting the observed clock times into the model and obtaining predicted y = depth values. We calculate the deviations of our observed values from these predicted values, and we calculate residuals. A residual is calculated by subtracting the values predicted by the model from the observed values, and may be positive or negative. The deviation is just the absolute value of the residual, which is always positive. If the residuals seem random, without any tendency to form of pattern, then our assumption of the quadratic model is probably appropriate. If the residuals are random and the deviations are small, then we probably have a good model.
We plot our observed data points, using the convention that y vs. x has y on the vertical axis and x values on the horizontal axis, so that they plot of depth vs. clock time will have a depth measured on the vertical axis and clock time on the horizontal. We generally choose a scale such that the clock time values span approximately the same distance as the depth values.
When we have obtain the model function we plot it on the same set of axes in order to see visually how the model compares to the data. We try to see in our plot how close our function is to our data points, whether there will be a pattern to the residuals.
A quadratic function is a function whose graph can be obtained from the basic y = t^2 function by vertically stretching that function then, without rotating it, shifting it horizontally and vertically by any chosen displacements. Any such graph will be the graph of function y = a t^2 + b t + c, where a, b and c our real numbers. Furthermore, any such function is a quadratic function.
The zeros of any function y = y(t) are those points where y = 0. The zeros of the quadratic function y = a t^2 + bt + 0 are at t = [ -b +- `sqrt(b^2 - 4 * a * c) ] / (2a). The solutions, if any, are the t coordinates where the graph of the parabola crosses the t axis; the graph and crossed the t axis only at these points.
Depending on whether b^2 - 4 * a * c is positive, zero or negative, the square root in the expression will be positive, zero or nonexistent, so the formula will give us either two zeros (corresponding to + or - a positive square root), one zero (+ and - a zero square root both give the same result), or no zeros (we can't take the square root of the negative).
If we have model y = y(t) and wish to find the value of y for a given value of t, we simply substitute t into the form of the function. For example if y = y(t) = .02 t^2 - 12 t + 50 and we wish to find y when t = 30, we simply calculate y(30) = .02 (30)^2 - 12(30) + 50.
If we have model y = y(t) and wish to find the value of t for a given value of y, we substitute y into the form of the function. For example if y = y(t) = .02 t^2 - 12 t + 50 and we wish to find t when y = 30, we write 30 = .02 t^2 - 12 t + 50. We then solve this equation as a quadratic equation (being sure to subtract the 30 from both sides), obtained two values of t. Often only one of the values will have a meaning in the context of the real-world situation we are modeling.
The average rate at which a quantity changes in time is the difference in the quantity divided by the length of time required for the difference to occur. In the context of depth vs. clock time, this average rate is the change in depth divided by the length of time required for the change. If depth is symbolized by the variable y clock time by the variable t, we use `dy (standing for 'delta' y, with the Greek delta) and `dt ('delta' t) to stand for the differences in depth and clock time, and state that ave. rate = `dy / `dt.
We call this the average rate because over any finite time interval the actual rate is most likely changing and `dy / `dt expresses the average of all the rates that occur.
The figure below depicts the average rate as a slope, and notes that in the limit we get the derivative dy/dx.
When we represent depth vs. time data, or more generally any y vs. t data, on a graph, then between any two points the rise `dy represents the change in y (e.g., the change in depth) and the run `dt represents the change in t (e.g., the changing clock time). The slope `dy / `dt therefore represents the average rate at which y changes with respect to t (e.g., the average rate which depth changes with respect to time, which is the average velocity of the water surface).
A depth chronicle can be thought of as a table of depth vs. time data. Two consecutive depth vs. time listings give us two depths at two clock times. The difference in the two depths use of the change `dy in depth and the difference between the two clock times gives us the time interval `dt. The associated average rate of depth change is thus ave. rate = `dy / `dt. This average rate is associated more appropriately with the midpoint of the time interval than with the beginning or end of the time interval. If the depth changes at a constant rate, so that the depth vs. time function is linear, then the average rate will occur exactly at the midpoint of the time interval. Otherwise the average rate occurs at approximately the midpoint, but usually not precisely at the midpoint.
