Exercise 5.1

Examine the continuity of f, where f is defined by sin cos , if 0( ) 1, if 0

x x xf	x	x

− ≠= − = Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29. 26. cos , if2 2( ) 3, if 2 k x xxf x x π ≠ π −=  π = at x = 2 π 27. 2, if 2( ) 3, if 2 kx xf x x  ≤=  > at x = 2 28. 1, if( ) cos , if

kx xf	x	x	x

≤ π=  > π at x = π 29. 1, if 5( ) 3 5, if 5 | kx xf | x | x | x | |---|---|---|---|
≤=  − > at x = 5 30. Find the values of a and b such that the function defined by 5, if 2 | ( ) , if | 2 | 10 | |---|---|---|

21, if 10 x

f x ax	b	x
x / ≤ = + < <  ≥ is a continuous function. 31. Show that the function defined by f (x) = cos (x2) is a continuous function. 32. Show that the function defined by f (x) =	cos x	is a continuous function. 33. Examine that sin

( ) ( )lim h

f	c	h	f	c
h→ / + − f (x) x n sin x cos x tan x f ′(x) nx n - 1 cos x - sin x sec² x provided this limit exists. Derivative of f at c is denoted by f ′(c) or ( ( ))	c d f xdx . The function defined by

( ) ( )( ) lim h f x h f xf x h→ + −′ = wherever the limit exists is defined to be the derivative of f. The derivative of f is denoted by f ′ (x) or ( ( ))d f xdx or if y = f (x) by dy dx or y′. The process of finding derivative of a function is called differentiation. We also use the phrase differentiate f (x) with respect to x to mean find f ′(x). The following rules were established as a part of algebra of derivatives: (1) (u ± v)′ = u′ ± v′ (2) (uv)′ = u′v + uv′ (Leibnitz or product rule) (3) u u v uv v v ′ ′ − ′  =   , wherever v ≠ 0 (Quotient rule). The following table gives a list of derivatives of certain standard functions: Table 5.3 Whenever we defined derivative, we had put a caution provided the limit exists. Now the natural question is; what if it doesn’t? The question is quite pertinent and so is its answer. If ( ) ( )lim h

f	c	h	f	c

h→ + − does not exist, we say that f is not differentiable at c. In other words, we say that a function f is differentiable at a point c in its domain if both -0 ( ) ( )lim h

f	c	h	f	c

h→ + − and ( ) ( )lim h

f	c	h	f	c

h / +→ + − are finite and equal. A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b]. As in case of continuity, at the end points a and b, we take the right hand limit and left hand limit, which are nothing but left hand derivative and right hand derivative of the function at a and b respectively. Similarly, a function is said to be differentiable in an interval (a, b) if it is differentiable at every point of (a, b). MATHEMATICS120

f	x	x
x x / + ≤ − = − − <  + ≥ 8.		, if 0( ) 0, if 0
x xf	x	x
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x x / + ≤ − = − − <  + ≥ 8.

, if 0( ) 0, if 0

x xf

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