Download Exercises for Thursday and Friday

The University of Sydney
MATH1111 Introduction to Calculus
Semester 1
Week 3 Exercises (Thurs/Fri)
2017
Important Ideas and Useful Facts:
√
β 2 = α. In particular,
(i) Square
roots: If α ∈ R then √α = β if and only if β ≥ 0 and √
√
0 = 0. If α > 0 then we call α the positive square root and − α the negative square
root of α. For example, 2 is the positive square root of 4, and −2 is the negative square
root of 4.
√
1
We also write α 2 for α, and note the following laws hold, where α, β ≥ 0 and γ > 0:
1
1
1
α 2 β 2 = (αβ) 2
and
1
1
1
α 2 /γ 2 = (α/γ) 2 .
Negative real numbers do not have square roots within the real number system R (though
they do have square roots in the larger complex number system C).
(ii) Absolute value: If α ∈ R then the absolute value or magnitude of α, denoted by |α|, is the
distance from α to 0, regarded as points on the real number line, so that
(
α if α ≥ 0 ,
|α| =
−α if α < 0 .
The following laws hold, for all α, β, γ ∈ R, with γ 6= 0:
√
| − α| = |α| = α2 , |αβ| = |α||β| , |α/γ| = |α|/|γ| ,
|α + β| ≤ |α| + |β| ,
the last of which is known as the triangle inequality. If α, β ∈ R then |α − β| is the
distance between α and β, as points on the real number line.
(iii) Theorem of Pythagoras: If a and b are the lengths of the shorter sides of a right angled
triangle, and c is the length of the hypotenuse, then
a2 + b2 = c2 .
(iv) Squares and differences: The following identities hold for all a, b in any number system:
(a + b)2 = a2 + 2ab + b2 ,
(a − b)2 = a2 − 2ab + b2 ,
a2 − b2 = (a + b)(a − b) ,
√
(v) Irrational surds: If n ∈ N then n is rational √
if and
is a perfect square, that is
√ only
√ if n √
2
n = m for some m ∈ N. Thus, for example, 2, 3, 5 and 6 are irrational.
√
(vi) Rationalising the denominator: If a ∈ Z, b ∈ N and a ± b 6= 0 then
√ √
1
1
a∓ b
a∓ b
√ =
√
√
.
= 2
a −b
a± b
a± b
a∓ b
1
(vii) Cartesian plane: The Cartesian plane or xy-plane consists of all ordered pairs (x, y) as x
and y range over all reals numbers, denoted by
R2 = { (x, y) | x, y ∈ R } ,
containing a horizontal x-axis { (x, 0) | x ∈ R } and a vertical y-axis { (0, y) | y ∈ R }.
The axes are perpendicular and intersect at the origin (0, 0).
If P = P (a, b) is a point in the xy-plane, then we call a the x-coordinate and b the ycoordinate of P , obtained by projecting P to the closest points on the x-axis and y-axis
respectively. Together, we call the pair (a, b) the Cartesian coordinates of P .
(viii) Lines: A line in the xy-plane is the set of points (x, y) satisfying an equation either of the
form y = mx + c (the nonvertical case), where m is the slope and c is the y-intercept, or
x = k (the vertical case), where k is the x-intercept. All lines may be put in the form
ax + by = c
for some constants a, b and c.
(ix) Slope and distance: Let P (x1 , y1), Q(x2 , y2 ) be points in the plane. The slope of the line
joining P to Q is
y2 − y1
,
x2 − x1
if x1 6= x2 , and infinite if x1 = x2 (when the line is vertical). An equation of the line
through P and Q is
y2 − y1
(x − x1 )
y − y1 =
x2 − x1
p
if x1 6= x2 . The distance from P to Q is (x2 − x1 )2 + (y2 − y1 )2 .
(x) Parallel and perpendicular lines: Two nonvertical lines in the xy-plane with slopes m1 and
m2 respectively are parallel if m1 = m2 , and perpendicular if m1 m2 = −1, in which case
m1 and m2 become negative reciprocals of each other.
