If the force required to remove the coatings is plotted against film thickness, a graph as illustrated schematically in Fig. 5 may characteristically result.

If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If this triangulated graph is colorable using four colors or less, so is the original graph since the same coloring is valid if edges are removed.

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

If the graph is directed, the direction is indicated by drawing an arrow.

If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis ( where, y is zero ), is − C / A.

If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis ( where x is zero ), is − C / B, and the slope of the line is − A / B.

* If asked to imagine the graph of the relationship y = ax < sup > 2 </ sup >, one typically visualizes a range of values of x, but only one value of a.

( If the graph has no weights, 1 indicates an edge, 0 indicates no edge.

If no path exists between two cities, adding an arbitrarily long edge will complete the graph without affecting the optimal tour.

If a variable undergoes regular oscillations, and a graph of the system is drawn with the oscillating variable as the vertical axis and time as the horizontal axis, the amplitude is visually represented by the vertical distance between the extrema of the curve and the equilibrium value.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths.

If there are n vertices in the graph, then each tree has n-1 edges.

If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If birds don't belong in a Square or Park, what does??

If private brand competition hasn't been felt in your product field as yet, have you thought how you will cope with it if and when it does appear??

If Af leads to an address Af that is equal to the address computed from Af, even though Af does not match Af, the chain of information cells is extended from Af by storing the address of the next available cell in the Y-region, Af, in Af.

If both George and a piece of information George does not have are prominent in the context, but the idea of telling George is new, then dominant stress will probably be on tell in why not tell George??

If, on the other hand, it is not settled, or while it is being debated and resolved, does the Federal government have a responsibility in situations like that in Prince Edward County??

If this threefold division of costs were to have its counterpart in the actual rates of charge for service, as it actually does have in some rates, there would result a three-part rate for any one class of service.

If this attitude is seriously questioned in the Soviet Union, it does not necessarily follow that the majority of the society in which I live is too aware of the necessity for clarity on this ethical as well as aesthetic point of view.

If a continuation card follows a statement that does not permit continuation cards, the compiler will generate a NOP and issue an error message.

If the compiler does encounter such statements, a warning message will be issued.

If the fourth dimension is a physical concept and not purely metaphysical, through what medium does it extend??

If a child does lose his first teeth prematurely because of decay -- and if no preventive steps are taken -- the other teeth may shift out of position, become overcrowded and malformed.

If he does, it's still better than an even chance he won't notice the transposition of the numbers, and if he should notice it, the thing can be passed off as an honest mistake.

If the administration does not succeed in passing the sales tax bill, or any other tax bill, it could very well be faced this spring at the fiscal session of the Legislature with an interesting dilemma.

If Bultmann's own definition of myth is strictly adhered to ( and it is interesting that this is almost never done by those who make such pronouncements ), the evidence is overwhelming that he does not at all exaggerate the extent to which the mythological concepts of traditional theology have become incredible and irrelevant.

If you know that he ( God ) is just ( righteous ), know that everyone also who does what is just ( righteous ) has been born of him.

If I get a low M. R., I do so because I want it that way or my dark self does.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the appellate court does find a legal defect in the decision " below " ( i. e., in the lower court ), it may " modify " the ruling to correct the defect, or it may nullify (" reverse " or " vacate ") the whole decision or any part of it.

If it does, ocean levels would rise by a few metres in a very short period of time.

If, however, there are no shares left when the chain is founded, then the founding player does not receive the free share.

If Anne formed an attachment to Weightman it does not imply that he was attracted to her.

If the numbering does not start at 0, the constant B may not be the address of any element.

If abiotic stress does not have a strong presence in an area, the biodiversity will be much higher.

If light moves instantaneously, the telescope does not move, and the true direction of the star relative to the observer can be found by following the line ES.