EXERCISES ON CALCULUS Exercise A. Differentiations

EXERCISES ON CALCULUS Exercise A. Differentiations

EXERCISES ON CALCULUS
K.C. Chung (AMA)
Exercise A.
Differentiations
Differentiate y with respect to x. (No. 1–100.)
Constant multiplications, sums and differences
A1.
y = x5
Ans: y 0 = 5x4
A2.
y = 3x4
Ans: y 0 = 12x3
A3.
A4.
y = x2 + x
√
y= x
A5.
y = x3 + 1/x3
A6.
y = 3x2 − 4x + 3
A7.
y = 3(x3 − 2x − 1/x)
A8.
y = 1 + x−1 + 3x−2 − 2x−3
A9.
y = sin x + cos x
Ans: y 0 = 2x + 1
√
Ans: y 0 = 1/(2 x)
Ans: y 0 = 3x2 − 3/x4
Ans: y 0 = 6x − 4
Ans: y 0 = 9x2 − 6 + 3/x2
Ans: y 0 = 6x−4 − 6x−3 − x−2
Ans: y 0 = cos x − sin x
A10.
y = 4 cos x − 3 tan x
A11.
y = sec x + 3 tan x
A12.
y = tan−1 x + sin−1 x
A13.
y = sin−1 x + cos−1 x
A14.
y = sin−1 x − cos−1 x
A15.
y = 2 ln x + 3ex −
A16.
y = 2 tan x + 3ex − 1/x
Ans: y 0 = −4 sin x − 3 sec2 x
Ans: y 0 = sec x tan x + 3 sec2 x
1
1
+√
Ans: y 0 = 2
x +1
1 − x2
Ans: y 0 = 0
2
Ans: y 0 = √
1 − x2
2
1
Ans: y 0 = + 3ex − 2/3
x
3x
0
2
x
Ans: y = 2 sec x + 3e + 1/x2
√
3
x
Products and quotients
Ans: y 0 = 12x + 1
A17.
y = (2x − 1)(3x + 2)
A18.
y = (2x − 1)2
A19.
y = (2x2 + x − 1)ex
A20.
y = (3x + 2) sin x
Ans: y 0 = 3 sin x + (3x + 2) cos x
A21.
y = (2x − 1) cos x
Ans: y 0 = 2 cos x − (2x − 1) sin x
A22.
y = sin x cos x
A23.
y = ex tan x
Ans: y 0 = 4(2x − 1)
Ans: y 0 = (5 + 2x)xex
Ans: y 0 = cos 2x
Ans: y 0 = ex (sec2 x + tan x)
A−1
A24.
y = ex sin x
A25.
A37.
y = (x2 + 1) tan−1 x
x+2
y=
x−2
1
y= 2
x +1
ax + b
y=
cx + d
1
y=
4x + 3
x+1
y= 2
x −x−2
x2 − 1
y= 2
x +1
sin x
y=
x
cos x
y= 2
x +1
ex
y=
x
x+1
y=
ex
ln x
y=
x
y = (x + 1)(x + 2)(x + 3)
A38.
y = (x + 2)2 (x − 3)
A39.
y = x2 ex sin x
A40.
x2
y = x sin x
e
A26.
A27.
A28.
A29.
A30.
A31.
A32.
A33.
A34.
A35.
A36.
Ans: y 0 = ex (cos x + sin x)
Ans: y 0 = 2x tan−1 x + 1
Ans: y 0 = −4(x − 2)−2
−2x
+ 1)2
ad − bc
Ans: y 0 =
(cx + d)2
−4
Ans: y 0 =
(4x + 3)2
Ans: y 0 =
(x2
Ans: y 0 = −(x − 2)−2
4x
(x2 + 1)2
cos x sin x
Ans: y 0 =
− 2
x
x
2x cos x
sin x
Ans: y = − 2
−
(x + 1)2 x2 + 1
Ans: y 0 =
Ans: y 0 = (1/x − 1/x2 )ex
Ans: y 0 = −x/ex
Ans: y 0 = (1 − ln x)/x2
Ans: y 0 = 3x2 + 12x + 11
Ans: y 0 = 3x2 + 2x − 8
Ans: y 0 = xex (2 sin x + x cos x + x sin x)
Ans: y 0 = xe−x (2 sin x + x cos x − x sin x)
Compositions of functions and other miscellaneous types
A41.
