MATH0111 & MATH0131
Elementary Differential Calculus (Versions 1 & 3)
Supplementary Materials

Definition 1   A zero can be characterised by the property that its addition to any natural number does not changed it:

0+n=n.     (1)

Theorem 2  

1. The zero is unique, i.e. any two numbers satisfying (1) are equal.
2. For any natural number n we have 0· n=0.

Exercise 3  

1. Without using a calculator evaluate:
   

   (a) 6+(−3)	(b) 6−(−3)	(c) 16+(−5)	(d) −16−(−5)
   (e) 27−(−3)	(f) 27−(−29)	(g) −16+3	(h) −16+(−3)
   (i) −23+52	(j) −23+(−52)

   
   

   (k) 3× (−8)

   (l) (−4)× 8

   (m) 15×(−2)

   (n) (−2)× (−8)

   
   

   (o) 15/−3

   (p) 21/7

   (r) −21/7

   (s) 21/−7

   (t) −12/2

   (u) −12/−2

2. Evaluate the following expressions:
   

   (a) 6−2× 2	(b) (6−2)× 2	(c) 6−(2+3) × 2	(d) 6−2+3 × 2
   (e) 6÷ 2− 2	(f) (6÷ 2)− 2	(g) (6−2)+3 × 2	(h) −6×(−2)×(−3)

3. Place brackets in the following expressions to make them correct:
   

   (a) 6 × 12 −3 +1 =55	(b) 6 × 12 −3 +1 =68	(c) 6 × 12 −3 +1 =60
   (d) 5× 4 − 3 +2 =7	(e) 5× 4 − 3 +2 =15	(f) 5× 4 − 3 +2 =−5

4. Evaluate:
   

   (a) 6÷ 2 +1	(b) 6÷ (2 +1)	(e) 6−2+4÷ 2	(f) 6−(2+4)÷ 2
   (c) 12+ 4÷ 4	(d) (12+ 4)÷ 4	(g) 2× 6÷ (3−1)	(h) 2× (6÷ 3−1)

Exercise 4   In the following calculations digits are replaced by letters. Within each calculation any digit is always replaced by the same letter, no two different digits are covered by the same letter.
Recover original calculations:

1. +THREE

   FOUR

   SEVEN

   +SEVEN

   SEVEN

   TWENTY

   -NINE

   FOUR

   FIVE

       
   

2. -FIVE

   FOUR

   +ONE

   ONE

   TWO

   
   
3. TWO × TWO = THREETOC× TOC= ENTRE    

Example 5  

1. Product of 142857 by any digit from 2 to 6 has an interesting property, e.g. 142857× 2= 285714.
2. Product of 123456789 by 9, 18, …, 81 is easy to calculate due to 123456789× 9 =1111111101

Definition 1   Equation of any straight line on the coordinate plane is

ax+by+c=0,     (2)

where at least one number a or b is non-zero.

Definition 2   Equation of straight non-vertical line graph is

y=mx+c.       (3)

Here m is called slop or gradient. The value c is the intersection of the straight line with vertical axis.

Exercise 3  

1. There are some chickens and rabbits on a farm. In total they have 18 heads and 58 legs.
   How many chickens and how many rabbits on the farm?
2. A car left Leeds at 9:00 and arrived to London at 14:15 on the same day. Assume that the distance is 320 kilometres.
   What was the car’s speed? What was the car’s average speed?
3. A car left Leeds towards London at 9:00 and drives with a constant speed 105 km/hour. Another car left London to Leeds twenty minutes later with a constant speed 95 km/hour.
   What time cars will meet each other? Where is the meeting point?
4. There are 13 pupils in a class. Boys have as many teeth as girls have fingers (a child has 32 teeth and 20 fingers).
   How many boys and how many girls are in the class?

Exercise 4  

1. Nine points are arranged into a square. Cross all nine points by a single stroke made out of only four segments of straight lines.
   

   •	•	•
   •	•	•
   •	•	•

2. The distance between Leeds and Bradford is 18 km. At 9am two students started walk along the same road but in the opposite directions: Lionel walked from Leeds to Bradford at a constant speed 4 km/h and Brian went from Bradford to Leeds with the constant speed 5 km/h.

