Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 8

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Weekly Quiz Competition

Question 1
1 / -0.25

The dervative of sin(x² ) is

A
2x sin (x² )

B
cos (x² )

C
2x cos x²

D
-2x cos x²
Question 2
1 / -0.25

If where V = x² - 2x + 3 then dy/dx is

A
V² (2x - 2)

B
V²

C
2 (x - 1) (x² - 2x + 3)²

D

Solution

y = V^3/3
V = x² - 2x + 3
y = 1/3*v ³
dy/dx = d/dx{1/3*(x² -2x+3)³ }
= 1/3*3*(x² -2x+3)² *(2x-2)
{d/dx(xⁿ )=nx^n-1*dx/dx}
= 2(x-1)*(x² -2x+3)²
Question 3
1 / -0.25

F(x) = tan (log x)
F '(x) =

A
sec² (logx)

B

C

D
-x sec² (logx)

Solution

∫tan(log x) dx
log x = t
x = e^t
dx = e^t dt
f(t) = ∫tan t dt
f ’(t)= sec² t
= sec² (log x)
Question 4
1 / -0.25

Which of the following function is not differentiable at x = 0?

A
x²

B
|x|

C
cos x

D
ax + b
Question 5
1 / -0.25

A
√π/2

B
√π

C
2 √π

D
2
Question 6
1 / -0.25

y = log(sec + tan x)

A
sec x tan x –1

B
sec x

C
sec x tan x + 1

D
tan x

Solution

y = log(secx + tanx)
dy/dx = 1/(secx + tanx){(secxtanx) + sec^2x }
= secx(secx + tanx)/(secx + tanx)
= secx
Question 7
1 / -0.25

If y=sin 3x . sin³ x then dy/dx is

A
3 cos² x. cos 4x

B
cos 3x. cos³ x

C
-9 cos 3x. cos² x sinx

D
3 sin² x.sin 4x

Solution

Given: y = sin 3x . sin³

= 3 sin² x.sin 4x
Question 8
1 / -0.25

A
differentiable but not continuous at x=0

B
continuous but not differentiable at x=0

C
is differentiable at x=0

D
neither differentiable nor continuous at x=0

Solution

LHD = limh →0 ((−h)² sin(−1/h) −0)/−h
limh →0 −h² sin(1/h)/−h
= limh →0 h ×sin(1/h)
= 0
RHD = limh →0 (h² sin(1/h) −0)/h
= limh →0 h ×sin(1/h)
= 0
RHD=LHD
∴f(x) is differentiable at x= 0
Question 9
1 / -0.25

Find the derivate of y = sin⁴ x + cos⁴ x

A
–sin 2x

B
4 sin³ x + 4 cos³ x

C
–sin 4x

D
4 sin x cos x cos 2x

Solution

y=sin4x, and z=cos4x
So by using chain rule
df(x)/dx = dsin⁴ x/dx + dcos⁴ x/dx
=dy⁴ /dy * dy/dx + dz⁴ /dz * dzdx
=dy⁴ /dy * dsinxdx + dz⁴ /dz * dcosx/dx
=4y^{(4 −1)} ⋅cosx+4z(4 −1)⋅(−sinx)
=4sin³ xcosx −4cos³ xsinx
=4sinxcosx(sin² x −cos² x)
=2sin2x(−cos2x)
=−2sin2xcos2x
=−sin4x
Question 10
1 / -0.25

If is equal to

A
1 / s

B
1 / s2

C
1 / s3

D
1 / s4

Solution

Concept:
Calculation:
Hence, option (c) is correct.