y = sin 5x
Ans: y 0 = 5 cos 5x
A42.
y = tan 3x
Ans: y 0 = 3 sec2 3x
A43.
y = e−3x
A44.
y = sin2 x
Ans: y 0 = sin 2x
A45.
y = cos2 x
A46.
x
y = sin−1 ,
a
A47.
y = u3 − 3u + 6, u = x2 − 2x − 1
Ans: y 0 = − sin 2x
1
Ans: y 0 = √
2
a − x2
Ans: y 0 = 6x(x − 1)(x − 2)(x2 − 2x − 2)
A48.
y = u3 , u = x2 + 2x − 3
A49.
y = (2x + 1)100
Ans: y 0 = −3e−3x
(a 6= 0, |x| < |a|)
Ans: y 0 = 6(x + 1)(2x + x2 − 3)2
Ans: y 0 = 200(2x + 1)99
A−2
1
− 3x + 3
A50.
y=
A51.
y=
A52.
√
y = 1/ 4x2 + 1
A53.
y=
A54.
y=
A55.
A56.
y = (x − 3)(x2 + x + 1)9
√
y = (x − 1) 4x2 + 3
A57.
y=
x2
3 − 2x
− 3x + 3)2
x
Ans: y 0 = √
2
x +4
−4x
Ans: y 0 =
2
(4x + 1)3/2
2x
Ans: y 0 =
2
3(x + 4)2/3
1
Ans: y 0 = √
2x ln x
0
2
2
Ans: y = (19x − 44x − 26)(x + x + 1)8
Ans: y 0 =
√
x2 + 4
√
3
x2 + 4
√
A58.
A59.
A60.
A61.
ln x,
(x > 1)
A63.
y = sin(x2 − 2x + 3)
A64.
y = cos(3x + 2) sin(3x − 2)
A65.
y = exp(2x − 1) sin 2x
A66.
y = exp(x2 + 2x) sin x2
A67.
y = exp(4x2 − 3x + 2)
A68.
y = sin cos(x3 − 1)
A69.
A73.
y = ln(x2 + 4)
x−1
−1
y = sin
, (x > 0)
x+1
2
x −1
−1
y = tan
x2 + 1
sin x
y=
1 + cos x
√
y = cot 4x
A74.
y=
A71.
A72.
8x2 − 4x + 3
√
4x2 + 3
Ans: y 0 = (x − 2)−5 (x + 1)4 (x − 14)
1
Ans: y 0 = p √
4 x( x + 4)
y = (x + 1/x)6
6
2
x +1
y=
x2 − 1
√
x − x2 + 1
√
y=
x + x2 + 1
y = sin2 x − sin4 x
A70.
Ans: y 0 =
(x + 1)5
(x − 2)4
q
√
y = 4+ x
A62.
(x2
Ans: y 0 = 6(x + 1/x)5 (1 − 1/x2 )
Ans: y 0 = −24x(x2 − 1)−7 (x2 + 1)5
2
√
Ans: y 0 = √
x2 + 1(x + x2 + 1)2
Ans: y 0 = sin 2x cos 2x
Ans: y 0 = (2x − 2) cos(x2 − 2x + 3)
Ans: y 0 = 3 cos 6x
Ans: y 0 = 2(cos 2x + sin 2x)e2x−1
Ans: y 0 = 2ex
2 +2x
[x cos x2 + (x + 1) sin x2]
Ans: y 0 = (8x − 3)e4x
2 −3x+2
Ans: y 0 = −3x2 sin(x3 − 1) cos(cos(x3 − 1))
2x
Ans: y 0 = 2
x +4
1
Ans: y 0 = √
x(x + 1)
2x
+1
1
Ans: y 0 =
1 + cos x
−2
Ans: y 0 = √
cot 4x sin2 4x
−2
Ans: y 0 =
(sin x + cos x)2
Ans: y 0 =
1 − tan x
1 + tan x
A−3
x4
A75.
