Cengage Maths Solutions Algebra Limits and Derivatives

CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER LIMITS AND DERIVATIVES || ALGEBRA  Download Doubtnut Today

Ques No.

Question CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous

1

Let f(x) = x +

1 2x +

. ′

1 2x +

Compute the value of f(50)f (50)

1 2x + ∞

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

If 2

x2 + y 2 = R2 (whereR > 0)andk y' ' = 3 (1 + y'2 ) then find k in terms of R alone.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If 3

x2 1 y= + x √x 2 + 1 2 2

+ (log) e √x + √x2 + 1 , prove that 2y = xy ′ + (log)e y ′ , wherey' denotes the derivative w.r.t x.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

y =

4

2

√a 2 − b2 a − b tan x {tan − 1( ) } a+b 2

, then show that

d 2y dx2

=

b sin x (a + b cos x)2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 5

Differentiate

tan − 1 x 1 + √(1 −

x2 )

+ {2 tan

−1

1−x . . . √( )}sin wrtx.

1+x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

6

1 n−1 If y = ( ) cos(n cos − 1 x), then prove that y satisfies the differential 2 2 d y dy 2 2 equation (1 − x ) − x + n y=0 2 dx dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If x ∈ (0,

π ), thenshowthat 2 7 d cos − 1 { (1 + cos 2x) 2 dx

7

+ (√sin2 x − 48 cos2 x)sin x}

=1+

7 sin x √sin 2 − 48 cos 2 x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

f(x) = cos − 1( − 3 sin x) 8

+ sin − 1 (

1

√13

1 √13

)(2 cos x

)(2 cos x

..

+ 3 sin x)wrt√1 + x2 , df(x) 3 then find at x = .

4 dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous If 9

|a1 sin x + a2 sin 2x + + an sin nx| ≤ |sin x| for x ∈ R, then prove that |a1 + 2a1 + 3a + 3 + nan| ≤ 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous .

10

.

cos x cos x cos x sin x Given that = .

2 4 8 x 1 sec 2 x 1 sec2 x + 4 +.

2 2 4 2 2

Then

find

the

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous If 0 < x < 1, then="" prove="" that=""

(1 − 2x) = (1 − x = x2 ) 11

2x − 4x3 4x3 − 8x7 + + + 2 4 4 8 1−x +x 1−x +x 1 + 2x .... ∞ = 1 + x + x2  Watch Free Video Solution on Doubtnut

sum

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If 12

d [(xm − A 1xm − 1 + A 2xm − 2 − ... dx m .... + ( − 1) A m )ex ] = xm ex , find the value of A r  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

13

AND

DERIVATIVES_Higher

Order

Let f(x)andg(x) be two function having finite nonzero third-order derivatives f x andg x for all x ∈ R. If f(x)g(x) = 1 for all x ∈ R, then prove that

f / f ′ − g / g ′ = 3(f / f − g / g).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Determinants If

f(x) , (x − a)(x − b)(x − c) wheref(x) is polynomial of degree < 3, then prove that dg(x) −2 = ∣∣ 1af(a)(x − a) 1bf(b)(x dx − b) − 2 1cf(c)(x − c) − 2∣∣ g(x) =

14

÷ ∣∣a2 a1b2 b1c 2c1 ∣∣  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

1

15

f(x) = e − x , wherex > 0, Let for each positive integer n, Pn be the polynomial d nf(x) 1 − 1x such that = P ( )e for all x > 0. Show that n n x dx d Pn + 1 (x) = x2 [Pn (x) − Pn (x)] dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

16

AND

DERIVATIVES_Higher

Order

AND

DERIVATIVES_Higher

Order

→ Let f : R R be a function satisfying condition f(x + y 3 ) = f(x) + [f(y)]3f or allx, y ∈ R .

If f ′ (0) ≥ 0, find f(10) .  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 17

Let f(x + y) = f(x) + f(y) + 2xy − 1 for all real xandy and f(x) be a differentiable function. If f ′ (0) = cos α, the prove that f(x) > 0 ∀ x ∈ R.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

18

AND

DERIVATIVES_Higher

Order

2 + f(x) + f(y) x+y If f( )= for all real xandy and f ′ (2) = 2, then 3 3 determine y = f(x).  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Determinants If f, gandh are differentiable functions of xand

19

(x) = ∣∣fgh(xf)' g' (xh)' (x2 f)' (x2g) ' (x2h)' ∣∣ then prove that

^ (' )(x) = ∣∣fghf' g' h' (x3f □ ) ' (x3g □ )' (x3 h □ )' ∣∣  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative

AND

DERIVATIVES_Geometrical

If 20

y = f(ax )andf ′ (sin x) = (log)e x, dy thenf ∈ d , dx

if it exists, where `pi/2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

.

