How to check if function is continuous and differentiable? Show that the function, f(x) = \left\{\begin{matrix} (x^2 + 1), & if & x \leq -1\-2x, & if & x > -1 \end{matrix}\right. is continuous and differentiable at the value x = -1. ...
For a function y=f(x), if the left-hand derivative f′(a−)and the right-hand derivative f′(a+) are equal at a point, the function is said to be differentiable at that point. The limit-definition of derivative of f(x) is as follows: ...
What is a continuous function? Different types (left, right, uniformly) in simple terms, with examples. Check continuity in easy steps.
where multiplication needn't be commutative). In Lean, we like to make every lemma as general as possible. A part of the reason is that we can make reasoning "by lemma XYZ" but not reasoning "by the
Afunctionthat has a continuous derivative isdifferentiable; It’s derivative is acontinuous function. How do I know if I have a continuous derivative? As the definition of a continuous derivative includes the fact that the derivative must be a continuous function, you’ll have tocheck for contin...
forhto 0, these points will lie infinitesimally close together; therefore, it is the slope of the function in the pointx.Important to note is that this limit does not necessarily exist. If it does, the function is differentiable; if it does not, then the function is not differentiable. ...
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Let g(x) and h(y) be differentiable functions, and let f(x, y) = h(y)i+ g(x)j. Can f have a potential F(x, y)? If so, find it. You may assume that F would be smooth. (Hint: Consider the mixed partial derivatives of F.) How to answer this question? Wha...
is never exactly equal to L), just a value arbitrarily close to L?! PeroK said: All I'd add is that if that limit does not exist at some point x0x_0 then ff is not differentiable at x0x_0. And, such functions do exist! Good point, I had meant to put that in my ...
aTaylor ' s Theorem, which will be introduced in this section, is a basic theorem to show how to approximate a given differentiable function by means of polynomials which has important applications in theoretical research and approximate calculations. 泰勒‘s定理,在这个部分将被介绍,是显示如何的一...