Saying f (x) = 0 can mean different things depending on context If the function being defined, then it means that f (x) = 0 for all x, in which case your reasoning is correct Sometimes, though, it means that for a particular x, f (x) takes the value of zero, in which case f ′ (x) also being zero is notableA function may be thought of as a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image x → Function → y A letter such as f, g or h is often used to stand for a functionThe Function which squares a number and adds on a 3, can be written as f(x) = x 2 5The same notion may also be used to show how a function affects particular valuesGiven f (x) = 3x 2 – x 4, find the simplified form of the following expression, and evaluate at h = 0 This isn't really a functionsoperations question, but something like this often arises in the functionsoperations context
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F(x).g(x) 0 meaning-The expression f(0) represents the yintercept on the graph of f(x) The yintercept of a graph is the point where the graph crosses the yaxis This occurs where x is equal to 0 Therefore, if we plug x = 0 into a function, denoted f(0), the result will be the yinterceptThe output f (x) is sometimes given an additional name y by y = f (x) The example that comes to mind is the square root function on your calculator The name of the function is \sqrt {\;\;} and we usually write the function as f (x) = \sqrt {x} On my calculator I input x for example by pressing 2 then 5 Then I invoke the function by pressing
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Differentiable Meaning A function is said to be differentiable if the derivative of the function exists at all points in its domain Particularly, if a function f (x) f ( x) is differentiable at x = a x = a, then f ′(a) f ′ ( a) exists in the domainThe Meaning of the First Derivative At the end of the last lecture, we knew how to differentiate any polynomial function Polynomial functions are the first functions we studied for which we did not talk about the shape of their graphs in detail To > 0, then f(x) is an increasing function at x = pFor example if F(x) = x 2x 2 = (x2)(x1) = 0 then the graph is all points (x,y) in the plane that satisfy F(x) = 0 This graph is two vertical lines at x = 1 and x = 2 Of course F(x,y) may not actually have y occurring in the equation it may be the second case in disguise The second case may not have any x it may be 0=0 (the entire
We set the denominator,which is x2, to 0 (x2=0, which is x=2) When we set the denominator of g (x) equal to 0, we get x=0 So x cannot be equal to 2 or 0 Please click on the image for a better understanding To find the function value for f (0) you substitute 0 for each occurrence of x in the function f (x) = − 4x for 0 becomes f (0) = − 4 × 0 f (0) = 0 Answer linkAsymptotes Definition of a horizontal asymptote The line y = y 0 is a "horizontal asymptote" of f(x) if and only if f(x) approaches y 0 as x approaches or Definition of a vertical asymptote The line x = x 0 is a "vertical asymptote" of f(x) if and only if f(x) approaches or as x approaches x 0 from the left or from the right
0) = 0 x→x 0 This is the same as saying that the function is continuous, because to prove that a function was continuous we'd show that lim f(x) = f(x 0) x→x 0 We prove lim f(x) − f(x 0) = 0 by multiplying and dividing it by the same x→x 0 number – this won't change its value lim f(x) − f(x 0) = lim f(x) − f(x 0) (x − x 0The known derivatives of the elementary functions x 2, x 4, sin(x), ln(x) and exp(x) = e x, as well as the constant 7, were also used Definition with hyperreals Relative to a hyperreal extension R ⊂ ⁎ R of the real numbers, the derivative of a real function y = f ( x ) at a real point x can be defined as the shadow of the quotient ∆ yShow more Tbh thats about as good as most people would be able to do and i didnt know that f' (x) was dy/dx 0
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Assuming that f is integrable on compact sets, if f (x)= ∫ 0x f (t)dt, then f ′(x) = f (x), and f (0) = 0 The (unique) solution is f (x) = f (0)ex, hence f (x)= 0 for all x)f(y) f(x) f0(x)(y x) Now to establish (ii) ,(iii) in general dimension, we recall that convexity is equivalent to convexity along all lines;\(\lim_{x\rightarrow a0}f(x)=\pm \infty\) or \(\lim_{x\rightarrow a0}f(x)=\pm \infty\) Otherwise, at least one of the onesided limit at point x=a must be equal to infinity For Oblique asymptote of the graph function y=f(x) for the straightline equation is y=kxb for the limit \(x\rightarrow \infty\) if and only if the following two limits