A77.
y = sin 3x cos 4x
1
1
1
y = sin − cos
x x
x
2
2
y = x tan x
A78.
y = (sin 2x − cos 4x)2
A79.
y = cos5 (2 sin x)
A80.
y = sin 3x − 4 ln 5x
1
y = cos
x−2
r
x−1
y=
x+1
A76.
A81.
A82.
Ans: y 0 = 3 cos 3x cos 4x − 4 sin 3x sin 4x
1
1
Ans: y 0 = − 3 sin
x
x
0
2
2
2
Ans: y = 2x tan x + 2x (tan x)(sec x)
Ans: y 0 = 4(sin 2x − cos 4x)(cos 2x + 2 sin 4x)
Ans: y 0 = −10 cos x cos4 (2 sin x) sin(2 sin x)
4
Ans: y 0 = 3 cos 3x −
x
1
1
Ans: y 0 =
sin
(x − 2)2
x−2
Ans: y 0 =
1
(x − 1)1/2 (x + 1)3/2
A85.
y = exp(x4) − x sin(x3)
3x − 4
y = cos
5x + 3
√
y = ln( x2 + 1 − x)
A86.
y = ln(x −
A87.
y = xn sin(ax2 + c)
A88.
y = cos(x2 + 3x − 1) sin(x2 + 3x + 1)
A89.
y = exp(x3 + 2x − 1) sin(x2 + 3x)
Ans: y 0 = exp(x3 + 2x − 1)[(3x2 + 2) sin(x2 + 3x) + (2x + 3) cos(x2 + 3x)]
A90.
y=
A91.
y = ln ln ln x,
A83.
A84.
√
x2 − 1)
Ans: y 0 = 4x3 exp(x4) − sin(x3) − 3x3 cos(x3 )
3x − 4
0
−2
Ans: y = −29(5x + 3) sin
5x + 3
−1
Ans: y 0 = √
1 + x2
−1
Ans: y 0 = √
x2 − 1
Ans: y 0 = 2axn+1 cos(ax2 + c) + nxn−1 sin(ax2 + c)
cos(x2 + 3x − 1)
sin(x2 + 3x + 1)
Ans: y 0 = (2x + 3) cos(2x2 + 6x)
−(2x + 3) cos 2
sin2 (x2 + 3x + 1)
1
Ans: y =
x(ln x)(ln ln x)
Ans: y =
(x > ee )
Implicit functions
A92.
y 2 = x2 + 1
Ans: y 0 = x/y
A93.
x2 + y 2 = 9
A94.
x2 + xy + y 2 = 9
A95.
y + sin x = cos y
Ans: y 0 = −x/y
2x + y
Ans: y 0 = −
x + 2y
− cos x
Ans: y 0 =
1 + sin y
A96.
xy = exp(x + y)
A97.
y 2 + 2y = sin xy
A98.
y = f (sin x)
ex+y − y
Ans: y =
x − ex+y
y cos xy
Ans: y 0 =
2 + 2y − x cos xy
0
Ans: y = (cos x)f 0 (sin x)
A99.
y = f (f (x))
Ans: y 0 = f 0 (f (x))f 0 (x)
0
A−4
A100.