21

If P n is the sum of a GP . upto n terms (n ≥ 3), then prove that . dP n (1 − r) = (1 − n)Pn + nPn − 1, where r is the common ratio of GP . dr

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

22

AND

DERIVATIVES_Theorems

On

f(x) f(y) If f(x) = + holds for all real x and y greater than 0andf(x) is a y x 1 differentiable function for all x > 0 such that f(e) = , thenf ∈ df(x). e  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 23

`(dy)/(dx)fory=tan^(-1){sqrt((1+cosx)/(1-cosx))},w h e r e0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions 24

If f(x) = ∣∣x2 − 5x + 6 ∣∣, then f' (x) equals  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

25

⎛ log( xe2 ) ⎞ y = tan − 1 ⎜ ⎟ 2) log(ex ⎝ ⎠

d2y 3 + 2 log x + tan ( ), then 2 1 − 6 log x dx is 2 (b) 1 (c) 0 (d) − 1 −1

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 26

AND

DERIVATIVES_Theorems

On

If f(0) = 0, f ′ (0) = 2, then the derivative of y = f(f(f(x))) at x = 0 is 2 (b) 8 (c) 16 (d) 4  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions

27

If y = axn + 1 + bx − n , thenx 2

n y

2 d y 2

dx

2

is equal to n(n − 1)y (b) n(n + 1)y ny (d)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

28

x2 x3 xn y = 1+x+ + + ... + , 2!

3! n! dy then dx xn xn xn is equal to (a) y (b) y + (c) y − (d) y − 1 − n! n! n!  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

29

2 dy If y = a sin x + b cos x , then y 2 + ( ) is a (a)function of x (b) function ofy dx

(c)function of x and y (d) constant  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions 30

dy 1 − sin 2x π = (√ ) is equal to, 0 < x < 1 + sin 2x 2 dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

31

n

If y = (x + √x + a ) , then 2

nx

√x2 + a2

(d) −

2

nx

√x2 + a2

dy ny ny is (a) (b) − (c) 2 2 2 2 dx √x + a √x + a

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

f(x) = √1 + cos2(x2), 32

thenf ′ ( is

√π ) 2

√π (b) − √π /6 1 / √6 (d) π / √6 6

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

33

1 d (cos − 1 √cos x) is equal to (a) √1 + secx (b) √1 + sec x (c) 2 dx 1 − √1 + sec x (d) − √1 + sec x 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm

AND

DERIVATIVES_Differentiation

If

y = (log) sin x (tan x), 34

dy )) is equal to π dx 4 4 −4 (a) (b) −4 log 2 (c) (d) none of these log 2 log 2 then((

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

35

If y =

xy − 2

sin − 1 x √1 −

x2

, then

(1 − x2)dy dx

is equal to x + y (b) 1 + xy 1 − xy (d)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

y

36

= cot − 1[

√1 + sin x + √1 − sin x √1 + sin x − √1 − sin x

] , (0 < x <

dy π ), then = 2 dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

37

If y = ae

mx

+ be

(d) none of these

− mx

, then

d 2y dx2

AND

DERIVATIVES_Theorems

On

− m2y is equal to m2 (aemx − be − mx ) 1 (c) 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 38

AND

DERIVATIVES_Higher

Order

(n − 1)! (n − 2) ! n! dn n − 1 (n − 1) ! (b) n (c) (d) ( − 1) n (log x) = (a) n n x x x xn dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

39

y = √log x + √log x + √log x + ∞, dy is dx 1 1 x x (a) (b) (c) (d) 2y − 1 2y + 1 x(2y − 1) x(1 − 2y) then

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions The differential coefficient of f((log)e x) with respect to x, where f(x) = (log)e x, 40

is (a)

1 1 x (b) (log)e x (c) (d) none of these x (log) e x x(log)e x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 41

If y = sec(tan

−1

dy cos π sin π sin π cos π x), then atx = 1 is (a) (b) (c) (d) 4 2 6 3 dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

f ′ (x) = √2x2 − 1andy = f(x2), 42

dy atx = 1 dx is 2 (b) 1 (c) − 2 (d) none of these then

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of One Function W.R.T. Other Function If

u = f(x3), v = g(x2), f ′ (x) 43

= cos x, andg ′ (x) = sin x, then

du dv

is `  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

44

x = t cos t, y = t + sin t. Then

d 2x dy

a= 2

AND

DERIVATIVES_Higher

Order

π+4 π+4 π is (b) − − 2 (d) none 2 2 2

of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

f(x) = √1 − sin 2x , then f ′ (x) is equal to π π π − (cos x + sin x), f or x ∈ ( , ) cos x + sin x, f or x ∈ (0, ) 4 2 4 π π π − (cos x + sin x), f or x ∈ (0, ) cos x − sin x, f or x ∈ ( , ) 4 4 2 If

45

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of One Function W.R.T. Other Function 46

If y = x − x2 , then the derivative of y 2 with respect to x2 is (a) 1 − 2x (b) 2 − 4x (c)3x − 2x2 (d) 1 − 3x + 2x2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

47

The first derivative of the function [cos − 1 (sin √ at x = 1 is (a)3 /4 (b) 0 (c)1 /2 (d) − 1 /2  Watch Free Video Solution on Doubtnut

1+x ) + xx ] with respect to x 2

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

48

AND

DERIVATIVES_Higher

Order

∣ y y1 y2 ∣ ∣ ∣ If y = sin pxandy n is the nth derivative of y, then ∣ y3 y4 y5 ∣is (a)1 (b) 0 (c) ∣ ∣ ∣ y6 y7 y8 ∣ − 1 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

49

A function f, defined for all positive real numbers, satisfies the equation f(x2 ) = x3

for every x > 0 . Then the value of f ′ (4) is (a)12(b)3 (c) 3 / 2 (d) cannot be determined  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

f(x) = eax + ebx , wherea ≠ b, and that '' f (x) − 2f ′ (x) − 15f(x) = 0 for all x . Then the product ab is (a)25 (b) 9 (c) − 15 (d) − 9 Suppose

