Exercise - 1
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1) Show that f(x)= | x | is not differentiable at x= 0
2) Show that the function
f(x)= x-1, if x < 2
2x -3, if x ≥ 2 is differentiable at x= 2.
3) Show that the function
f(x)= x² sin(1/x), if x≠ 0
0 , x = 0 is differentiable at x= 0 and f'(0)=0
4) Show that f(x)= x² is differentiable at x= 1 and find f'(1).
5) Discuss the differentiability of
f(x)= ₓₑ -(1/|x| + 1/x) x≠ 0
0 x= 0 at x= 0
6) If f(x) is differentiable at x= a, find lim ₓ→ₐ {x²f(a) - a²f(x)}/(x-a)
7) For what choice of a and b is the function f(x)= x² , x≤ c
ax+b , x> c is differentiable at x= c.
**8) If f(2)= 4 and f'(2)= 1, then find
lim ₓ→² {xf(2) - 2f(x)}/(x-2)
9) Discuss the differentiability of f(x)= x |x| at x= 0.
10) Show that the function
f(x)= x sin(1/x) , when x ≠ 0
0 , when x= 0 is continuous but not differentiable at x= 0.
EXERCISE --2
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1) Show that
f(x)= 12x - 13, if x ≤ 3
2x² +5, if x > 3 is differentiable at x= 3. Also, find f'(3). 12
2) Show that f(x)= x¹⁾³ is not differentiable at x= 0.
3) Show that function
f(x)= x ᵐ sin(1/x), x ≠ 0
0, x= 0 is
a) differentiable at x= 0, if m> 1
b) continuous but not differentiable at x= 0, if 0<m<1
c) neither continuous nor differentiable, if m ≤ 0.
4) Show that f(x)= | x - 2| is continuous but not differentiable at x= 2.
5) Find the values of a and b so that the function
f(x) = x²+ 3x+a, if x ≤ 1
bx +2, if x > 1 is differentiable at each x belongs to R. 3,5
6) Show that the function
f(x)= | 2x -3 | [x], x ≥ 1
sin(πx/2), x < 1 is continuous but not differentiable at x=1.
7) If f(x)= ax² - b, if |x| < 1
1/|x|, if |x| ≥ 1 is differentiable at x= 1, find a,b. -1/2, -3/2.
8) for what choice of a and b is the function f(x)= x², x≤ c
ax+b, x> c is differentiable at x= c. 2c, - c²
9) If f is Defined by f(x)= x², find f'(2). 4
10) If f is defined by f(x)= x² - 4x +7, Show that f'(5)= 2 f'(7/2)
11) Show that the derivative of the function f given by f(x)= 2x³ - 9x² + 12x +9, at x= 1 and x= 2 are equal.
12) If for the function $(x)= Kx² + 7x - 4, $'(5)= 97, find K. 9
13) If f(x)=x³+ 7x²+ 8x -9, find f'(4). 112
14) Discuss the continuity and differentiability:::
a) f(x)=| log | x | |. No at x=±1
b) ₑ | x |. No at x= 0
c) f(x)= (x-c) cos {1/(x-c)}, x ≠ c
0 , x= c
Not differentiable at x= c
15) Is | sin x | differentiable ? What about cos | x | ? Not differentiable at x=nπ, n belongs to Z,
is everywhere differentiable.
VERY SHORT ANSWER QUESTIONS
+++++++++++++++++++++++(
1) Is every differentiable function continuous ? Yes
2) Is every continuous function differentiable ? No
3) Give an example of a function which is continuous but not differentiable at a point. f(x)=|x| at x= 0
4) If f(x) is differentiable at x= c, then write the value of lim ₓ→꜀f(x). f(c)
5) If f(x)= | x -2 | write whether f'(2) exist or not. Doesn't exist
6) write the points where f(x)=|logₑx| is not differentiable. 1
7) write the points of non-differentiability of f(x)=|log|x||. ±1
8) write the derivatives of f(x)= |x|³ at x= 0. 0
9) Write the number of points where f(x)= |x|+|x-1| is continuous but not differentiable. 0,1
10) If lim ₓ→꜀{f(x) - f(c)}/(x-c) exists finitely,write the value of lim ₓ→꜀f(x) f(c)
11) Write the value of the derivatives of f(x)=|x-1| + |x-3| at x= 2. 0
12) If f(x)= √(x²+9), write the value of lim ₓ→₄ {f(x) - f(4)}/(x-4). 4/5
MULTIPLE CHOICE QUESTIONS
********************************
1) Let f(x)= | x | and g(x)= | x |³, then
A) f(x) and g(x) both are continuous at x= 0.
B) f(x) and g(x) both are differentiable at x= 0
C) f(x) is differentiable but g(x) is not differentiable at x = 0.
D) f(x) and g(x) both are not differentiable at x= 0. ∞∈ₑ⁺⁻ₑ
2) the function f(x)= sin⁻¹(cosx) is
A) discontinuous at x= 0
B) continuous at x= 0.
C) differentiable at x= 0 D) none
3) the set of points where the function f(x)= x |x| is differenciable is
A) (-∞,∞). B)(-∞,0)U(0,∞)
C) (0,∞) D) [0,∞]
4) If f(x)= {|x+2|}/{tan⁻¹(x+2)}, x≠-2
2 , x=-2
then f(x) is
A) continuous at x=-2
B) not continuous at x=-2.
