Download Exercises for Thursday and Friday

The University of Sydney
MATH1111 Introduction to Calculus
Semester 1
Week 4 Exercises (Thurs/Fri)
2017
Important Ideas and Useful Facts:
(i) Angles, degrees and radians: Angles in the plane are positive when measured anticlockwise,
and negative when measured clockwise. Angles may be measured in degrees (using the
symbol ◦ ) or in radians (expressed simply as real numbers). The angle of a full circle is
360◦ . The radian measure of an angle θ is the length of an arc on a unit circle subtended
by θ. In particular 360◦ corresponds to 2π, the perimeter of the unit circle. Some common
conversions between degrees and radians are as follows:
degrees
radians
1◦
30◦
45◦
60◦
90◦
120◦
135◦
150◦
π
180
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
180◦
π
270◦
3π
2
360◦
2π
(ii) The sine, cosine and tangent of an acute angle: Consider a right-angled triangle with acute
angle θ, adjacent side length a, opposite side length b and hypotenuse c. Define the sine,
cosine and tangent of θ to be, respectively:
b
adjacent
a
opposite
=
,
cos θ =
=
,
hypotenuse
c
hypotenuse
c
opposite
b
sin θ
tan θ =
=
=
.
adjacent
a
cos θ
sin θ =
c
b
θ
a
(iii) The sine, cosine and tangent of any angle: For an arbitrary angle θ, we put
y
sin θ
=
,
(cos θ, sin θ) = (x, y)
and
tan θ =
cos θ
x
where (x, y) lies on the unit circle satisfying the equation x2 + y 2 = 1, centred at the
origin (0, 0), and where the radius from (0, 0) to (x, y) subtends an angle θ:
1
1
(cos θ, sin θ)
sin θ
sin θ
tan θ
θ
−1
θ
cos θ
−1
1
−1
cos θ
1
−1
Then tan θ is the length of the line segment along the tangent to the circle, from the
x-axis to the intersection point with the line extending the radius, measured negatively
in the 2nd and 4th quadrants.
1
(iv) Equivalent angles: Angles are equivalent if they differ by integer multiples of 2π. For
and 13π
are equivalent; − π2 , 3π
and 7π
example, 0, ±2π, ±4π, . . . are equivalent; π6 , − 11π
6
6
2
2
are equivalent. The values of sin, cos and tan remain the same for equivalent angles.
(v) Common values for first quadrant angles:
sin 0 = 0 ,
sin π4 =
sin
π
2
√1
2
=1,
cos 0 = 1 ,
,
cos π4 =
cos
π
2
√1
2
tan 0 = 0 ,
,
=0,
sin π6 =
tan π4 = 1 ,
tan
π
2
1
2
,
sin π3 =
cos π6 =
√
3
2
,
√
3
2
,
cos π3 =
tan π6 =
1
2
,
√1
3
tan π3 =
,
√
3,
is undefined .
(vi) Circular identity: For any angle θ, the following identity holds:
cos2 θ + sin2 θ = 1 .
(vii) Double angle formulae: The following hold for any angles α and β:
cos(α + β) = cos α cos β − sin α sin β ,
sin(α + β) = sin α cos β + cos α sin β .
In particular, the following hold for any θ:
cos 2θ = cos2 θ − sin2 θ ,
sin 2θ = 2 sin θ cos θ .
Using the circular identity, cos 2θ = 1 − 2 sin2 θ = 2 cos2 θ − 1 , and so, after rearranging,
sin2 θ =
1 − cos 2θ
,
2
cos2 θ =
1 + cos 2θ
.
2
(viii) Functions: A function f is a process or rule that takes an input, typically a real number
x, and produces an output, also typically a real number f (x). It is common to refer to
the equation
y = f (x)
as representing a function f , and call x the independent variable and y the dependent
variable.
(ix) Domain and range: The domain of a function f is the set of inputs x such that the rule
f (x) is sensibly defined. The range of f is the set of outputs f (x) that are produced as
x varies over the domain. For example, the√interval [0, ∞) is both the domain and range
of the function f given by the rule f (x) = x.
