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An exclamation point denotes something called a factorial. The formal definition of n! (n factorial) is the product of all the natural numbers less than or equal to n. In math symbols: n! = n*(n-1)*(n-2)... Trust me, it's less confusing than it sounds. Say you wanted to find 5!. You just multiply all the numbers less than or equal to 5 until you get to 1: 5! = 5*4*3*2*1=120 Or 6!: 6! = 6*5*4*3 ...
Explanation: The points of continuity are points where a function exists, that it has some real value at that point. Since the question emanates from the topic of 'Limits' it can be further added that a function exist at a point 'a' if lim x→a f (x) exists (means it has some real value.) The points of discontinuity are that where a function ...
A function has a discontinuity if it isn't well-defined for a particular value (or values); there are 3 types of discontinuity: infinite, point, and jump. Many common functions have one or several discontinuities. For instance, the function y=1/x is not well-defined for x=0, so we say that it has a discontinuity for that value of x. See graph below. Notice that there the curve does not cross ...
A tangent line can be defined as the equation which gives a linear relationship between two variables in such a way that the slope of this equation is equal to the instantaneous slope at some (x,y) coordinate on some function whose change in slope is being examined. In essence, when you zoom into a graph a lot, it will look more and more linear ...
Answer: Boundedness is about having finite limits. In the context of values of functions, we say that a function has an upper bound if the value does not exceed a certain upper limit. More... Explanation: Other terms used are "bounded above" or "bounded below". For example, the function f (x) = 1 1 + x2 is bounded above by 1 and below by 0 in that:
The definition of limit of a sequence is: Given {an} a sequence of real numbers, we say that {an} has limit l if and only if. ∀ε> 0, ∃n0 ∈ N/ ∀n> n0 ⇒ |an −l|) <ε. F. Javier B. · 2 · Jun 10 2018.
The formal definition of derivative of a function y=f (x) is: y'=lim_ (Deltax->0) (f (x+Deltax)-f (x))/ (Deltax) The meaning of this is best understood observing the following diagram: The secant PQ represents the mean rate of change (Deltay)/ (Deltax) of your function in the interval between x and x+Deltax. If you want the rate of change, say ...
It tells you how distance changes with time. For example: 23 km/h tells you that you move of 23 km each hour. Another example is the rate of change in a linear function. Consider the linear function: y = 4x +7. the number 4 in front of x is the number that represent the rate of change. It tells you that every time x increases of 1, the ...
This simple equation is called the slope formula. If y = f (x +h) = 3(x +h)2, (Just plug x + h in for x). So, you get this: The instantaneous rate of change, or derivative, can be written as dy/dx, and it is a function that tells you the instantaneous rate of change at any point. y' = f '(x + h) = (d dx)(3 ⋅ (x)2) = 6x ⋅ 1 = 6x.
The derivative of a function f (x) is written f' (x) and describes the rate of change of f (x). It is equal to slope of the line connecting (x,f (x)) and (x+h,f (x+h)) as h approaches 0. Evaluating f' (x) at x_0 gives the slope of the line tangent to f (x) at x_0.