Download Problems - TCD Maths home

Course MA1S11: Michaelmas Term 2016.
Tutorial 9: Sample
December 6–9, 2016
Results that may be useful.
Differentiation by Rule
Let x be a real variable, taking values in a subset D of the real numbers, and let y, u and v dependent variables, expressible as functions of the
independent variable x, that are differentiable with respect to x. Then the
following results are valid:—
(i) if y = c, where c is a real constant, then
dy
= 0;
dx
(ii) if y = cu, where c is a real constant, then
(iii) if y = u + v then
dy
du
=c ;
dx
dx
dy
du dv
=
+ ;
dx
dx dx
(iv) if y = xq , where q is a rational number, then
(v) (Product Rule) if y = uv then
dy
= qxq−1 ;
dx
dv
du
dy
=u
+v ;
dx
dx
dx
du
dv
−u
v
u
dy
(vi) (Quotient Rule) if y = then
= dx 2 dx ;
v
dx
v
(vii) (Chain Rule) if y is expressible as a differentiable function of u, where u
dy
dy du
in turn is expressible as a differentiable function of x, then
=
.
dx
du dx
Definitions and Basic Properties of Trigonometric Functions
The sine function (sin) sends a real number x to the sine sin x of an angle
measuring x radians. The circumference of a circle of radius one is of length
2π. Radian measure corresponds to distance along the circumference of the
unit circle. Therefore four right angles equal 2π radians, and thus one right
angle equals 21 π radians.
1
The cosine function (cos) satisfies the identity cos x = sin( 21 π − x) for all
real numbers x.
The tangent function (tan), cotangent function (cot), secant function
(sec) and cosecant function (csc) are defined so that
tan x =
sin x
,
cos x
cot x =
cos x
,
sin x
sec x =
1
,
cos x
csc x =
1
sin x
for all real numbers x. These functions satisfy the following identities:—
cos2 x + sin2 x = 1,
1 + tan2 x = sec2 x,
cos(x + y) = cos x cos y − sin x sin y,
sin(x + y) = sin x cos y + cos x sin y,
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x = cos2 x − 1 = 1 − 2 sin2 x.
sin x sin y =
cos x cos y =
sin x cos y =
1
(cos(x − y) + cos(x + y)),
2
1
(cos(x + y) + cos(x − y)),
2
1
(sin(x + y) + sin(x − y)).
2
Derivatives of Trigonometric Functions
The derivatives of the sine, cosine and tangent functions are as follows:—
d
(sin x) = cos x,
dx
d
(cos x) = − sin x,
dx
d
1
(tan x) =
= sec2 x.
dx
cos2 x
Derivatives of Logarithm and Exponential Functions
The exponential ex of x is defined for all real numbers x, and es+t = es et
for all real numbers t. Moreover
d kx
(e ) = kekx
dx
for all real numbers k. The exponential ex of x
The natural logarithm function satisfies
d
1
(ln kx) =
dx
x
(x > 0 and k > 0)
for all positive real numbers k. The natural logarithm ln x of x is defined for
all positive real numbers x, and satisfies ln(uv) = ln u + ln v for all positive
real numbers u and v.
2
Properties of Integrals
Let f and g be integrable functions on a closed bounded interval [a, b],
and let c be a real number. Then
Z b
Z b
Z b
(f (x) + g(x)) dx =
f (x) dx +
g(x) dx
a
a
and
b
Z
a
Z
(cf (x)) dx = c
a
Also
Z
b
f (x) dx.
a
b
xq dx =
a
1
(bq+1 − aq+1 )
q+1
for all rational numbers q and for all positive real numbers a and b. This
identity is also valid for all real numbers a and b in the special case where q
is a non-negative integer.
Integrals of sine and cosine are as follows:
Z s
Z s
1
1
cos kx = sin ks.
sin kx dx = (1 − cos ks) and
k
k
0
0
Also
Z
s
ekx dx =
0
3
1 ks
(e − 1).
k
Problem 1
Determine the derivatives of the following functions.
2 −4x3 )
(a)
esin(x
(b)
x5
7 + 2 cos(ln x)
(c)
cos
x5
2 + (ln x)6
Problem 2
Determine the values of the following definite integrals.
Z
(a)
4
(18x2 − 8x + 6) dx
2
Z
1
π
12
(b)
sin 6x dx
0
Z
27
2
1
(25x 3 + 12x 3 ) dx
(c)
8
It is recommended that you show your working on the Additional Work
sheets attached to the tutorial sheet.
4
Additional Work
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
5
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
6
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
7
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
...........................................................................
8