(x) Graph of a Function: When x and y are real numbers, the graph of f is the collection of
ordered pairs (x, y) in the real plane such that y = f (x). Most graphs of functions that
you meet will be curves in the plane.
(xi) Shifting and Stretching: Here are some simple transformations applied to the graph of
y = f (x) :
(a) vertical shift c units upwards: y = f (x) + c
(b) horizontal shift k units to the right: y = f (x − k)
(c) vertical stretch by a factor of c : y = cf (x)
(d) horizontal stretch by a factor of k : y = f (x/k)
2
Tutorial Exercises:
1.
The angle θ is an acute angle of a right-angled triange. Solve each of the following
problems exactly by drawing an appropriate diagram.
(i) Find the length of the side adjacent to θ given that cos θ =
has length 6.
1
3
and the hypotenuse
(ii) Find sin θ and cos θ given that tan θ = 3.
(iii) Find sin θ and tan θ given that cos θ = 23 .
2.
A kite is flying above the ground at an angle of elevation of 50◦ , from the point of view
of the person holding the string. The string is of length 50 metres. Estimate the height
of the kite (to the nearest metre) above the person.
3.
The angle of elevation to the top of the Empire State Building in New York is estimated
to be 24◦ from the ground at a distance of one kilometre from the base of the building.
Estimate the height of the Empire State Building (to the nearest 10 metres).
4.
Graph the following functions and find the domain and range in each case:
1
(i) f (x) = x2 + 1 (ii) f (x) = x2 − 2x + 3 (iii) f (x) = x1 (iv) f (x) = x−1
√
√
√
1
(v) f (x) = x+1
(vi) f (x) = x (vii) f (x) = x − 1 (viii) f (x) = x + 1
∗
5.
The number of apples produced by trees in an apple orchard depends on the density of
the tree planting. It is known that if n trees are planted on one hectare of land then each
tree will produce approximately 3000 − 10n apples. Write down a function A(n) of n that
describes the total number of apples produced per hectare. Predict the number of trees
per hectare that should be planted to maximise the apple yield and estimate the maximal
yield per hectare.
∗
6.
Graph the following functions and find the domain and range in each case:
(i) f (x) = sin x (ii) f (x) = cos x (iii) f (x) = tan x (iv) f (x) =
1
cos x
Further Exercises:
∗
7.
Solve the following equations:
(i)
∗
8.
5
3
+
=2
x x+2
(ii) x4 − 5x2 + 4 = 0
The following equations in fact describe circles in the planes. Use the method of completing the square to rewrite the equations in a form that you can read off the centre and
radius in each case.
(i) x2 + y 2 − 2x − 4y − 11 = 0
(ii) 4x2 + 4y 2 − 16x − 24y + 51 = 0
(iii) x2 + y 2 + 10y + 24 = 0
∗∗
9.
√
(iii) 2x − 1 = − 2 − x
(iv) x2 + y 2 + 2x + 2y + 2 = 0
Recall that sin 30◦ = 21 . Now find exact values for cos 15◦ and sin 15◦ . Verify that
√
tan 15◦ = 2 − 3 .
3
Short Answers to Selected Exercises:
√3 , √1
10
10
(iii)
√
5
,
3
√
5
2
1.
(i) 2 (ii)
2.
38 metres
3.
450 metres
4.
(i) R , [1, ∞) (ii) R , [2, ∞) (iii) R\{0} , R\{0} (iv) R\{1} , R\{0}
(v) R\{−1} , R\{0} (vi) [0, ∞) , [0, ∞) (vii) [1, ∞) , [0, ∞) (viii) [−1, ∞) , [0, ∞)
5.
150 trees and 225, 000 apples per hectare
6.
, ± 5π
, . . .} , R
(i) R , [−1, 1] (ii) R , [−1, 1] (iii) R\{± π2 , ± 3π
2
2
π
3π
5π
(iv) R\{± 2 , ± 2 , ± 2 , . . .} , (−∞, −1] ∪ [1, ∞)
7.
(i) 3, −1 (ii) ±1, ±2 (iii) − 14
8.
(i) (1, 2), 4
(ii) (2, 3),
1
2
(iii) (0, −5), 1
4
(iv) (−1, −1), 0