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QUESTION: 1

Differentiate sin^{2}(θ^{2} + 1) with respect to θ^{2}

Solution:

y = sin^{2}(θ^{2}+1)

v = θ^{2}

dy/d(v) = dydθ/dvdθ

dy/dthη = sin^{2}(V+1)

= 2sin(V+1)⋅cos(V+1)dv/dθ

= 2sin(θ^{2}+1)cos(θ^{2}+1)

= sin2(θ^{2}+1).

QUESTION: 2

Differentiate with respect to x^{2}

Solution:

QUESTION: 3

Find dy/dx if x= a cos θ, y = b sin θ

Solution:

QUESTION: 4

Difference equation in discrete systems is similar to the _____________ in continuous systems.

Solution:

Difference equation are the equations used in discrete time systems and difference equations are similar to the differential equation in continuous systems.

QUESTION: 5

X = at² and y = 2at are parametric equations of

Solution:

Together the equations x = at^{2} and y = 2at (where t is the parameter) are called the parametric equations of the parabola y^{2} = 4ax.

QUESTION: 6

If x = 4(t + sin t), y = 4(1-cos t), Evaluate dy/dx at t= π/2

Solution:

x = a(t+sin t)

⟹dx/dt = a(1+cos t)

And y = a(1−cos t)

⟹dy/dt = a[0−(−sin t)]

=a sin t

Therefore, dy/dx = a sin t/(a(1+cos t))

= 2sin t/2 cos t/2)/(2cos^2 t/2)

=tan(t/2)

At π/2

tan(π/4) = 1

QUESTION: 7

Find dy/dx if x = cos^{3} θ, y = sin^{3} θ

Solution:

x = cos^{3}θ

⇒ dx/dθ = a(3cos^{2}θ)(−sinθ)

= −3sinθcos^{2}θ

y = sin^{3}θ

⇒dy/dθ = a(3sin^{2}θ)(cosθ)

= 3sin^{2}θcosθ

∴dy/dx = (dy/dθ)(dx/dθ)

= (3sin^{2}θcosθ)/(−3asinθcos^{2}θ)

= −sinθ/cosθ

= −tanθ

QUESTION: 8

Find ; x = 20 (cos t + sin t) and y = 20 ( sin t - t cos t)

Solution:

x = 20(cost + tsint)

differentiate x with respect to t,

dx/dt = 20{d(cost)/dt + d(tsint)/dt]

= 20[-sint + {t. d(sint)/dt + sint.dt/dt}]

= 20[ -sint + tcost + sint]

= -20t.cost

hence, dx/dt = -20t.cost -----(1)

y = 20(sint - tcost)

differentiate y with respect to t,

dy/dt = 20[d(sint)/dt - d(tcost)/dt ]

= 20[cost - {t.d(cost)/dt + cost.dt/dt}]

= 20[cost +tsint -cost]

= 20t.sint

hence, dy/dt = 20t.sint -------(2)

dividing equations (2) by (1),

dy/dx = 20t.sint/20t.cost

dy/dx = tant

now again differentiate with respect to x

d²y/dx² = sec²t. dt/dx ------(3)

now from equation (1),

dx/dt = 20t.cost

so, dt/dx =1/at.cost put it in equation (3),

e.g., d²y/dx² = sec²t. 1/20t.cost

d²y/dx² = sec³t/20t

QUESTION: 9

If f (x) = [x sin p x] { where [x] denotes greatest integer function}, then f (x) is

Solution:

If −1≤x≤1, then 0≤xsinπx≤1/2

∴ f(x)=[xsinπx]=0, for −1≤x≤1

If 1<x<1+h, where h is a small positive real number, then

π<πx<π+πh

⇒−1<sinπx<0

⇒−1<xsinπx<0

∴ f(x)=[xsinπx]=−1 in the right neighbourhood of x=1.

Thus, f(x) is constant and equal to zero in [−1,1] and so f(x) is differentiable and hence continuous on (−1,1).

QUESTION: 10

Differentiate sin x^{3} with respect to x^{3}

Solution:

Differentiation of sinx^{3} with respect to x^{3} is

d(sin(x^{3})/dx = cosx^{3}

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