(xi) Equation of a circle: A circle in the xy-plane, centred at P (x0 , y0 ) with radius r is the set
of points (x, y) satisfying the equation
(x − x0 )2 + (y − y0 )2 = r 2 .
(xii) Quadratic formula: A quadratic equation has the form ax2 + bx + c = 0 where a, b, c
are constants, a 6= 0, and has solutions (also called roots of the quadratic) given by the
quadratic formula:
√
−b ± b2 − 4ac
x =
2a
We call ∆ = b2 − 4ac the discriminant. If ∆ > 0 then there are exactly two real roots. If
∆ = 0 then there is exactly one real root. If ∆ < 0 then there are no real roots (though
exactly two complex roots exist in the complex number system C).
(xiii) Completing the square: The variable x in the expression ax2 + bx may be isolated by the
technique of completing the square:
2
b
b
b2
2
2
ax + bx = a x + x = a x +
.
−
a
2a
4a
For example, 2x2 + 3x = 2(x2 + 32 x) = 2(x + 43 )2 −
2
9
8
, yielding only one occurrence of x.
Exercises labelled with an asterisk are suitable for students aiming for a credit or higher.
Tutorial Exercises:
1.
Factorise the following expressions as much as possible:
(i) 2x + 4 (ii) 2x − 4 (iii) −2x + 4 (iv) −3x − 27 (v) ax + ay (vi) a2 x − ax2
(vii) x2 − 4 (viii) 9 − x2
(xiii) x2 − 4x + 4
2.
1
9
1
64
(x) x2 − 2 (xi)
(xiv) x2 + 8x + 15
(xv) x2 − x − 12
− x2
(xii) x2 + 4x + 4
(xvi) 3x2 + 5x − 12
Rationalise the denominators and simplify:
1
√
(i)
1+ 2
3.
(ix) x2 −
1
√
(ii)
1− 2
1
√
(iii)
2− 3
√
1+ 2
√
(iv)
1− 2
√
1+ 5
√
(v)
3− 5
Factorise the following quadratics and hence write down their roots:
√
2+ 3
√
(vi)
3− 2
(i) x2 + 3x + 2 (ii) x2 − 3x + 2 (iii) x2 − x − 2 (iv) x2 + 5x + 6 (v) x2 + 8x + 16
Check that you get the same roots by applying the quadratic formula.
4.
In each case, find the equation of the line passing through the two given points, either in
the form y = mx + c or in the form x = k :
(i) (1, 2) and (2, 5)
(iv) (1, 4) and (1, −4)
5.
(ii) (−1, 2) and (5, −3)
(v) (−3, −5) and (5, 3)
(iii) (−1, 4) and (5, 4)
(vi) (0, 1) and (9, −6)
There is a linear relationship between temperature C measured in degrees Celsius and
temperature F measured in degrees Fahrenheit. Find this relationship given that water boils at 212◦ Fahrenheit and 100◦ Celsius, and water freezes at 0◦ Celsius and 32◦
Fahrehnheit.
(i) Comfortable room temperature is commonly regarded as within the range 20 to 25
degrees Celsius. Convert this to a range in degrees Fahrenheit.
(ii) Convert the blistering 100◦ Fahrenheit to its Celsius equivalent.
(iii) Is there any temperature that has the same numerical value both in degrees Celsius
and degrees Fahrenheit?
∗
6.
Solve for x for each of the following equations:
(i) |x − 3| = 5
(ii) |2x − 1| = 6
(v) |x − 2| = |x + 4|
(iii)
1
=2
|x − 1|
(vi) |3x + 4| = |2x − 7|
Further Exercises:
7.
(iv) |x − 2| = |x − 4|
x + 1
=2
(vii) 2x + 7 = |x| (viii) 2 − x
Use the quadratic formula to solve the following quadratic equations or explain why no
real solution exists:
(i) x2 − x − 1 = 0
(iv) 3x2 − 5x − 1 = 0
(ii) x2 + 3x + 1 = 0
(v) 4x2 + 12x + 9 = 0
3
(iii) x2 − 2x + 3 = 0
(vi) x2 + 2x + 2 = 0
∗
8.