50

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

y = 51

(a − x)√a − x − (b − x)√x − b

,then

(√a − x + √x − b)

x + (a + b) dy 2x − a − b wherever it is defined is (a) (b) (c) dx √(a − x)(x − b) 2 √a − x√x − b

−

(a + b) 2√(a − x)(x − b)

(d)

2x + (a + b) 2 √(a − x)(x − b)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions The function f(x) = ex + x, being differentiable and one-to-one, has a 52

differentiable inverse f − 1 (x) . The value of

1 1 1 (b) (c) (d) none of these 3 4 1n2

d (f − 1 ) at the point f(log 2) is (a) dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions Let h(x) be differentiable for all x and let f(x) = (kx + ex )h(x) , where k is some constant. If 53

h(0) = 5, h ′ (0) = − 2, andf ′ (0) = 18, then the value of k is (a)5 (b) 4 (c) 3 (d) 2.2.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

y = tan 54

then

−1

(

2x 1+2

2x + 1

),

dy atx = 0 dx

is (a)1 (b) 2 (c) 1n 2 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 55

AND

DERIVATIVES_Higher

Order

1 The nth derivative of the function f(x) = [wherex ∈ ( − 1, 1)] at the 1 − x2 point x = 0 where n is even is (a)0 (b) n! (c) n nC 2 (d) 2n C 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

d 20 56

(2 cos x cos 3x)is 20

equal

AND

DERIVATIVES_Higher

(a)2 20 (cos 2x − 2 20 cos 3x)

to

Order

(b)

dx 2 20(cos 2x + 220 cos 4x) (c)2 20(sin 2x + 220 sin 4x) (d)220 (sin 2x − 220 sin 4x)  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form 57

(1 − x2 )dy 1−x 1 y 2 If y = √ , then is equal to y (b) (c) −y (d) − 1+x y dx x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm 58

AND

DERIVATIVES_Differentiation

The derivative of y = (1 − x)(2 − x).... (n − x) at x = 1 is (a)0 (b)

( − 1)(n − 1) ! (c)n! − 1 (d) ( − 1)n − 1 (n − 1)!  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

5 cos x − 12 sin x ), 13 π wherex ∈ (0, ), 2 dy then is. (a)1 (b) − 1 (c) 0 (d) none of these dx y = cos − 1 (

59

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

60

If y = tan − 1 √

x+1 dy −1 −1 1 , then is (b)

x−1 dx 2|x|√x2 − 1 2x√x2 − 1 2x√x2 − 1

(d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 61

If sin

−1

x2 − y 2 dy y x2 − y 2 y x ( 2 ) = log a, then is equal to (b)

(d) y x + y2 dx x x2 x2 + y 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 62

If y = cos

−1

dy 5π (cos x), then at x= is equal to (a)1 (b)-1 (c)0 (d)4 4 dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If

y = cos − 1 (cos x), then 63

=

5π 4

is (b) − 1 (c)

1 √2

dy atx dx

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

ex = 64

√1 + t − √1 − t

andtan

y 2

√1 + t + √1 − t 1−t =√ 1+t dy 1 1 1 ,then at t = is (a)− (b) (c) 0 (d) none of these 2 2 2 dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If

1 andx4 + y 4 = t2 t 1 3 dy + 2 , thenx y = dx t (a) 0(b)1 (c) −1 (d) none of these x2 + y 2 = t −

65

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If

y m = (x + √1 + x2), 1

66

then(1 + x2)y2 + xy 1 is (where y r represents the rth derivative of y w.r.t. x. (a)m2 y (b) my 2 (c) m2 y 2 (d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

67

Suppose the function f(x) − f(2x) has the derivative 5 at x = 1 and derivative 7atx = 2. The derivative of the function f(x) − f(4x)atx = 1 has the value equal to (a)19 (b) 9 (c) 17 (d) 14  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

68

AND

DERIVATIVES_Higher

If f(x) = sin − 1 cos x, then the value of f(10) + f ′ (10) is (a)11 −

7π 5π − 11 (c) − 11 (d) none of these 2 2

Order

7π (b) 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

AND

DERIVATIVES_Higher

Order

If 69

d 2y 1 π (sin x)(cos y) = , then 2 at( , 2 4 dx π ) 4 is (a) −4 (b) − 2 (c) − 6 (d) 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

70

A function f satisfies the condition f(x) = f ′ (x) + f ' ' (x) + f ' ' ' (x)...... ,where f(x) is a differentiable function indefinitely and dash denotes the order of derivative. If x f(0) = 1, thenf(x) is (a)e 2 (b) ex (c) e2x (d) e4x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

Let f(x) be a polynomial of degree 3 such that

f(3) = 1, f ′ (3) = − 1, f ' ' (3) = 0, 71

andf ' ' ' (3) = 12.

Then the value of f ′ (1) is (a)12(b)23 (c) − 13 (d) none of these  Watch Free Video Solution on Doubtnut

On

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

72

(3 − x) 1 3 2 d − 1 √x [tan ( )] = (b)

(d) 1 − 3x dx 2(1 + x) √x (1 + x) √x (1 + x)√x 3 2(1 + x) √x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Let g(x) be the inverse of an invertible function f(x) which is differentiable at x = c .

