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Limits in calculus?
A limit in calculus is a value which a function, f(x), approaches at particular value of x. They can be used to find asymptotes, or boundaries, of a function or to find where a graph is going in ambiguous areas such as asymptotes, discontinuities, or at infinity. There are many different ways to find a limit, all depending on the particular function. If the function exists and is continuous at the value of x, then the corresponding y value, or f (x), is the limit at that value of x. However, if the function does not exist at that value of x, as happens in some trigonometric and rational functions, a number of calculus "tricks" can be applied: such as L'Hopital's Rule or cancelling out a common factor.
Use the concept of a limit to explain how you could find the exact value for the definite integral value for a section of your graph?
The definite integral value for a section of a graph is the area under the graph. To compute the area, one method is to add up the areas of the rectangles that can fit under the graph. By making the rectangles arbitrarily narrow, creating many of them, you can better and better approximate the area under the graph. The limit of this process is the summation of the areas (height times width, which is delta x) as delta x approaches zero. The deriviative of a function is the slope of the function. If you were to know the slope of a function at any point, you could calculate the value of the function at any arbitrary point by adding up the delta y's between two x's, again, as the limit of delta x approaches zero, and by knowing a starting value for x and y. Conversely, if you know the antideriviative of a function, the you know a function for which its deriviative is the first function, the function in question. This is exactly how integration works. You calculate the integral, or antideriviative, of a function. That, in itself, is called an indefinite integral, because you don't know the starting value, which is why there is always a +C term. To make it into a definite integral, you evaluate it at both x endpoints of the region, and subtract the first from the second. In this process, the +C's cancel out. The integral already contains an implicit dx, or delta x as delta x approaches zero, so this becomes the area under the graph.
What does a graph tell you?
A graph of an equation (or function) helps to clarify the behavior of that equation. In this case, the behavior of the graph is just that: it describes how something acts-- for example:Whether it is a straight line or a bending curveHow many times it changes direction and whereWhether the y-value becomes greater or smaller (moves up or down), or stays constant, as it moves from left to rightIf it is discontinuous (skips around without warning, turns sharply, flies up into infinity for a while, or simply vanishes for a short time)What the equation must look like, such as a line for a linear equation (y = mx + b) or a parabola for a quadratic equation (y = ax2 + bx + c)When the equation crosses the x-axis, something that is very useful to know in Algebra and later mathematicsHow fast the equation is increasing or decreasingIn Calculus, a graph can be used to find the derivative of a function, which is a new function that describes the slope of a function at each pointIn general, a graph is a very useful tool to understand how an equation works, and can make encounters with new and unfamiliar forms of equations easier to understand.
What does a graph with infinitely many solutions look like?
they have same slop.then two linear equations have infinite solutions
What is the graph of a function when it has no gaps or holes?
It is a continuous line whose shape depends upon what expression it is meant to represent.The equation y = x would be a straight line passing through (0,0) and all the other points where the x and y co-ordinates were equal, including negative ones such as (-11,-11).But if the equation has x squared in it the shape would be a parabola, while the graph of an equation with y cubed in it would have something like an S shape in it. More complex equations could produce many differently shaped lines.
