NON-FINITES EXERCISE

1. Having learnt his lesson, he went out  to play cricket.
2. Walking along the street one day, I saw a dead cobra.
3. Leaving the forest we advanced into the open plain.
4. Driven out of his country, he sought asylum in a foreign land.
5. It being a very hot day, I remained in my tent.
6. A gipsy, wandering across the meadows, found the child.
7. I once saw a man walking on a rope.
8. Not knowing my way, I asked a policeman.
9. Hearing the noise, I woke up.
10. Being paralytic, he could not walk.
1. After he learnt his lesson, he went out to play cricket.
2. While I was walking along the street one day, I saw a dead cobra.
3. We left the forest and advanced into the open plain.
4. As he was driven out of his country, he sought asylum in a foreign land.
5. As it was a very hot day, I remained in my tent.
6. A gipsy found the child while he was wandering across the meadows.
7. I once saw a man who was walking on a rope.
8. As I did not know my way, I asked a policeman.
9. When I heard the noise, I woke up.
10. As he was paralytic, he could not walk.

DERIVATIVE OF THE EXPONENTIAL FUNCTION

The derivative of ex is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex!
$What does this mean?$ It means the slope is the same as the function value (the y-value) for all points on the graph.
Example: Let's take the example when x = 2. At this point, the y-value is e2 ≈ 7.39.
Since the derivative of ex is ex, then the slope of the tangent line at x = 2 is also e2 ≈ 7.39.
We can see that it is true on the graph:

Let's now see if it is true at some other values of x.

We can see that at x = 4, the y-value is 54.6 and the slope of the tangent (in red) is also 54.6.
At x = 5, the y-value is 148.4, as is the value of the derivative and the slope of the tangent (in green).

DIFFERENTIATION OF EXPONENTIAL FUNCTIONS
 The derivative of f(x) = b x is given by f '(x) = b x ln b Note: if f(x) = e x , then f '(x) = e x Example 1: Find the derivative of f(x) = 2 x Solution to Example 1: ·         Apply the formula above to obtain f '(x) = 2 x ln 2 Example 2: Find the derivative of f(x) = 3 x + 3x 2 Solution to Example 2: ·         Let g(x) = 3 x and h(x) = 3x 2, function f is the sum of functions g and h: f(x) = g(x) + h(x). Use the sum rule, f '(x) = g '(x) + h '(x), to find the derivative of function f f '(x) = 3 x ln 3 + 6x Example 3: Find the derivative of f(x) = e x / ( 1 + x ) Solution to Example 3: ·         Let g(x) = e x and h(x) = 1 + x, function f is the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2, to find the derivative of function f. g '(x) = e x h '(x) = 1 f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2 = [ (1 + x)(e x) - (e x)(1) ] / (1 + x) 2 Multiply factors in the numerator and simplify  f '(x) = x e x / (1 + x) 2 Example 4: Find the derivative of f(x) = e 2x + 1 Solution to Example 4: ·         Let u = 2x + 1 and y = e u, Use the chain rule to find the derivative of function f as follows. f '(x) = (dy / du) (du / dx) ·         dy / du = e u and du / dx = 2 f '(x) = (e u)(2) = 2 e u ·         Substitute u = 2x + 1 in f '(x) above f '(x) = 2 e 2x + 1 Exercises Find the derivative of each function. 1 - f(x) = e x 2 x 2 - g(x) = 3 x - 3x 3 3 - h(x) = e x / (2x - 3) 4 - j(x) = e (x2 + 2) solutions to the above exercises 1 - f '(x) = e x 2 x ( ln 2 + 1) 2 - g '(x) = 3 x ln 3 - 9x 2 3 - h '(x) = e x(2x -5) / (2x - 3) 2 4 - j '(x) = 2x e (x2 + 2)

Solve the following problem
Differentiate y=e^(ax+b) where a and b are constants
Let u = ax + b and y = e ^u
Use the chain rule to find the derivative of function as follows
dy/dx = (dy / du) (du / dx)
dy / du = e ^u and du / dx = a  ( Since the derivative of the constant b is 0)

Hence dy/dx = (e ^u)(a) = a e ^u
Substitute u = ax + b in dy/dx above

dy/dx = a e ^ax + b

VOLUME OF A TRIANGULAR PRISM

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How Do You Find the Volume of a Triangular Prism?