73

Then g ′ (f(x)) equal. (a)f ′ (c) (b)

1 (c) f(c) (d) none of these ′ f (c)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If f(x) = x + tan xandf is the inverse of g, then g ′ (x) equals (a) 74

1

2

1 + [g(x) − x]

(b)

1

2

2 − [g(x) − x]

(c)

1

2

2 + [g(x) − x]

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives If f √x2 + 1 = log(√x2 + 1 − x) then 75

(d) none of these  Watch Free Video Solution on Doubtnut

AND

DERIVATIVES_Theorems

(x2 + 1)dy dx

On

+ xy + 1 = 0 (b) 1 (c) 2

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If y =

√a + x − √a − x

√a + x + √a − x ay

76

x√a2

−

x2

,then

dy ay ay is equal to (a) (b) (c) 2 2 2 2 dx x√a − x √a − x

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

77

4

3

AND

DERIVATIVES_Higher

2

If f(x) = x tan(x ) − x1n(1 + x ), then the value of (b) 6 (c) 12 (d) 24

d 4 (f(x)) 4

dx

Order

at x = 0 is 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all 78

real x. Then g ' ' (f(x)) equals. (a)−

f ′ (x)f ' ' (x) − (f ′ (x))2 2

(f ′ (x))

f ' ' (x) 3

(f ′ (x))

(b)

f ′ (x)f ' ' (x) − (f ′ (x)) 3 f ′ (x)

(c)

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives If

AND

DERIVATIVES_Theorems

On

y = |cos x| + |sin x|, then 79

dy atx dx

2π 3 1 − √3 1 is (b) 0 (c) (√3 − 1) (d) none of these 2 2 =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If g is the inverse function of fandf ′ (x) = sin x, theng ′ (x) is (a)cos ec{g(x)} (b) 80

sin{g(x)} (c)−

1 (d) none of these sin{g(x)}

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Continuity If x = φ(t), y = ψ(t), then 81

φ

''

ψ' '

(d)

d 2y dx

2

is (a)

φ ′ ψ ' ' − ψ' φ' ' (φ ′ )2

(b)

φ ′ ψ ' ' − ψ' φ' ' (φ ′ )3

(c)

ψ'' φ''

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

82

AND

DERIVATIVES_Higher

Order

2 f(x) = ex − e − x − 2 sin x − x3 . Then the least value of n for which 3 n d n f(x)( ∣ ) x = 0 isnonzerois 5 (b) 6 (c) 7 (d) 8 dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If f(x) satisfies the relation

5f(x) − 3f(y) 5x − 3y )= ∀ x, 2 2 y ∈ R, f(

83

and f(0) = 3andf ′ (0) = 2, then the period of sin(f(x)) is (a)2π (b) π (c) 3π (d)

4π

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D ⋅ F(x) by the formula

84

D ⋅ f(x) = f 2 (x + h) − f 2 (x) lim , h→ 0 h wheref 2 (x) 2 mean [f(x)] and if f(x) = x log x ,then D ⋅ f(x)∣x = e has the value (A)e (B) 2e (c) 4e (d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If

f(x) = 2 sin − 1 √1 − x 85

+ sin − 1 (2√x(1 − x))

, where x ∈ (0,

π

1 2 2 ), thenf ′ (x) is (a) (b) zero (c)− (d) 2 √x(1 − x) √x(1 − x)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

x 2 x 2 If f' ' (x) = − f(x)andg(x) = f (x) and F(x) = (f( )) + (g( )) 2 2 and given that F(5) = 5, then F (10) is (a)5 (b) 10 (c) 0 (d) 15 ′

86

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS

AND

DERIVATIVES_Higher

Order

Derivaties

√ 1 + x2 − 1 2x√1 − x2 −1 The derivative of tan ( ) with respect to tan ( ) 2 x 1 − 2x 1 1 1 at x = 0 is (b) (c) (d) 1 8 4 2 −1

87

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 88

AND

DERIVATIVES_Theorems

On

The nth derivative of xex vanishes when (a)x = 0 (b) x = − 1 (c)x = − n (d)

x=n

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

89

2

2

If y = ax + bx + c ,then y

3

d2 y 2

AND

DERIVATIVES_Higher

Order

is (a)a constant (b)a function of x only (c)a

dx function of y only (d)a function of xandy  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties x

If y = sin x + e , then 90

sin x − (cos x +

ex ex )3

(d)

d2x 2

AND

DERIVATIVES_Higher

x −1

= (a)( − sin x + e )

dy sin x + ex

(b)

sin x − ex (cos x +

ex )2

Order

(c)

(cos x + ex )3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of

One Function W.R.T. Other Function If 91

u = x2 and x = s + 3t, y = 2s − t, d2u 5 5 then is (a) t (b) 20t8 (c) (d) non of these 2 6 2 16t ds  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

92

AND

DERIVATIVES_Higher

Order

2 2 dy dy x 3d y If y = x log( ), thenx = (a)x − y (b) (x − y) 2 a + bx dx dx dx 2 dy dy y − x (d) (y − x) dx dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

93

AND

DERIVATIVES_Theorems

On

u(x) Let u(x)andv(x) be differentiable functions such that = 7.If v(x) u ′ (x) u(x) ′ = pand( ) = q, ′ v (x) v(x) p+q then p−q has the value of to (a)1 (b) 0 (c)7 (d) − 7  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions 2