   At the same time (9am) a drake start to fly from Lionel to Brian with the constant speed 10 km/h. As soon as it reached Brian it turned around and flied to Lionel (with the same speed), when it reached Lionel it turned to Brian and so on till Lionel met Brian on the road.

   Which distance the drake flied?

Exercise 5  

1. Let two lines are given by equations

   a1 x+ b1 y +c1=0       and       a2 x+ b2 y +c2=0      (4)

   Show that they are parallel if and only if:

   a1 b2 − a2 b1 = 0.     (5)

2. Let two lines are given by the above equations (4). Show that they are perpendicular if and only if:

   a1 a2 + b1 b2 = 0.     (6)

3. Use algebra and equations (5) and (6) to show that if two lines are perpendicular to a third line then they are parallel to each other.
4. Let three lines passing the same point are given by the equations: y=m₁ x +d₁, y=m₂ x + d₂, and y=m₃ x+d₃. Express d₃ through m₁, m₂, m₃, d₁, and d₂.

Example 1   Powers oftenly appeared in the real life:

1. Evaluation of areas involves squares and volumes needs cubes.
2. Gravitation, electromagnetic forces, light, sound are decreasing proportionally to the square of the distance. I.e. if distance is twice bigger the effect is four times low.
3. Kepler’s law: the square of period for a planet rotating around the Sun is proportional to cube of the distance between them.
4. Engineering calculations of hardness and durability involve fourth powers and sometimes sixth power (e.g. diameter of high-pressure pipeline).
5. Fluid dynamics also used sixth power: if flow in one river is 4 times faster than in another river, then it cone role stones which are 4⁶=4096 times heaver.
6. The brightness of incandescence bulb is proportional to twelfth power of the temperature in the white spectrum and even to thirtieth power for the red spectrum. In other words rising temperature from 2000^∘ to 4000^∘ produces 2¹2=4096 times brighter source.

Theorem 2   Laws of exponents:

1. (a b)ⁿ=aⁿ  bⁿ, (a/b)ⁿ=aⁿ/bⁿ
2. a ^n+m=aⁿ  a^m, aⁿ/a^m=a^n−m
3. (aⁿ)^m=a^{n m}
4. a^1/n=(a)^1/n for n∈ℕ
5. 1^x=1, for all x∈ℝ
6. 0^x=0, for all non-zero x∈ℝ
7. x⁰=1, for all non-zero x∈ℝ

Remark 3   The expression 0⁰ cannot be assigned a clear mathematical meaning due to inconsistency between Rules 6 and 7.

Exercise 4  

1. The following pictures are usually used for a visual proof of the Pythagoras Theorem. Reconstruct this proof.
         
2. Count areas of the two rectangles made by an rearrangement of the “same” four pieces. Explain the difference.
         
   Are visual proofs of the Pythagorean theorem convincing you now?

Exercise 5  

1. We know that 3²+4²=5². Check also that 3³+4³+5³=6³. Can it be extended further?
2. A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Show that
   1. If m and n are two positive integers with m > n, then a = m² − n², b= 2mn, c = m² + n² is a Pythagorean triple.
   2. Exactly one of a, b is odd; c is odd. The area A = ab/2 is an integer.
   3. Exactly one of a, b is divisible by 3. Exactly one of a, b is divisible by 4. Exactly one of a, b, c is divisible by 5.
   4. For any Pythagorean triple, ab is divisible by 12, and abc is divisible by 60.
   5. Exactly one of a, b, a + b, a · b is divisible by 7. At most one of a, b is a square.
   6. Every integer greater than 2 is part of a Pythagorean triple.

