Rajasthan Board RBSE Class 12 Maths Chapter 8 Application of Derivatives Miscellaneous Exercise
If radius and height of a cylinder is r and A, then find the rate of change of surface area with respect to its radius.
Radius of cylinder = r
and Height of cylinder = h
Rate of change of surface area w.r.t. r, =
Surface area of cylinder S = 2πr2 + 2πrh
Find the value of x and y for function, y = x3 + 21 where, the rate of change ofy is three times as the rate of change of x.
Given, function y = x3 + 21
Diff w.r.t. t,
= 3x2 …..(i)
According to question
= 3 …..(ii)
From equation (i) and (ii)
3x2 = 3
⇒ x = ± 1
In equation y = x3 + 21
Putting x = 1
y = 13 + 21
= 1 + 21 =22
Putting x = -1
y = (- 1)3 + 21 = -1 + 21 = 20
So, x = ± 1 and y = 22, 20
Prove that exponential function (ex) is an increasing function,
y = ex
Then, = ex
= + ve ∀ x ∈ R
So, for x ∈ R, ex is an increasing function.
Prove that f(x) = log (sinx) is increasing in (0,) and decreasing in (,π)
f(x) = log (sin x)
Diff w.r.t. t,
If tangent OX and OY of curve – = at some point cut the axis at point P and Q, then show that OP + OQ – a, where O is origin.
Equation – = …..(i)
Let, tangent cut the axis OX and OY at point P and Q, point on curve contact with tangent is (h, k).
Since, (h, k) situated on curve.
So, – = …..(ii)
DifF. (i) w.r.t. x,
So, from y – y1 = m(x – x1) equation of tangent at the point (h, k).
Find equation of tangent of curve y = cos (x + y), x ∈ [- 2π, 2π] which is parallel to the line x + 2y = 0
Hence, equation (i) and (ii) are the required equations of tangents.
Find the percentage error in calculating the volume of a cubical box if an error of 5% is made in measuring the length of edge of the cube.
Let side of cube is x and volume is V.
Percentage error in volume = 3 × 5% = 15%
Hence, required percentage error is 15%
A circular metal plate expands under heating so that its radius increased by 2%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.
Let radius of circular plate is r and surface area S.
Hence, approximate increase in the area of plate is 4π cm2.
Prove that the volume of the largest cone inside a sphere is of the volume of sphere.
Let r and h be radius and height of the cone respectively inscribed in a sphere of radius R.
Prove that semivertical angle of a cylinderical cone of given surface area and maximum volume is sin-1 ().
Let, Slant height of cone = l
Height = h
radius = r
T.S.A. = S
Volume = V
and Semi vertical angle = α
It is clear that, for S when V is maximum or minimum, then V2 be also maximum or minimum
Hence, when semi-vertical angle of cone is sin-1 () then its volume will be maximum.