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Ie, f Rn!Ris convex if g( ) = f(x 0 v) is convex 8x 0 2dom(f) and 8v2Rn We just proved this happens i g00( 2) = vTrf(x 0 v)v 0; 1 Answer George C Use definition f '(a) = lim h→0 f (a h) −f (a) h to find f '(x) = 1 √1 2xBut let's use "f" We say "f of x equals x squared" what goes into the function is put inside parentheses after the name of the function So f(x) shows us the function is called "f", and "x" goes in And we usually see what a function does with the input f(x) = x 2 shows us that function "f" takes "x" and squares it
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How to tell where f(x) greater than 0 or f(x) less than 0 How to tell where f(x) greater than 0 or f(x) less than 0If f''(x) >0 on an interval, then f is concave upward on that interval d) If f''(x)A key observation is that a sentence like f(x) =0 f ( x) = 0 or f(x) >0 f ( x) > 0 is a sentence in one variable, x x To solve such a sentence, you are looking for value (s) of x x that make the sentence true The function f f is known, and determines the graph that you'll be investigating
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Get the free "Solve f(x)=0" widget for your website, blog, Wordpress, Blogger, or iGoogle Find more Mathematics widgets in WolframAlphaAnswer and Explanation 1 When presented with the information that f(x) = 2, f ( x) = 2, this means that there exists some function where some input x x produces an output f(x), f ( x), and thatF (x,y)=0 is a two variable function where x and y are variables whereas in the second and third cases they are single variable functions and here we are simply substituting the xy and x/y in the function f 585 views
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Summary "Function Composition" is applying one function to the results of another (g º f) (x) = g (f (x)), first apply f (), then apply g () We must also respect the domain of the first function Some functions can be decomposed into two (or more) simpler functionsIf x is close enough to 0 For the formal definition, suppose f(x) and g(x) are two functions defined on some subset of the real numbers We write f(x) = O(g(x)) (or f(x) = O(g(x)) for x> to be more precise) if and only if there exist constants N and C such that First let's find the derivative f ′ ( x) = 3 x 2 4 x − 1 f ′ ( x) = 3 x 2 4 x − 1 Now, to find the numbers that satisfy the conclusions of the Mean Value Theorem all we need to do is plug this into the formula given by the Mean Value Theorem
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P ( X > x a X > a) = P ( X > x) If X is exponential with parameter λ > 0, then X is a memoryless random variable, that is P ( X > x a X > a) = P ( X > x), for a, x ≥ 0 From the point of view of waiting time until arrival of a customer, the memoryless property means that it does not matter how long you have waited so far If you Answers and Replies is the amplitude of a sinusoidal forcing function, is the undamped natural frequency of the system, and is the frequency of the forcing function, which looks something like or The definition of shown is also the damped natural frequency of the system So are the units for F_0 meters?Hence, the name "piecewise" function When I evaluate it at various x values, I have to be careful to plug the argument into the correct piece of the function
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A function f f f is differentiable at a point x 0 x_0 x 0 if 1) f f f is continuous at x 0 x_0 x 0 and 2) the slope of tangent at point x 0 x_0 x 0 is well defined At point c c c on the interval a, b a, b a, b of the function f (x) f(x) f (x), where the function is continuous on a, b a, b a, b, there is a To break it down to its components, let's define each part of the formula Y the outcome or outcomes, result or results, that you want X the inputs, factors or whatever is necessary to get the outcome (there can be more than one possible x) F the function or process that will take the inputs and make them into the desired outcome Simply put, the Y=f(x) equationFor example, if f is a function that has the real numbers as domain and codomain, then a function mapping the value x to the value g(x) = 1 / f(x) is a function g from the reals to the reals, whose domain is the set of the reals x, such that f(x) ≠ 0 The range of a function is the set of the images of all elements in the domain