2

If ax + 2hxy + by = 1, then 94

h2 + ab 2

(hx + by)

d 2y 2

is (a)

dx

h2 − ab (hx + by)

2

(b)

ab − h2 (hx + by)

2

(d) none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of One Function W.R.T. Other Function 95

2

3

If x = t , y = t , then

d2y

3 3 3 3t = (a) (b) (c) (d) 2 2 (4t) 2(t) dx2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 96

x

If y = x + e , then

d 2x dy2

ex

x

is (a)e (b) −

AND

(1 + ex ) 3

DERIVATIVES_Higher

(c)−

ex (1 + ex ) 3

(d)

Order

−1 (1 + ex ) 3

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

If f(x) = |sin x − |cos x ∣ ∣ , then the value f(x) ′ atx = 97

1 − √3 (c)0 (d) none of these 2

On

7π is (a)positive (b) 6

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

If graph of y = f(x) is symmetrical about the y-axis and that of y = g(x) is symmetrical about the origin and if .

98

h(x) = f(x) g(x), then =0

d 3 h(x) 3

atx

dx

.

is (a)cannot be determined (b) f(0)g(0) (c)0 (d) none of these  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

99

AND

DERIVATIVES_Higher

d2y 1 If x = log pandy = , then (a) 2 − 2p = 0 (b) p dx

d2y dx

2

Order

+ y = 0 (c)

d 2y

d2y dy dy + = 0 (d) − =0 2 2 dx dx dx dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 2

AND

Let y = 1n(1 + cos x) . Then the value of 100 (c)

4 −4 (d) 1 + cos x (1 + cos x) 2

d2y dx

2

DERIVATIVES_Higher

+

2 y

equal (a)0 (b)

e2

Order

2 1 + cos x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative

AND

DERIVATIVES_Geometrical

Let

f(x) =( 101

lim )h → 0 .

Then f(

(sin(x + h)) 1n ( x + h ) − (sin x) 1nx h

π π ) equal to (a)0 (b) equal to 1 (c)In (d) non-existent 2 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of

Implicit Functions A function f : R → R satisfies

102

sin x cos y(f(2x + 2y) − f(2x − 2y) = cos x sin y(f(2x + 2y) + f(2x − 2y)).

1 If f ′ (0) = , then (a)f ′ (x) = f(x) = 0 2 x f + f(x) = 0 (d)4f x − f(x) = 0

(b)4f x + f(x) = 0

(c)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives If

(lim)t→ x 103

=

etf(x) − ex f(t) 2

(t − x)(f(x))

AND

DERIVATIVES_Theorems

On

= 2andf(0)

1 , 2

then find the value of f ′ (0). 4 (b) 2 (c) 0 (d) 1  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

dy e√x e√x − e − √x If y = e +e , then is equal to (a) (b)

dx 2x 2 √x 1 1 √y 2 − 4 (d) √y 2 + 4 2√x 2√x √x

104

− √x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

105

Let y = √x + √x + √x + ∞ ,

1

√1 + 4x

(d)

y 2x + y

dy 1 x is equal to (a) (b) (c) 2y − 1 x + 2y dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions 106

If 1 is a twice repeated root of the equation ax3 + bx2 + bx + d = 0, then (a) a = b = d (b) a + b = 0 (c)b + d = 0 (d) a = d  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions

107

If x3 − 2x2 y 2 + 5x + y − 5 = 0 and y(1) = 1, then (a)y ′ (1) =

y' (1) = −

4 22 2 (c)y' ' (1) = − 8 (d) y ′ (1) = 3 27 3

4 (b) 3

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CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

108

f(x) = ∣∣ x2 − 3∣∣x| + 2|. Then which of the following is/are true f ′ (x) = 2x − 3f or x ∈ (0, 1) ∪ (2, ∞) f ′ (x) = 2x + 3f or x ∈ ( − ∞, − 2) ∪ ( − 1, 0) f ′ (x) = − 2x − 3f or x ∈ ( − 2, − 1) noneofthese  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If

y= 109

x4 − x2 + 1 x2 + √3x + 1

and

dy = ax dx

+ b, then the value of a-b is (a)cot(

5π 5π π π ) (b) cot( ) (c) tan( ) (d) tan( ) 8 12 12 8

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

110

√x − 2√x − 1

3 x. Then (a)f ′ (10) = 1 (b)f ′ ( ) = − 1 2 √x − 1 − 1 (c)domain of f(x) is x ≥ 1 (d)range of f(x)is( − 2, − 1) ∪ (2, ∞) Let

f(x) =

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm

DERIVATIVES_Differentiation

dy y is (a) (1nx ∞ x − 1) + 21nx1n(1nx)) dx x y y log ( log x ) (log x) (2 log(log x) + 1) (c) [(1nx) 2 + 21n(1nx)] x x1nx y log y [2 log(log x) + 1] x log x

If 111

.

AND

y=x

( log x ) log ( log x)

, then

(b) (d)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Which of the following is/are true?