Ans: y 0 =
f (xy) = f (x + y)
f 0 (x + y) − yf 0 (xy)
xf 0(xy) − f 0 (x + y)
d2 y
Find 2 for the following:
dx
A101.
y = x3 − 3x2 − x + 2
A102.
y = x3 (x + 1)2
A103.
y = e2x(x2 + 2x − 1)
A104.
y = (x2 − 1) sin 2x
Ans: y 00 = 6x − 6
Ans: y 00 = 2x(10x2 + 12x + 3)
Ans: y 00 = 2e2x(2x2 + 8x + 3)
Ans: y 00 = 2(−2x2 + 3) sin 2x + 8x cos 2x
Show that:
A106.
If y = 2 cos ax + 3 sin ax, where a is a constant, then y 00 + a2 y = 0.
√
If y = x + x2 − 1 then (x2 − 1)y 00 + xy 0 − y = 0.
A107.
If x2 + y 2 = 2 then 1 + yy 00 + (y 0)2 = 0.
A108.
If sin x + cos y = 1/2 then y 00 sin3 y + sin x sin2 y + cos y cos2 x = 0.
A109.
If y = (x2 − 1)n , where n is a positive integer, then
A105.
(a) (x2 − 1)y 0 − 2nxy = 0, and
(b) (x2 − 1)y (n+2) + 2xy (n+1) − n(n + 1)y (n) = 0.
A110.
If y = exp(sin−1 x) then
(a) (1 − x2)y 00 − xy 0 − y = 0, and
(b) (1 − x2)y (n+2) − (2n + 1)xy (n+1) − (n2 + 1)y (n) = 0 for all n ≥ 0.
A111.
Hence find the value of y (6)(0).
√
If y = 1 − x2 sin−1 x then
Ans: 85
(a) (1 − x2)y 0 + xy = 1 − x2 , and
(b) (1 − x2)y (n+1) − (2n − 1)xy (n) − n(n − 2)y (n−1) = 0 for all n ≥ 3.
Hence find the value of y (7)(0).
Ans: −384
Find the local maxima and minima by (i) the first derivative test, and, if
possible, (ii) the second derivative test.
A112.
f (x) = x3 − 3x2 + 2
A113.
f (x) = x4 + 4
Ans: Loc. max= f (0) = 2, loc. min= f (2) = −2
Ans: Loc. min= f (0) = 4
A−5
A114.
f (x) = x3 − 12x − 4
A115.
f (x) = x4 + 2x3
A116.
f (x) = 2 − (x − 1)2/3
A117.
f (x) = x + 1/x, x 6= 0
A118.
f (x) = x − 2 sin x,
A119.
f (x) =
sin x
,
2 + cos x
Ans: Loc. max= f (−2) = 12, loc. min= f (2) = −20
Ans: Loc. min= f (−3/2) = −27/16
Ans: Loc. max= f (1) = 2
Ans: Loc. max= f (−1) = −2, loc. min= f (1) = 2
0 < x < 2π.
Ans: Loc. max= f (5π/3) ≈ 6.968, loc. min= f (π/3) ≈ −0.685
0 < x < 2π.
Ans: Loc. max= f (2π/3) =
√
√
3/3, loc. min= f (4π/3) = − 3/3
Find the (global) maxima and minima.
A120.
f (x) = x3 − 3x + 1 on [0, 2]
A121.
√
f (x) = 2 x − x on [1/4, 3]
A122.
f (x) = x5 − 5x3 on [0, 2]
Ans: Max= f (1) = 1, min= f (3) ≈ 0.464
√
Ans: Max= f (0) = 0, min= f ( 3) ≈ −10.4
A123.
f (x) = ex (x − 2) on [0, 2]
Ans: Max= f (2) = 0, min= f (1) = −e ≈ −2.718
A124.
f (x) = ex cos x on [0, π/2]
Ans: Max= f (π/4) ≈ 1.55, min= f (π/2) = 0
A125.
f (x) = ex sin x on [0, π]
A126.
f (x) = ex sin 2x on [1, π/2]
Ans: Max= f (2) = 3, min= f (1) = −1
Ans: Max= f (3π/4) ≈ 7.46, min= f (π) = 0
Ans: Max= f (1.0172) ≈ 2.474, min= f (π/2) = 0
Problems on maxima and minima.