As an example the table below shows two consecutive depth vs. time listings and the average rate of depth change (`dy = -5 cm, `dt = 10 sec, so `dy / `dt = -5 cm / 10 sec = -.5 cm / sec; midway between clock times 40 sec and 50 sec is 45 sec; so we say that the rate of depth change is -.5 cm / sec at approximately clock time 45 seconds).
clock time depth rate of depth change midpoint clock time 40 sec 80 cm (45 sec--see last column) -- -5 cm/sec 45 sec 50 sec 75 cm
This process can be applied to the entire depth vs. time chronicle, and represented as the first two columns in such a table. The last two columns represent a chronicle of approximate depth change rate vs. clock time.
The figure below is a schematic sketch of the process. Note that the midpoint time and `dy / `dt columns have been reversed. We also note that the graph of `dy / `dt vs. tmid is an approximate graph of y' vs. t; if we look at a graph of y vs. t and visualize the sort of table we might get from the graph, we can then visualize the graph of y' vs. t. This image at least might be helpful in learning to 'see' the graph of y' in a graph of y.
On a graph of depth change rate vs. clock time, the vertical coordinate measures depth change rate and the horizontal coordinate measures clock time. Between any two given clock times the vertical height of the graph at the first clock time represents the rate of depth change at the beginning of the time interval, and the height at the second clock time represents the rate of depth change of the end of the time interval. The average of these two vertical heights therefore provides an approximation of the average rate of depth change over the time interval. Multiplying the approximate average rate of depth change by the length of the time interval we obtain the approximate depth change corresponding to that time interval.
Geometrically, we can form a trapezoid whose base lies along the horizontal axis and represents the length of the time interval, and whose altitudes are the initial and final depth change rates. The average altitude of the trapezoid corresponds to the approximate average depth change rate and, when multiplied by the width (representing the life of the time interval), gives us the area of the trapezoid. This area represents average rate of depth change multiplied by time interval, which is the depth change corresponding to the time interval.
The figure below depicts a typical trapezoid.
If we have a chronicle of depth change rate vs. clock time, we can take any two successive depth change rates and find the average rate which depth changes. We can then multiplied this by the corresponding difference in clock times to obtain the approximate change in depth. The depth changes can be successively added to obtain the total depth change relative to the first clock time represented in the chronicle. This gives us a relative chronicle of depth vs. time.
For example the table below gives the rates of depth change at clock times 30, 40, and 50 seconds. Average rates are given for the interval immediately preceding the given clock time. Since no interval precedes clock time 30 seconds, no average rate is given. The first average rate given is for the interval ending at 40 seconds, and is the average of the rates at clock times 30 seconds and 40 seconds. Depth changes are obtained by multiplying rate of depth change by the corresponding 10-second time interval. The total depth change up to clock time 40 seconds is the -35 cm change in the interval from 30 to 40 seconds. The total depth change up to clock time 50 seconds is obtained by adding the -26 cm change in the interval from 40 to 50 seconds to the total change up to the 40 second clock time.
clock time depth change rate ave rate for preceding interval depth change total depth change to clock time 30 sec -4 cm / sec ** ** 0 40 sec -3 cm / sec -3.5 cm / sec -35 cm -35 cm 50 sec -2.2 cm/sec -2.6 cm / sec -26 cm -61 cm
The figure below depicts this process, with the numbers in the last four columns offset to be 'between' the numbers in the first two columns from which they are derived.
If we wish to find the precise rate at which quantity y is changing with respect to clock time t at a specific clock time, we can calculate average rates starting at the specific time, with the average calculated over smaller and smaller time intervals. The smaller the time interval the closer will be the calculated average rate to the precise rate at the specified clock time. In the limit as the time interval approaches 0, the average rate approaches the instantaneous rate at the specified clock time.
This process can be visualized graphically by imagining smaller and smaller 'slope triangles' starting at the specified point on a graph of depth vs. clock time. The slopes of these triangles approach the slope of the line tension to the depth vs. clock time curve.
