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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.
How many non-verticle asymptotes can a rational function have?
Not sure what non-verticle means, but a rational function can have up to 2 non-vertical asymptotes,
Can the graph of a rational function have more than one vertical asymptote?
Assume the rational function is in its simplest form (if not, simplify it). If the denominator is a quadratic or of a higher power then it can have more than one roots and each one of these roots will result in a vertical asymptote. So, the graph of a rational function will have as many vertical asymptotes as there are distinct roots in its denominator.
How many vertical asymptotes can there be in a rational function?
Factoring is usually helpful in identifying zeros of denominators. If there are not common factors in the numerator and the denominator, the lines x equal the zeros of the denominator are the vertical asymptotes for the graph of the rational function. Example: f(x) = x/(x^2 - 1) f(x) = x/[(x + 1)(x - 1)] x + 1 = 0 or x - 1 = 0 x = -1 or x = 1 Thus, the lines x = -1 and x = 1 are the vertical asymptotes of f.
Sketch a Tangent Functions?
A tangent function is a trigonometric function that describes the ratio of the side opposite a given angle in a right triangle to the side adjacent to that angle. In other words, it describes the slope of a line tangent to a point on a unit circle. The graph of a tangent function is a periodic wave that oscillates between positive and negative values. To sketch a tangent function, we can start by plotting points on a coordinate plane. The x-axis represents the angle in radians, and the y-axis represents the value of the tangent function. The period of the function is 2π radians, so we can plot points every 2π units on the x-axis. The graph of the tangent function is asymptotic to the x-axis. It oscillates between positive and negative values, crossing the x-axis at π/2 and 3π/2 radians. The graph reaches its maximum value of 1 at π/4 and 7π/4 radians, and its minimum value of -1 at 3π/4 and 5π/4 radians. In summary, the graph of the tangent function is a wave that oscillates between positive and negative values, crossing the x-axis at π/2 and 3π/2 radians, with a period of 2π radians.
What is the graph combination of line segments that passes the vertical line test?
Many to one function
What is vertical line test in a function?
If, at any time, a vertical line intersects the graph of a relationship (or mapping) more than once, the relationship is not a function. (It is a one-to-many mapping and so cannot be a function.)
How many asymptotes can a bounded function have?
I believe the maximum would be two - one when the independent variable tends toward minus infinity, and one when it tends toward plus infinity. Unbounded functions can have lots of asymptotes; for example the periodic tangent function.
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.
Which function has no horizontal asymptote?
Many functions actually don't have these asymptotes. For example, every polynomial function of degree at least 1 has no horizontal asymptotes. Instead of leveling off, the y-values simply increase or decrease without bound as x heads further to the left or to the right.
How many asymptotes does a ellipse have?
None.
How to identify the graph of a function?
Every function is a graph. So the only thing is to distinguish functions from other graphs. One formal convention actually define function as its graph, and a graph is the set of all ordered pairs (x, y) A function is a special graph where it's set set of all ordered pairs (x, y) where y = f(x). f(x) is unique (or rather one goes in only one comes out), meaning for each x, there is one and only one y. (Note: For each y, there might be many x) So to test this, we use a "vertical line test". The idea is for all x in the domain of f, say A, we draw a vertical line (x = a for some a in A), it only intersect the graph of f one and only once. Of course, there are infinity many points, you have to do it infinitly many times. Therefore, you can do it generacally: Let A:= dom f For all a in A, f is a function if and only if (x = a implies f(x) = f(a) and nothing else)
Which kind of graph is always a function?
Any graph of a mapping which is one-to-one or many-to-one but not one-to-many.