112

dy f or y = sin − 1(cos x), wherex dx ∈ (0, π), is − 1 dy f or y = sin − 1(cos x), wherex dx ∈ (0, 2π), is1 dy f or y = cos − 1(sin x), wherex dx π π ∈ ( − , ), is − 1 2 2 dy f or y = cos − 1(sin x), wherex dx π 3π ∈( , ), is − 1 2 2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

113

If f(x − y), f(x)f(y), andf(x + y) are in A.P. for all x, y, andf(0) ≠ 0, then (a) f(4) = f( − 4) (b)f(2) + f( − 2) = 0 (c) f ′ (4) + f ′ ( − 4) = 0 (d)

f ′ (2) = f ′ ( − 2)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions

114

If y = cos − 1 (

|x| < 1 (c)

2x dy −2 −2 ), then is (a) for all x (b) for all 2 2 2 1+x dx 1+x 1+x

2 for |x| > 1 (d) none of these 2 1+x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions

115

f : R + → R is a continuous function satisfying x f( ) = f(x) − f(y) ∀ x, y ∈ R + .If f'(1)=1,then (a)f is unbounded (b) y f(1 + x) 1 lim f( ) = 0 (c) lim = 1 (d) lim x. f(x) = 0 x→0 x→0 x→0 x x  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions

116

fn (x) = efn − 1 ( x ) for all n ∈ Nandf0(x) = x, d then {f n(x)} dx . fn (x)d is (a) {fn − 1 (x)} (b) f n(x)fn − 1 (x) (c)f n(x)fn − 1 (x)....... f 2(x) f 1 (x) dx (d)none of these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS

AND

DERIVATIVES_Higher

Order

Derivaties Suppose fandg are functions having second derivative f' ' and g' ' everywhere. If .

117

′

f(x)g(x) = 1 for all xandf andg' are never zero, then − 2f ′ (x) 2g ′ (x) − f ′ (x) 2f ′ (x) (a) (b) (c) (d) f g(x) f(x) f(x)

f ' ' (x) f ′ (x)

−

g ' ' (x) g ′ (x)

equal

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

If 118

y = e − x cos xandyn + kny = 0, d ny whereyn = n andkn dx are constants ∀n ∈ N, then (a)k4 = 4 (b) k8 = − 16 (c)k12 = 20 (d) k16 = − 24  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties If 119

x=

a

function

1 + (log) e t

true? (a)y

t2 ''

is

;y=

represented

3 + 2(log) e t t

AND

DERIVATIVES_Higher

parametrically

be

the

Order

equations

, then which of the following statements are 2

2

(x − 2xy ′ ) = y (b)yy ′ = 2x(y ′ ) + 1 (c)xy ′ = 2y(y ′ ) + 2 (d)

y ' ' (y − 4xy ′ ) = (y ′ )

2

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 120

AND

DERIVATIVES_Theorems

On

Statement 1: If f(x) is an odd function, then f ′ (x) is an even function. Statement 2: If f ′ (x) is an even function, then f(x) is an odd function.

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

121

→ Statement 1: Let f : R R be a real-valued function ∀ x, y ∈ R such that |f(x) − f(y)| ≤ |x − y| 3 . Then f(x) is a constant function. Statement 2: If the derivative of the function w.r.t. x is zero, then function is constant.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 122

f(x) = sin x, f ′ (π) = f ′ (3π) . f(x) = sin x, f(π) = f(3π) .

Statement

1:

For

Statement

2:

For

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives Statement

differentiable function f(x) + f(x − 2) = 0 ∀ x ∈ R, and if 123

1:

AND

If

DERIVATIVES_Theorems

f(x)

satisfies

the

On

relation

((

d f(x))) = b, dx x=a d then(( f(x))) = b. dx a + 4000 Statement 2: f(x) is a periodic function with period 4.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions

124

If for some differentiable function f(α) = 0andf ′ (α) = 0, Statement 1: Then sign of f(x) does not change in the neighbourhood of x = α Statement 2: α is repeated root of f(x) = 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous Suppose the function f(x) satisfies the relation 3

3

.

f(x + y ) = f(x) + f(y ) ∀ x, y 125

∈R

and is differentiable for all x. Statement 1: If f ′ (2) = a, thenf ′ ( − 2) = a Statement 2: f(x) is an odd function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous

126

If y = cos − 1 (

2x dy −2 −2 ), then is

for all x (b) for all |x| < 1 2 2 2 1+x dx 1 + x 1+x

2 for |x| > 1 (d) none of these 2 1+x

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 127

AND

DERIVATIVES_Theorems

On

f ′ (x) = ϕ(x) and ϕ ′ (x) = f(x) for all x.Also,f(3)=5 and f ′ (3) = 4. Then the 2 2 value of [f(10)] − [ϕ(10)] is _____  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 128

AND

DERIVATIVES_Theorems

On

If y = f(x) is an odd differentiable function defined on ( − ∞, ∞) such that f ′ (3) = − 2, then∣ f ′ ( − 3) ∣ equals_________.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 129

AND

DERIVATIVES_Theorems

On

2

If x3 + 3x2 − 9x + c is of the form (x − α) (x − β), then the positive value of c is_________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

130

If graph of y = f(x) is symmetrical about the point (5, 0) and f ′ (7) = 3, then the value of f ′ (3) is__________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties 131

AND

DERIVATIVES_Higher

Order

Let g(x) = f(x)sin x, wheref(x) is a twice differentiable function on ( − ∞, ∞) such that f( − π) = 1. The value of ∣g − π ∣ equals __________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Let 132

f(x) = (x − 1)(x − 2)(x − 3)(x − n), n ∈ N, andf(n) = 5040.