If y is a function of x, y = y(x), then the derivative of y at a specified value of x is the instantaneous rate which y is changing at that value of x. This is the limit as `dx approaches 0, of the ratio `dy / `dx as calculated starting from the specified value of x.
By taking the limit as `dt approaches zero of the expression `dy / `dt = [ y(t + `dt) - y(t) ] / `dt, for y(t) = a t^2 + b t + c, we obtain dy / dt = 2 a t + b. This is the derivative of the specified quadratic function at clock time t. This derivative can also be notated as y'(t), or simply y'. Thus when we know the specific quadratic function, we can obtain a formula for its derivative, or rate-of-change function, by substituting the values of the parameters a and b into the formula dy / dt = 2 a t + b.
This formula for the derivative of a quadratic is the first in a series of formulas for derivatives of a variety of functions, including exponential, power, logarithmic and trigonometric functions.
For flow from a uniform cylinder through a uniform hole near the bottom, the depth vs. clock time function is a quadratic function y(t) = a t^2 + b t + c. The rate of depth change function is the rate at which y(t) changes with respect to clock time, and is therefore the derivative y'(t) = 2 a t + b.
The general form for a linear function is often expressed as value = m t + b. If the value of this function represents the rate y'(t) at which some quantity y is changing with respect clock time, then we have the equation y'(t) = m t + b (this is incidentally called a differential equation, and you would do well to remember the terminology). We wish to find the function y(t) for which the derivative y'(t) is m t + b.
We note that if y(t) is a quadratic function y(t) = a t^2 + b t + c, then its derivative is y'(t) = 2 a t + b. This would agree with y'(t) = m t + b if we had a = m / 2. So we conclude that the quadratic function whose derivative is m t + b is y(t) = .5 m t^2 + b t + c.
We note that this quadratic function contains the parameter c, which is unrelated to anything in the given linear function y'(t) = m t + b. This parameter c can therefore have any value at all without affecting the fact that the derivative of y(t) = .5 m t^2 + b t + c is y'(t) = m t + b. There are therefore infinitely many functions, one for each possible value of the parameter c, for which y'(t) = m t + b.
Each of the infinitely many quadratic functions y(t) = .5 m t^2 + b t + c is said to be an antiderivative of y'(t) = m t + b.
If we have a linear rate of depth change function y'(t) = m t + b, then the depth function is an antiderivative y(t) = .5 m t^2 + b t + c of this function. If we wish to find the changing depth between two clock times t= t1 and t = t2, we need only find the difference between the corresponding depths y(t1) and y(t2):
Depth change from clock time t1 to clock time t2 = y(t2) - y(t1) = (.5 m t2^2 + b t1 + c) - (.5 m t1^2 + b t1 + c) = .5m (t2^2 - t1^2) + b (t2 - t1).
We note that this expression does not involve the arbitrary parameter c. While the depth function y(t) depends on the value of c chosen when we take the antiderivative of the rate function, the depth change is affected at all since to calculate the depth change we subtract one depth from another and c 'cancels out'.
We have obtained a formula y' = 2 a t + b for the derivative of the quadratic function y = a t^2 + b t + c. This formula is symbolic, and can be applied to any quadratic function by substituting the values of the parameters a, b and c. We have also seen that if y' = m t + b, the quadratic function of which it is the rate of change is y = .5 m t^2 + b t + c for an arbitrary value of c. Again the formula is symbolic and can be applied to any linear function as long as we know the values of m and b. These are examples of symbolic derivatives and integrals.
Later in the course we will obtain formulas for derivatives of other common functions. We will then devise techniques for turning the process of taking derivatives around in order to find antiderivatives to use in the process of integration.
Examples of formulas we will obtain include, among others:
If we integrate a rate of depth change function to get a depth vs. time function, then take the derivative of this function, we end up with the exact same rate of depth change function we started with.
If we take the derivative of a depth vs. clock time function to get a rate of depth change function, then integrate this function between two clock times, we obtain a depth change which is identical to that predicted by the original depth vs. clock time function. The antiderivative of our rate of depth change function is identical to the original depth vs. clock time function, except that the antiderivative contains and arbitrary constant parameter c which might differ from the original.