Exercise 6  

1. Calculate and spot the pattern of results:
   (a)  

   12× 9+3	=
   123× 9+4	=
   1234× 9+5	=
   12345× 9+6	=
   ……	…

      (b)  

   9× 9 +7	=
   97× 9 +8	=
   978× 9 +6	=
   9786× 9 +5	=
   ……	…

   
   Can you explain the pattern?
2. 1. Is there a digit such that any number finished by this digits has a square finished by the same digit?
   2. Which digit is at the end of cube of the digit described above? Fourth power? Fifth?
   3. If a number finished by 25, what are two last digits of its square? Cube?
   4. Can you found another two digits which has the same property as 25 from above?
   5. Are there triples of digits with the similar property?
3. Write statements “neither a or b is zero” and “either a or b is not zero” as formulae. [Chose from these: a+b≠ 0, a²+b²≠0, ab≠0.]

Example 7   The big numbers are oftenly called astronomically big. Here is an illustration:

1. The distance from us to Andromeda Galaxy is about 9.5· 10²³cm.
2. The weight of our Sun is about 1 983· 10³⁰ grams.

Example 8   According to U.S. National Debt Clock The Outstanding Public Debt as of 02 Oct 2006 at 11:06:24 AM GMT is: about $8.5 trillions. The estimated population of the United States is 300 millions. Then per capita:

8.5· 1012 300· 106

=

85 3

·

1012−1 106+2

> 28· 1011−8=28 000.

So each citizen’s share of this debt is more than $28 000.

Rule 9  

1. If a>b and b>c then a>c.
2. If a>b then a+c>b+c for all numbers c.
3. If a>b and c>0 then a· c>b· c.
4. If a>b and c<0 then a· c<b· c.

Rule 10  

1. If a>1 then n>m implies aⁿ > a^m.
2. If a<1 then n>m implies aⁿ < a^m.
3. If a<b then n>0 implies aⁿ < bⁿ.
4. If a<b then n<0 implies aⁿ > bⁿ.

Example 11   How to write the biggest possible number using only three digits 9?
Answer: 9⁹⁹ (this is really a hu-uuu-ge number!)

Exercise 12  

1. 1. How to write the biggest possible number using only three digits 2? (Hint: it is bigger than 16).
   2. How to write the biggest possible number using only three digits 3?
   3. How to write the biggest possible number using only three digits 4?
   4. Let a be a digit. For which values of a the biggest possible number formed by these three digits is a^aa?
2. 1. How to write the biggest possible number using only four digits 1?
   2. Which number is bigger 222², 22²² or 2²²²?
   3. How to write the biggest possible number using only four digits 2?

Remark 13   It is possible to use any number integer greater or equal to 2 as a number bases. For example in computer science the most useful number system use bases which are powers of 2, e.g. 2 itself or 16=2⁴. Can you explain why?

Example 14   Take a three digit number made out of the different digits. Subtract a number done out of the same digits in the reverse order. Tell me the last digit only and I will tell you the entire result.

Exercise 15  

1. On the last meeting of the Foundation Year Math Soc members shacked hands with each other. It was 66 handshakes in total.
   How many member in FYMS?
2. A ball is thrown upward with the speed 25 meters per second. When it will be 20 meters above the initial position?
3. (L.Euler’s problem) Two peasants brought apples to a local market. There were 100 apples in total. By the end of the day peasants sold all their apples and received equal amounts of money. First peasant told to second one: “I could get 15 escudos in total if I had your apples for sale as well”. The second peasant replied: “However I would obtain 62/3 escudos for your apples alone”.
   How many apples had each peasant?
4. There are two different loudspeakers: 400 Watts and 900 Watts. Where between them is the position to get the proper stereo sound?
5. Check the identity: 10²+11²+12²=13²+14².
   Are their other five consecutive integers having the same property?

Exercise 1  

1. Give an example of two functions f(x) and g(x) such that f∘ g ≠ g ∘ f.
2. Give an example of two functions f(x) and g(x) such that f∘ g = g ∘ f.
3. A function f(x) is called one-to-one function if an identity f(x₁)=f(x₂) always implies the identity x₁=x₂.
   1. Give an example of one-to-one function.
   2. Give an example of function which is not one-to-one.
   3. Show that the composition of one-to-one functions is again a one-to-one function.
4. Let function f(x) have the property

   f(x+y)=f(x)· f(y),       for all reals  x, y.

   Show that its inverse function g(x) has the property:

   g(x· y)= g(x) + g(y),       for all reals  x, y.