Then the value of n is________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions y=f(x), where f satisfies the relation

f(x + y) = 2f(x) + xf(y) 133

+ y√f(x) , ∀x, y → R and f'(0)=0.Then f(6)is equal ______  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

134

If function f satisfies the relation f(x) ⋅ f ′ ( − x) = f( − x) ⋅ f ′ (x) for all x ,and f(0)=3, and if f(3)=3, then the value of f( − 3) is ______________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 135

If y =

a + bx x

5 4

3 2

AND

DERIVATIVES_Theorems

andy ′ = 0atx = 5, then the value of

a2 b2

On

is_______

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Let

y

1

2 ( log ) 2 4 x − 3 log _ (27)(x2 + 1) 136

=

− 2x

3

and

7 4 ( log ) 49x − x − 1

dy dx

= ax + b,

then the value of a + b is_______.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative 137

( lim )

(e + h)1n( e + h ) − e → h0

h

is _ _

 Watch Free Video Solution on Doubtnut

AND

DERIVATIVES_Geometrical

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives If the function

f(x) = − 4e 138

1 −x 2

AND

DERIVATIVES_Theorems

On

DERIVATIVES_Theorems

On

x2 +1+x+ 2

x3 + andg(x) = f − 1(x), 3 −7 then the reciprocal of g ′ ( ) is_________ 6  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 139

AND

Suppose that f(0) = 0andf ′ (0) = 2, and let g(x) = f( − x + f(f(x))) . The value of g' (0) is equal to _____  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Suppose f(x) = eax + ebx , where a ≠ b, and that f x − 2f ′ (x) − 15f(x) = 0 140

|ab| for all x. Then the value of is ___ 3

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CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties A 141

nonzero

polynomial

.''

with

real

AND

DERIVATIVES_Higher

coefficient

has

the

Order

property

that

f(x) = f ′ (x)f (x). If a is the leading coefficient of f(x), then the value of is____

1 2a

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CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

A function is reprersented parametrically by the equations

x= 142

1+t 3 ; y = t3 2t2

∣ dy 2 . + T henthevalueof ∣ t ∣ dx 3 dy ∣ − x( ) ∣ dx ∣

is__________

On

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

143

Let

5

z = (cos x) andy = sinx.

Then

AND

the

DERIVATIVES_Higher

value

of

2

Order

d 2z

2π atx = 9 dy2

is____________.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Let 144

x2 + x tan x − x tan 2x g(x) = { ,x ax + tan x − tan 3x ≠ 00, x = 0 If g ′ (0) exists and is equal to nonzero value b, then 52

b is equal to _____ a

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CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative 145

AND

DERIVATIVES_Geometrical

Find the derivative of sin(x2 + 1) with respect to x for first principle.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives Find the derivative of 146

AND

DERIVATIVES_Theorems

On

x−1 f(x) = { 2 whenx ≠ 1 2x − 7x + 5 1 = whenx = 1atx = 1.

3  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives Given

y= 147

5x

(1 − x) 23 dy ∈d dx

AND

DERIVATIVES_Theorems

On

+ cos 2(2x + 1), f

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm 148

x

3

x

.

Let y = e s ∈ x + (tanx) F ∈ d

AND

DERIVATIVES_Differentiation

dy . dx

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

Let f be a twice differentiable function such that 149

f x = − f(x), andf ′ (x) = g(x), h(x) = [f(x)]2 + [g(x)]2.

Find h(10) if h(5) = 11

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DERIVATIVES_Higher

Order

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Determinants If α is a repeated root of a quadratic equation f(x) = 0andA(x), B(x), andC(x) are polynomials of degree 3,4, and 5, respectively, then show that 150

∣A(x)B(x)C(x)A(α)B(α)C(α)A ′ (α )B ′ (α)C ′ (α)∣ is divisible by f(x), where prime denotes the derivaties.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives Find 151

the

derivative

with

((log) cos x sin x)((log)sin x cos x)

AND

respect −1

+ sin − 1 (

DERIVATIVES_Theorems to

x

of

the

2x π )atx = 4.

1 + x2

On

function

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If 152

x = cos ecθ − sin θandy = cos ecn θ − sinn θ, 2 dy then show that (x2 + 4)( ) = n2(y 2 + 4). dx  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Miscellaneous Find

153

dy sin ( π x ) 2 atx = − 1, when(sin y) dx √3 + sec − 1 (2x) 2 x + 2 tan(log(x + 2)) = 0

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CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm

154

AND

DERIVATIVES_Differentiation

x2 y2 y x If + = √2 touches the ellipses 2 + 2 = 1, then fin the ecentric angle θ a b a b of point of contact

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CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If

2x + 1 ′ )andf (x) x2 + 1 dy = s i n x2, t h e n =_ _ dx

y = f( 155

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Determinants If f r(x), g r(x), hr (x), r = 1, 2, 3 are polynomials such that

156

fr (a) = gr (a) = hr (a), r = 1, 2, 3and ∣ f1(x) f2(x) f3(x) ∣ ∣ ∣ F(x) = ∣ g1(x) g2(x) g3(x) ∣ then F ′ (x)atx = a is____________________ ∣ ∣ ∣ h1(x) h2(x) h3(x) ∣  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If 157

f(x) = (log)x (log x), thenf ′ (x)atx =e is _ _ _ _ _  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of One Function W.R.T. Other Function