Find the exact length of the hypotenuse of a right angled triangle, given the following
lengths of the shorter two sides:
√
√ √
√
√
(i) 6, 8 (ii) 5, 12 (iii) 4, 5 (iv) 7, 3 (v) 2, 3 (vi) 2 + 1, 2 − 1
9.
Use interval notation to describe the solution sets for each of the following inequalities:
(i) |x − 3| ≤ 5
(vi) 1 ≤ |x| ≤ 2
∗
10.
(ii) |7 − x| > 5
(iii) |2x − 1| < 6
(vii) 3 < |x − 2| ≤ 7
(iv)
1
|x−1|
(viii) |x − 2| < |x + 4|
<2
(v)
1
|x−1|
≥2
(ix) 2x + 7 ≥ |x|
Graph the line y = 3x − 6. How would you alter this equation to move this line
(i) vertically upwards 5 units?
(ii) vertically downwards 2 units?
(iii) horizontally 1 unit to the right?
(iv) horizontally 3 units to the left?
(v) parallel to itself so that the point (2, 0) is translated to the point (−5, 7)?
∗
11.
Prove each of the following:
(i) The points (1, 1), (−2, −8) and (4, 10) lie on a straight line.
∗∗
12.
(ii) The points (0, −2), (−4, 8) and (3, 1) lie on a circle with centre (−2, 3).
√
√
√
√
√
√
Given that 2 and 3 are irrational, explain why 1 + 2 , 2 − 3 and 2 + 3 are
also irrational.
Short Answers to Selected Exercises:
1.
2.
(i) 2(x + 2) (ii) 2(x − 2) (iii) −2(x − 2) (iv) −3(x + 9) (v) a(x + y) √
(vi) ax(a√− x)
1
1
(vii) (x + 2)(x − 2) (viii) (3 + x)(3 − x) (ix) (x + 3 )(x − 3 ) (x) (x + 2)(x − 2)
(xi) ( 18 − x)( 81 + x) (xii) (x + 2)2 (xiii) (x − 2)2 (xiv) (x + 3)(x + 5) (xv) (x + 3)(x − 4)
(xvi) (3x − 4)(x + 3)
√
√
√
√
√
√
√
√
6+2 2+3 3+ 6
(i) 2−1 (ii) −1− 2 (iii) 2+ 3 (iv) −3−2 2 (v) 2+ 5 (vi)
7
3.
(i) (x + 2)(x + 1), −2, −1 (ii) (x − 2)(x − 1), 2, 1
(iv) (x + 2)(x + 3), −2, −3 (v) (x + 4)2 , −4
4.
(i) y = 3x − 1 (ii) y = − 56 x +
5.
C = 59 F −
6.
(i) −2, 8 (ii) − 25 , 72 (iii)
7.
(iii) no solution (iv) 5±6 37 (v) − 32 (vi) no solution
√
√
√
(i) 10 (ii) 13 (iii) 41 (iv) 4 (v) 5 (vi) 6
8.
9.
10.
(i)
√
1± 5
2
160
9
(ii)
7
6
(iii) (x − 2)(x + 1), 2, −1
(iii) y = 4 (iv) x = 1 (v) y = x − 2 (vi) y = − 97 x + 1
(i) 68 to 77 (ii) approximately 38 (iii) −40
√
−3± 5
2
1 3
,
2 2
(iv) 3 (v) −1 (vi) −11, 35 (vii) − 73 (viii) 1, 5
√
(i) [−2, 8] (ii) (−∞, 2) ∪ (12, ∞) (iii) (− 52 , 27 ) (iv) (−∞, 12 ) ∪ ( 23 , ∞) (v) [ 12 , 1) ∪ (1, 23 ]
(vi) [−2, −1] ∪ [1, 2] (vii) [−5, −1) ∪ (5, 9] (viii) (−1, ∞) (ix) [− 73 , ∞)
(i) y = 3x − 1 (ii) y = 3x − 8 (iii) y = 3x − 9 (iv) y = 3x + 3 (v) y = 3x + 22
4