alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

An we note that value xa fx an we can differentiate

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Unformatted text preview: taking the δ timit we can find the INSTANTANEOUS →0 RATE OF CHANGE at any par ticular instant x2 or x3 or x4, in the time interval par eneral x0 to xn. We may drop the subscript and let x be a general instant in [x0 , xn]. f(x) f(x) (0 , 0 ) L imit δx → 0 f (x + δ x) -- f (x) = ( x + δ x) -- x x1 x0 L imit δx → 0 x2 x3 . . . xn x f (x + δ x) -- f (x) δx Definition : instantaneous rate of change of f(x) f ’ ( x) = δL imit0 x→ f (x + δ x) -- f (x) δx This is called the FIRST DERIVATIVE of f(x1). There are several other notations for the FIRST DERIVATIVE : . df --- , f’ ( x), D f ( x), f (x) dx So far we have only a NOTATION, some symbols that express the idea of what we are trying to do: find an expression for the INSTANTANEOUS RATE OF CHANGE AT ANY INSTANT. 63 We shall use the words DIFFERENTIATE or FIND THE DERIVATIVE to denote the process of finding the expression for the INSTANTANEOUS RATE OF CHANGE of a function. Properly speaking we should say : LEFT DERIVATIVE of f(x) = δL imit0 x→ f (x -- δ x) -- f (x) -- δ x f (x + δ x) -- f (x) δx If the LEFT DERIVATIVE = the RIGHT DERIVATIVE and is something well-defined, ell-defined then and only then we may combine both the LEFT DERIVATIVE and the RIGHT DERIVATIVE into one expression and say : L imit RIGHT DERIVATIVE of f(x) = DEFINITION : δx → 0 f ’ ( x) = δL imit0 x→ f (x + δ x) -- f (x) δx Even with SINGLE VALUED and CONTINUOUS functions, it is quite possible when taking the LIMITS to find the derivatives, we may end up with something undefined such as +∞, −∞, division by zero and ∞/∞ , or the LEFT DERIVATIVE ≠ RIGHT DERIVATIVE. In this case we say the function f(x) is not DIFFERENTIABLE at that instant. Note : In A LITTLE MORE CALCULUS the student will learn the MEAN VALUE THEOREM which states that : INSTANTANEOUS RATE OF CHANGE at some MEAN point (we do not know which) = denoted by -x in the interval [a,b] f ( -- ) = x 64 AVERAGE RATE OF CHANGE over the interval [a,b] f(b)...
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