If we start with a function f(t) and find its derivative function f'(t), then integrate this derivative function to find an antiderivative F(t), then the original function f(t) will differ from the antiderivative F(t) by a constant amount. The graph of F(t) will be identical in shape to the graph of f(t), and the graphs will be separated by a constant vertical distance.
The geometric representation of such an integration is shown below. The nearly trapezoidal 'slivers' into which the area beneath the graph of the rate function is divided each represent a close approximation of the small depth change over the corresponding time interval. The depth total change is the total of these areas and, as shown above, is equal to the integral of the function f(t) from t = a to t = b.
If a function Q(t) represents some quantity as a function of clock time t, then the average rate at which the quantity Q changes between t clock times t1 and t2 is represented as the slope of the trapezoid connecting the points (t1,0), (t1,Q(t1)), (t2, Q(t2)) and (t2, 0). The rise `dQ = Q(t2) - Q(t1) of the trapezoid represents the change in the quantity Q, and the run `dt = t2 - t1 represents the time interval over which the quantity changes, so the slope rise / run = `dQ / `dt represents the average rate of change of the quantity.
If a function R(t) represents the rate which some quantity changes as a function of clock time t, then the approximate change in that quantity between clock times t1 and t2 is represented as the area of the trapezoid (t1,0), (t1,R(t1)), (t2, R(t2)) and (t2, 0). The average altitude (R(t1) + R(t2)) / 2 of the trapezoid represents the approximate average rate Rave at which the quantities changing, and the width `dt = t2 - t1 represents the time interval over which this approximate rate applies. The area, which is equal to the average altitude multiplied by the width, is therefore Rave * `dt and represents the change in the quantity.
The figure below depicts on the left a trapezoid from a Q vs. t graph and its slope, which is an average rate; the figure on the right depicts a trapezoid from a rate vs. t graph and its area, which is an approximate change in the quantity.
To interpret the slopes of a trapezoid, note that the slope is the rise divided by the run. The change in whatever quantity is represented by the vertical coordinate of the trapezoid is represented by the rise and the change in the quantity represented by the horizontal coordinate (often but not always the clock time t) is represented by the run. Using common sense and a little imagination, we interpret by asking ourselves what it means to divide whatever is represented by the rise of whatever is represented by the run. The result is usually interpreted as the average rate at which the quantity represented by the vertical coordinate changes with respect to the quantity represented by the horizontal coordinate. For example of the vertical coordinate represents depth in the horizontal coordinate represents clock time, we know that the average slope represents the average rate at which depth changes with respect to clock time.
To interpret the areas of a trapezoid, note that the area is the average altitude multiplied by the width. This calculation makes sense when the quantity represented by the vertical coordinate of the graph can be interpreted as a rate. The average altitude then represents an approximate average of whatever rate is represented by the vertical coordinate a graph. The width represents the time interval over which this approximate average rate applies. The area therefore represents the product of the average rate by the time interval over which it applies, which gives the change in the quantity represented by the rate.
The trapezoidal approximation graph of a function over an interval of the independent variable (e.g., the time) is constructed by first dividing the interval into a specified number of subintervals, usually but not necessarily of equal length, and determining the value of the independent variable at the end of each subinterval. The function is then evaluated in each of these values, and the corresponding points are plotted. These points are connected by straight line segments, and vertical segments are constructed from each and of each subinterval to the plotted point above.
The straight line segments stay more or less near the curve representing the graph of the actual function, more so if the subintervals are small. The area under the graph formed by the straight line segments is similarly more or less equal to the area under the graph of the actual function between the endpoints of the original interval, again with 'more' applying to short subintervals.
The area and slope corresponding to each trapezoid are easily calculated. These are labeled by placing the numerical value of the area in a circle within the trapezoid and a slope within a rectangular frame just above the sloping top segment of the trapezoid.