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But, I still need t0 understand this in term of mean value theorem So here it goes Suppose f is differentiable on ( 0, ∞) and lim x → ∞ f ′ ( x) = 0 Prove that lim x → ∞ f ( x 1) − f ( x) = 0 Here, can we assume f ′ ( x) = f ( x 1) − f ( x) ( x 1) − x and proceed to conclusion?18B Local Extrema 2 Definition Let S be the domain of f such that c is an element of S Then, 1) f(c) is a local maximum value of f if there exists an interval (a,b) containing c such that f(c) is the maximum value of f on (a,b)∩S 2) f(c) is a local minimum value of f if there exists an interval (a,b) containing c such that f(c) is the minimum value of f on (a,b)∩S 3) f(c) is a local In order to find what value (x) makes f (x) undefined, we must set the denominator equal to 0, and then solve for x f (x)=3/ (x2);
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The functions f(x) and g(x) are defined as f(x) = 3x 1 and g(x) = 4x 2 fog(x) = f(g(x)) = f(4x 2 ) = 3(4x 2) 1 = 12x 6 1 = 12x 7 fog(0) = 12*0 7 = 7Chapter 2 21 Functions definition, notation A function is a rule (correspondence) that assigns to each element x of one set , say X, one and only one element y of another set, Y The set X is called the domain of the function and the set of all elements of the set Y that are associated with some element of the set X is called the range of the functionF (x) basically means y, and f' (x) means dy/dx The x can have a value, so for example, f (x) = 2x 1, then f (1) = 3 that is as good as I can explain it!!!
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The zero of a function is the xvalue that, when plugged into a function, makes the function equal to 0 f (x) means that we have a function of x The zero of the function f (x)=2x2 is the xvalue that makes f (x) = 0 So we can just set the function equal to 0, and then solve for x! The derivative of f (x) f ( x) with respect to x is the function f ′(x) f ′ ( x) and is defined as, f ′(x) = lim h→0 f (x h)−f (x) h (2) (2) f ′ ( x) = lim h → 0 f ( x h) − f ( x) h Note that we replaced all the a 's in (1) (1) with x 's to acknowledge the fact that the derivative is really a function as wellF(x) = (sinx)/x An approximate graph is indicated below Looking at the graph, it is clear that f(x) ≤ 1 for all x in the domain of f Furthermore, 1 is the smallest number which is greater than all of f's values o y=(sin x)/x 1 Figure 1 Loosely speaking, one might say that 1 is the 'maximum value' of f(x)
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It says, for example, that if x< 0, then f' (x)> 0 The derivative, f' (x), can be interpreted as "the slope of the tangent line" f' (x)> 0 means all tangent lines have positive slope are going up to the right Also "if x> 0, then f' (x)< 90" so for x positive, the derivative is negative which means tangent lines are going down to the right17 Inverse Functions Notation The inverse of the function f is denoted by f 1 (if your browser doesn't support superscripts, that is looks like f with an exponent of 1) and is pronounced "f inverse" Although the inverse of a function looks like you're raising the function to theBy the derivative of a function f (x), we mean the following limit, if it exists We call that limit the function f ' (x) " f prime of x " and when that limit exists, we say that f itself is differentiable at x, and that f has a derivative And so we take the limit of the difference quotient as h approaches 0
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8x 0 2dom(f);8v2Rn and 8 st x 0 v2dom(f)Hence, fis convex i >>>is the unsigned bitwise rightshift operator 0x0F is a hexadecimal number which equals 15 in decimal It represents the lower four bits and translates the the bitpattern 0000 1111& is a bitwise AND operation (x >>> 4) & 0x0F gives you the upper nibble of a byte So if you have 6A, you basically end up with 06 6A = ((0110 1010 >>> 4) & 0x0F) = (0000 0110 & 0x0F) =Given the function f (x) as defined above, evaluate the function at the following values x = –1, x = 3, and x = 1 This function comes in pieces;
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