This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nth 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....
View
Full
Document
This note was uploaded on 02/22/2011 for the course MA 1a taught by Professor Borodin,a during the Winter '08 term at Caltech.
 Winter '08
 Borodin,A
 Derivative

Click to edit the document details