The total area accumulated from the left end of the original interval through the end of a given trapezoid is called the 'accumulated area' and is obtained by adding the area of each trapezoid to the accumulated area through the preceding trapezoid. It is represented by placing its value within parentheses inside each trapezoid. The total area between any two interval endpoints is then equal to the difference in accumulated area between the last trapezoid contain between those endpoints, and the trapezoid to the left of the first endpoint.
At every interval endpoint except the last, there is a change in slope from the subinterval on the left to the subinterval on the right. If the lengths of the subintervals are uniform, as they usually but not always are, then the 'rate of slope change' associated with the endpoint is equal to the change in slope divided by the length of a subinterval. If the lengths of the subintervals are not uniform, then we divide the change in slope by the average of the lengths of the two subintervals involved. We label the rate of slope change within tags of the form < >, and place the label directly above the point to which it applies.
A trapezoidal approximation graph with labeling conventions is shown below. Note that when y represents a rate, areas represent changes in some quantity Q, whereas when y represents a quantity, the slopes represent average rates of change over the various intervals. The accumulated area of the graph up to a point represents the approximate integral of the function from t1 to that point. The slope change rates approximate the second derivative at each point.
Differential equations can express proportionalities between rates of change and the independent and dependent variables. Examples include:
The idea of the differential: Near t = t0, `df / `dt is near f'(t0), so `df is near f'(t0) `dt.
Tangent-line approximation: (y - f(t0) ) / (t - t0) = f'(t0), so y = f'(t0) (t - t0) + f(t0).
Numerical solutions of certain common differential equations can follow the scheme illustrated below. We use y' and `dt to predict `dy and the new y, contining to build our prediction step by step. Every step, of course, increases the probable error in our predictions; but smaller increments make prediction errors smaller.
A geometric depiction of this process is shown below. Starting at (t0, y0), the derivative y' = f(t0, y0) is calculated. We then follow the corresponding tangent line for a distance `dt, arriving at a point we call (t1, y1). This point approximates the t = t1 point of the solution function. At this point we repreat the process, calculating y' = f(y1, t1) to obtain a new tangent line, which we follow for another interval `dt.
At each iteration of this process (an iteration is a repetition of the process from the new point), the probable error of our approximation tends to grow. After a large enough number of iterations, we usually cease to expect that our results are meaningful.
A smaller `dt will allow us to not only make less error per step, but less error over whatever t distance we want to 'travel' to obtain our final approximation.
Predictor-corrector methods use the previous idea to predict the next y, then evaluate the average y' and use this to correct the prediction.
The derivative function has a value at every point that is equal to the limit of `dy / `dt at that point. The value of f'(t) is at every t value equal to the slope of the y = f(t) graph.
We algebraically determine the derivative of a function by writing the definition of the derivative, then simplifying the expression and taking the limit.
Derivative and second derivative estimates may be made based on the y values for three consecutive values of t.
Riemann Sums have the advantage that the difference between the right-hand and left-hand limits is easily quantified as ( f(b) - f(a) ) `dt, which approaches zero as `dt -> 0.
Exponential functions have growth rate r, growth factor b = e^k = 1+r. The doubling time `dt is log 2 / log(b), or ln 2 / k.
Quadratic functions have zeros and vertex easily found from the quadratic formula.
Logarithmic functions are defined as inverse functions to exponential functions and can be visualized by reversing the columns of a table of values for an exponential function. The graphs of a pair of inverse functions are symmetric with respect to the line y = x.
The sine function is defined in terms of a reference circle, with A sin(`theta) equal to the y coordinate of a point on the circle whose radial line makes angle `theta with the positive x axis.
The inverse sine of y is the angle whose sine is y, as illustrated below. The graph of the inverse sine is made by reflecting the graph of the sine function between -`pi/2 and `pi/2 through the line y = x; its domain is from -1 to 1 and its values range from -`pi/2 to `pi/2.
A polynomial function factors into linear and irreducible quadratic factors. It is zero wherever one of its factors is zero, and at no other value of the variable.
A rational function has a polynomial function for its numerator and another polynomial function for its denominator. Its value is zero when the numerator is zero and the denominator isn't; it has a vertical asymptote when the denominator is zero and the numerator isn't.
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