A127.
Find the largest possible value for xy given that x and y are both non-negative and
x + y = 20.
Ans: 100
A128.
Find the smallest sum of two positive real numbers such that their product is 100.
Ans: 20
A129.
Find the largest possible area for a rectangle inscribed in a circle of radius 4 m.
Ans: 32 m2
A130.
A rectangular playground is to be fenced off and divided in two by another fence
parallel to one side of the playground. 120 metre of fencing is to be used. Find the
dimensions of the playground that will enclose the greatest area.
Ans: 20 m × 30 m
A−6
A131.
A triangle is formed by the positive coordinate axes and a line through the given
point (a, b) where a > 0 and b > 0. Find the slope s of this line if the area of the
triangle is to be minimised. Show that the minimum area is 4ab.
Ans: s = −b/a
A132.
Find the volume of the largest right circular cone that can be inscribed in a sphere
πR3
of radius R.
Ans: 32
81
A133.
Find the volume of the largest right circular cylinder that can be inscribed in a
√
sphere of radius R.
Ans: 4 3πR3 /9
A134.
Find the dimensions of a cylinder of given volume V if its surface area is a minimum.
The cylinder has a closed top and bottom.
p
Ans: Radius r = 3 V/2π, height h = 2r
A135.
A wire of length 100 cm is cut into two pieces; one, of length x cm, is bent to form
an equilateral triangle, the other a square. What is the value of x if the sum of the
two area is to be (a) a minimum (b) a maximum. (Allow the possibility of no cut.)
√
Ans: (a) x = 900/(4 3 + 9) (b) x = 0
Sketch the graphs of the functions by considering the x-, y-intercepts, local
extrema, points of inflection and asymptotes.
A136.
y = (x − 1)2
A137.
y = x3 + 6x2 , x ∈ [−4, 4]
A138.
y = 2x3 + 3x2 − 12x, x ∈ [−4, 4]
A139.
y = 3x4 − 4x3
A140.
y = 3x5 + 5x3
√
y =x 1−x
A141.
A142.
A143.
y = x + sin 2x, x ∈ [0, π]
√
y = 3x − cos 2x, x ∈ [0, π]
A144.
y=
1
1−x
A145.
y=
x+1
x−1
A146.
y = x2 +
1
x+1
A−7
Differentials and approximations.
A147.
Find dy for the function y = x2 + 3x − 2.
A148.
Find dy for the function y = sin(cos x).
A149.
For the function y = x3 + 3x + 1, find dy with x = 2 and dx = 0.1.
Ans: dy = (2x + 3)dx
Ans: dy = −(sin x) cos(cos x)dx
Ans: dy = 1.5
A150.
For the function y = x2 + x + 1, find dy and ∆y with x = 1 and dx = 0.1.
Ans: dy = 0.3, ∆y = 0.31
A151.
For the function y = sin(2x + 1), find dy and ∆y with x = 0 and dx = 0.1.
Ans: dy = 0.10806, ∆y = 0.0906
A152.
For the function y = x3 − cos x, find dy and ∆y with x = 0 and dx = 0.2.
Ans: dy = 0, ∆y = 0.279
A153.
All six sides of a cubical metal box are 0.25 cm thick. The volume of the interior of
the box is 125 cm3. Use differentials to find the approximate volume of the metal
used to make the box.
Ans: 37.5 cm3
A154.
A tank has the shape of a cylinder with hemispherical ends. The cylindrical part is
100 cm long and has a radius of 10 cm. Use differentials to find roughly how much
paint is required to coat the outside of the tank to a thickness of 1 mm ?
Ans: 240π cm3
A155.
We are trying to determine the area of a circle by measuring the diameter. How
accurately must we measure the diameter if our estimate is to be correct within
1% ?
Ans: 0.5%
A−8