Chapter4

Chapter4 - 4 Differential Calculus 4.1 Basic Notions...

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Unformatted text preview: 4 Differential Calculus 4.1 Basic Notions Definition 4.1 The function f defined on a neighborhood of a ∈ R is called differentiable at a if the limit lim h → f ( a + h ) − f ( a ) h exists. It is called the derivative of f at a and is denoted as f ′ ( a ) (or as df dx ( a ) .) Recall that a neighborhood of a point a contains some open interval I = ( a − t,a + t ), and the limit above makes sense because for h suﬃciently close to 0, a + h lies in I . Note that h can be positive and negative. If one restricts to h > 0, then the corresponding limit is called the right derivative of f at a , and similarly for the left derivative (where h < 0). The derivative exits at a iff both the right and left derivatives exist and are equal. Examples : 1) Derivative of a constant function exists at any a and equals 0. 2) f ( x ) = mx + b ⇒ f ′ ( a ) = m for any a ∈ R . 3) f ( x ) = x n . Recall the binomial formula : ( a + h ) n = n ∑ j =0 ( n j ) a n − j h j = a n + na n − 1 h + O ( h 2 ) , where O ( h 2 ) denotes the sum of terms of order at least h 2 . Hence f ′ ( a ) = lim h → ( a + h ) n − a n h = lim h → { na n − 1 + O ( h ) } = na n − 1 . 4) f ( x ) = sin x . Since sin y − sin x = 2 sin y − x 2 cos y + x 2 , we have, for any a ∈ R , f ′ ( a ) = lim h → sin( a + h ) − sin( a ) h = lim h → sin( h/ 2) h/ 2 cos ( a + h 2 ) = cos a. 1 Similarly, (cos x ) ′ = − sin x . 5) f ( x ) = x 1 n , for n ∈ N . Set u = ( a + h ) 1 n and v = a 1 n . Then f ( a + h ) − f ( a ) h = u − v u n − v n = 1 u n − 1 + u n − 2 v + ··· + v n − 1 , which goes to 1 n a 1 n − 1 as h → 0. We say that f is differentiable on an open interval I iff f is differen- tiable at every point in I . In this case, we may treat f ′ as a function on I . In general f ′ may not be continuous, and if it is so at a (resp. on I ), we will say that f is C 1 , meaning it is continuously differentiable at a (resp. on I ). We will say that f is twice differentiable at a iff f and f ′ are both differentiable at a . We put f ′′ = ( f ′ ) ′ and call it the second derivative . This way we can define the third and fourth derivatives, and in fact, for any n ∈ N , the n-th derivative of f , denoted by f ( n ) , as the derivative of f ( n − 1) . f is said to be C n at a (or on I ) iff all the derivatives f ( j ) exist for j ≤ n and are continuous at a (resp. on I ). Finally, one says that f is C ∞ , or infinitely differentiable , iff f ( n ) exists for every n . The simplest examples of infinitely differentiable functions on all of R are polynomials and the sine and cosine functions. The same holds for rational functions and other trigonometric functions at the set of points where they are defined....
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Chapter4 - 4 Differential Calculus 4.1 Basic Notions...

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