C) differentiable at x=-2
D) continuous but not differentiable at x=-2
5) let f(x)=(x + |x|) |x|. then, for all x
A) f is continuous.
B) f is differentiable for some x.
C) f' is continuous.
D) f" is continuous
6) the function f(x)= ₑ -|x| is
A) continuous everywhere but not differentiable at x= 0.
B) continuous and differentiable everywhere
C) not continuous at x= 0 D) none
7) the function f(x)= | cosx| is
A) everywhere continuous and differentiable.
B) everywhere continuous but not differentiable at (2n+1)π/3, n ∈Z.
C) neither continuous nor differentiable at +2n+1)π/2, n∈ Z
D) none
8) If f(x)=√{1 - √(1-x²)}, then f(x) is
A) continuous on [-1,1] and differentiable on (-1,].
C) continuous on [-1,1] and differentiable on (-1,0)U (0,1).
D) continuous on [-1,] D) none
9) if f(x)= a |sinx| + b ₑ |x| + c|x|³ and if f(x) is differentiable at x= 0, then
A) a= b=c=0 B) a=0,b=0, c∈ R.
C) b=c=0, a ∈ R
D) c=0,a=0, b∈ R
10) If f(x)= x² + x²/(1+x²) + x²/(1+x²)² + ....+ x²(1+x²)ⁿ +....., then at x= 0, f(x)
A) has no limit
B) is discontinuous.
C) is continuous but not differentiable
D) is differenciable
11) If f(x)= | logx | then
A) f'(1⁺)=1. B) f'(1⁻)= -1.
C) f'(1)=1 D) f'(1)= -1
12) If f'(x) = |log |x| |, then
A) f(x) is continuous and differentiable for all x in its domain
B) f(x) is continuous for all x in its domain but not a differentiable at x=±1.
C) f(x) is neither continuous nor differenciable at x= ±1 D) none
13) let f(x)= 1/|x | for | x|≥1
ax²+b for |x|<1
If f(x) is continuous and differentiable at any point, then
A) a= 1/2 , b= -3/2
B) a= -1/2 , b= 3/2.
C) a= 1 , b= - 1 D) none
14) The function f(x)= x - [x], where [.] denote the greatest integer function is
A) continuous everywhere
B) continuous at integers points only
C) continuous at non-integers points only.
D) differentiable everywhere
15) Let f(x)= ax²+1, x> 1
x+ 1/2, x≤ 1.
Then, f(x) derivable at x=1, if
A) a= 2 B) a= 1 C)a= 0 D) a=1/2.
16) Let f(x)= | sinx |. Then
A) f(x) is everywhere differentiable
B) f(x) is everywhere continuous but not differentiable at x=nπ, n ∈Z.
C) f(x) is everywhere continuous but not differentiable at x=(2n+1)π/2, n ∈Z D) none
17) Let f(x)= | cosx |. Then,
A) f(x) is everywhere differentiable
B) f(x) is everywhere continuous but not differentiable at x=nπ, n ∈Z
C) f(x) is everywhere continuous but not differentiable at x=(2n+1)π/2, n ∈Z.
D) none
18) the function f(x)= 1+ | cosx | is
A) continuous nowhere
B) continuous everywhere.
C) not differentiable at x= 0
D) not differentiable at x=nπ, n ∈Z
19) the function f(x)= | cosx | is
A) differentiable at x=(2n+1)π/2, n ∈Z
B) continuous but not differentiable at x=(2n+1)π)2 n ∈Z.
C) neither differentiable nor continuous at x=nπ, n ∈Z
D) none
20) the function f(x)={sin(π[x-π])}/(4+[x²]) , where [.] denotes greatest integer function, is
A) continuous as well as the differenciable for all x ∈R.
B) continuous for all x but not differentiable at some x
C) differenciable for all but not continuous at some D) none
21) Let f(x)= a+ b | x | + c| x |⁴, where a,b,c are real constants. Then (x) is differentiable at x= 0, If
A) a= 0 B) b= 0. C) a= 0 D) none
22) If the function f(x)= | 3- x | + (3+x), where (x) denotes the least integer greater than or equal to x, then f(x) is
A) continuous and differentiable at x= 3
B) continuous but not differentiable at x= 3
D) differentiable but not continuous at x= 3
D) neither differentiable nor continuous at x= 3.
23) If f(x)= 1/(1+ₑ 1/x), x ≠ 0
0 , x= 0
then f(x) is
A) continuous as well as differentiable at x= 0
B) continuous but not differentiable at x= 0
C) differentiable but not continuous at x= 0 D) none.
24) If f(x)= (1-cosx)/x sinx, x ≠ 0
1/2 , x= 0
then at x= 0, f(x) is
A) continuous and differentiable.
B)differenciable but not continuous
C) continuous but not differentiable
D) none
25) The set of points where the function f(x) given by
f(x)= |x - 3| cosx is differenciable, is
A) R B) R -{3} C)(0,∞) D) none
26) Let f(x)= 1 , x ≤ -1
|x|, -1<x<1
0 , x≥ 1
then, f is
A) continuous at x= -1.
B) differentiable at x= -1
C) everywhere continuous
D) everywhere differentiable
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