158

The derivative of sec − 1 ( is___________

1 1 ) with respect to √1 − x2 at x = 2 2x2 − 1

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CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative

AND

DERIVATIVES_Geometrical

If

f(9) = 9, f ′ (9) = 4, 159

then(lim)x → 9

√f(x) − 3

___________.

√x − 3

=

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If 160

f(x) = |x − 2|andg(x) = f[f(x)], theng ′ (x) = ______ for x>20

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CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions 161

If xe

xy

dy = y + sin x, then at x = 0, = _ _ _ __ _ dx 2

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CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

Let F(x) = f(x)g(x)h(x) for all real x, wheref(x), g(x), andh(x) are differentiable functions. At some point 162

x0, F ′ (x0 ) = 21F(x0), f ′ (x0 ) = 4f(x0 ), g ′ (x0 ) = − 7g(x0 ), and h ′ (x0 ) = kh(x0 ). Then k is________  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 163

AND

DERIVATIVES_Theorems

On

The derivative of an even function is always an odd function.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

If 164

f(a) = 2, f ′ (a) = 1, g(a) = − 1, g ′ (a) = 2, g(x)f(a) − g(a)f(x) 1 then the value of lim is (a)− 5 (b) (c) 5 (d) none of x→a x−a 5 these

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

AND

DERIVATIVES_Higher

Order

2 d 3d y If y = P (x), a polynomial of degree 3, then 2 (y ) = (a) 2 dx dx ''' P (x) + P ' (x) (b) P ' ' (x)P ' ' ' (x) (c)P (x)P ' ' ' (x) (d) constant 2

165

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivaties

166

AND

DERIVATIVES_Higher

Order

Let f(x) be a quadratic expression which is positive for all the real values of x. If g(x) = f(x) + f ′ (x) + f x , then for any real x. g(x) < 0 (b) g(x) > 0 g(x) = 0 (d) g(x) ≥ 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Using Logarithm

AND

DERIVATIVES_Differentiation

167

dy = (a)(sin x) tan x (1 + sec2 x log sin x) (b) dx tan x − 1 tan x(sin x) cos x (c)(sin x)tan x (d)sec2 x log sin x tan x(sin x)tan x − 1 If

y = (sin x)tan x , then

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CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Determinants

168

∣ x3 sin x cos x ∣ d3 ∣ ∣ Let f(x) = ∣ 6 −1 0 ∣ , wherep is a constant. Then 3 (f(x)) at x = 0 ∣ ∣ dx ∣ p p2 p3 ∣ is (a) p (b) p − p3 (c)p + p 3 (d) independent of p  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

f : R → R be such that f(1) = 3andf ′ (1) = 6. f(1 + x) 1 / x 1 lim ( ) = (a)1 (b) e 2 (c) e2 (d) e3 x→0 f(1)

Let 169

On

Then

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CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative

(lim)

→ h0

170

f(2h + 2 + h2) − f(2)

AND

giventhatf ′ (2)

DERIVATIVES_Geometrical

f(h − h2 + 1) − f(1) = 6andf ′ (1) = 4 3 3 does not exist (b) is equal to − is equal to (d) is equal to 3 2 2  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Meaning Of A Derivative

AND

DERIVATIVES_Geometrical

If f(x) is differentiable and strictly increasing function, then the value of 171

(lim)

→ x0

f(x2) − f(x) f(x) − f(0)

is 1 (b) 0 (c) − 1 (d) 2

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CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions 172

If y is a function of xand log(x + y) − 2xy = 0, then the value of y ′ (0) is (a)1 (b) − 1 (c) 2 (d) 0  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

2

173

On

2

If x2 + y 2 = 1, then (a)yy ' ' − 2(y ′ ) + 1 = 0 (b)yy ' ' + (y ′ ) + 1 = 0 (c)

yy ' ' + (y ′ )

−2

2

− 1 = 0 (d)yy ' ' + 2(y ′ ) + 1 = 0

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

AND

DERIVATIVES_Theorems

On

174

(

2

d y 2

)

−1

(b)

dx dy 2 dx d y − ( 2 ) ^ ( − 1) dx

(

dy

2

d y dx

2

)

dx

^ ( − 3)

−(

dy

2

d y dx

2

)

dx

^ ( − 2)

(d)

 Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives

175

AND

DERIVATIVES_Theorems

On

Let f(x) = 2 + cos x for all real x. , Statement 1: For each real t, there exists a point c in [t, t − π] such that f ′ (c) = 0 because Statement 2: f(t) = f(t + 2π) for each real t.  Watch Free Video Solution on Doubtnut

CENGAGE_MATHS_ALGEBRA_LIMITS AND DERIVATIVES_Differentiation Of One Function W.R.T. Other Function

176

Let f(θ) = sin(tan − 1 (

d f(θ)is d(tan θ)

sin θ √cos 2θ

)),w h e r e−

π π < θ < then the value of 4 4

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CENGAGE_MATHS_ALGEBRA_LIMITS Derivatives 177

AND

DERIVATIVES_Theorems

2

On

2

The slope of the tangent to the curve (y − x5 ) = x(1 + x2 ) at the point (1, 3) is.  Watch Free Video Solution on Doubtnut

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