Exercise 2  

1. Find the gradient of the functions:

   (a)  y=t3/2;     (b)  y=t−5/11;     (c)  y=

   ∛

   t2

   ;     (d)  y=

   t2/7+ √ t ⎛ ⎝ t3 ⎞ ⎠ 1 5

   .

2. Find the gradient of the graph of the function y=f(x) at the given point x₀:

   (a)  y=x2, x0=1;     (b)  y=x2/3, x0=

   7 11

   ;     (c)  y=

   ∛

   x

   , x0=0.

3. Find the equation of the straight line touching the graph of the function y=f(x) at the given point x₀:

   (a)  y=x2, x0=1;     (b)  y=x2/3, x0=

   7 11

   ;     (c)  y=

   ∛

   x

   , x0=0.

4. A particle moving along a straight line such that its distance S (in meters) from the origin is depend from time t as follows:

   S= 2t3−21t2+120t+49.

   What is the distance from the reference point to the particle when its speed is equal 60 meters per second?

Exercise 3  

1. Giving that (e^x)′=e^x find derivatives of the functions:

   (a)  y=ekx,    (b)  y=2x,    (c) y=ax   (use that  elna=a)    (d)  y=xx

2. 1. From graphs of sinx and cosx verify that their slops at x=0 are 1 and 0 correspondingly. Find values of sin′ 0 and cos′0.
   2. Using the previous results and trigonometric identities:
      sin(a+b) = sina cosb + cosa sinb, cos(a+b) = cosa cosb − sina sinb
      find derivatives of the functions sinx and cosx.
   3. Find derivatives of y=tanx and y=cotx.
3. 1. Differentiate the identity g(x)· 1/g(x)=1 and show [1/g(x)]′=−g′(x)/g²(x).
   2. Derive the quotient rule: [u/v]′=u′· v−u· v′/v².
4. 1. Let x=g(y) be the inverse function of y=f(x), i.e. g(f(x))=x for all x. Using the chain rule show that g′(f(x))=1/f′(x).
   2. Using the above property and the fact that lnx is the inverse function of e^x show that (lnx)′= 1/x.
   3. Find derivatives of functions arcsin, arccos and arctan.

Exercise 4  

1. You are given 100 meters of a rope.
   1. What is the maximal area of a rectangular shape which can be bounded by this rope?
   2. The same for triangular shape? (Hint: use Heron’s formula)
2. You are again given 100 meters of a rope and need to make a maximal area as in the previous exercise, however you can now use a straight river’s bank as one side of your area which do not need a rope. What would be answer to both questions above?
3. What would be the maximal area bounded by the rope if you are not limited in the shape?
   1. Have you got physical evidence to support the answer?
   2. Can you prove it mathematically?
4. You are permitted to make a garden of 100 square meters and wish to chose it shape to minimise the length of fences.
   1. What are sides of a rectangular garden? A triangular one?
   2. What if one side is river’s bank which do not need a fence?
   3. Garden of an arbitrary shape?

Exercise 5  

1. Show that:

   sinα ± sinβ = 2sin ⎛ ⎜ ⎜ ⎝ α ± β 2 ⎞ ⎟ ⎟ ⎠ cos ⎛ ⎜ ⎜ ⎝ α ∓ β  2 ⎞ ⎟ ⎟ ⎠         cosα + cosβ = 2cos ⎛ ⎜ ⎜ ⎝ α + β 2 ⎞ ⎟ ⎟ ⎠ cos ⎛ ⎜ ⎜ ⎝ α − β  2 ⎞ ⎟ ⎟ ⎠         cosα − cosβ = 2sin ⎛ ⎜ ⎜ ⎝ α − β 2 ⎞ ⎟ ⎟ ⎠ sin ⎛ ⎜ ⎜ ⎝ α + β  2 ⎞ ⎟ ⎟ ⎠         cos4 α−sin4α = cos2α

2. Express sinα± cosβ as a product of two trigonometric functions.
3. Solve the equation for 0^∘≤ α ≤ 360^∘:
   1. 2tan²α−5secα−1=0.
   2. sin2α = −0.6
   3. cos(α+40^∘)=−0.25