Find the volume of the triangular prism below

SUMMARY

1. B stands for the area of the base
2. Since our bases are triangles, we can use (1/2)bh for B
3. We can call the height of the triangle h1 and the height of the prism h2 to avoid confusion
4. Volume is measured in units cubed, so we have 50 cm3

NOTES

1.
1. The volume of a prism is V=Bh
2. B is the area of the base of the prism
3. h is the height of the prism
4. The base of our prism is a triangle, so we can use (1/2)bh, the area of a triangle, for B
5. We can call the height of the triangle h1 and the height of the prism h2 to avoid confusion
2.
1. V is the volume, which is what we are trying to find
2. b is the base of the triangle, which is 5 cm
3. h1 is the height of the triangle, which is 2 cm
4. h2 is the height of the prism, which is 10 cm
3.
1. We can plug 5 in for 'b', 2 in for 'h1', and 10 in for 'h2' in our equation
4.
1. We need to use the order of operations to simplify the right hand side
2. First multiply 5•2 in the innermost parentheses
3. Then multiply (1/2)•10 within the next set of parentheses
4. This is the same as 10/2, or 5
5. Then multiply 5•10 to get 50 cm3

Monday, February 15, 2016

INTRANSITIVE VERBS AS TRANSITIVE VERBS

INTRANSITIVE VERBS AS TRANSITIVE VERBS
When an intransitive verb is used in a causative sense, it becomes transitive.
Study the examples given below.
The horse walks. (Intransitive)
The man walks the horse. (Transitive – here the man causes the horse to walk.)
Birds fly. (Intransitive)
The boys fly their kites. (Transitive)
The spellings of certain common verbs indicate whether they are transitive or intransitive.
Many trees fell in the storm. (Intransitive)
Woodcutters felled the trees. (Transitive)
Lie still. (Intransitive)
Lay the basket on the table. (Transitive)
Rise early with the lark. (Intransitive)
Sit here.
Set the lamp on the table.
Some intransitive verbs become transitive when a preposition is added to them.
His friends laughed at him. (Laugh is an intransitive verb and does not take any object. Laugh at is a transitive verb and takes the object ‘him’.)
He soon ran through his fortune. (The verb run is intransitive whereas run through is transitive. To run through one’s fortune is to use it rapidly.)
Please look into the matter carefully.
wish for nothing.
Sometimes the preposition is prefixed to the verb.
He overcame his enemy. (Come is an intransitive verb, but overcome is transitive.)
He bravely withstood the attack. (Stand is an intransitive verb, but withstand is transitive.)

COMPARISON EXERCISE

1. The elephant is ……………………. land animal.
a) the big
b) the bigger
c) the biggest
2. The cheetah is …………………….. animal.
a) the fast
b) the faster
c) the fastest
3. The tortoise has a ………………. lifespan than any other animal.
a) long
b) longer
c) longest
4. Mt. Everest is ………………… peak in the world.
a) the high
b) the higher
c) the highest
5. Asia is …………………… than Africa.
a) big
b) bigger
c) biggest
6. China has ………………… population in the world.
a) large
b) the largest
c) the larger
7. One gram is …………………. than a kilo.
a) light
b) lighter
c) lightest
8. The Nile is ……………….. than any other river.
a) long
b) longer
c) longest
9. Greenland is …………….. island.
a) the large
b) the larger
c) the largest

1. The elephant is the biggest land animal.
2. The cheetah is the fastest animal.
3. The tortoise has a longer lifespan than any other animal.
4. Mt. Everest is the highest peak in the world.
5. Asia is larger than Africa.
6. China has the largest population in the world.
7. One gram is lighter than a kilo.
8. The Nile is longer than any other river.
9. Greenland is the largest island.

COMBINE THE FOLLOWING SENTENCES

1. They caught the thief. They handed him over to the police. (Use having)
2. The thief heard a noise. He ran away. (Use hearing)
3. He is very fat. He cannot walk fast. (Use too)
4. She was very tired. She could not stand up. (Use so that)
5. She left home at 4. She hasn’t arrived yet. (Use although)
6. The tortoise was slow. He beat the hare in the race. (Use though)
7. You will not be able to meet him. You have to be there before 3 pm. (Use unless or if)
8. You may work very hard. Still you cannot please him. (Use however)
9. He is ill. He attends office regularly. (Use in spite of)
10. I bought some apples yesterday. They were good. (Use which)

1. Having caught the thief, they handed him over to the police.
2. Hearing a noise, the thief ran away.
3. He is too fat to walk fast.
4. She was so tired that she could not stand up.
5. Although she left home at 4, she hasn’t arrived yet.
6. Though the tortoise was slow, he beat the hare in the race.
7. Unless you reach there before 3 pm, you will not be able to meet him. / If you do not reach there before 3 pm, you will not be able to meet him.
8. However hard you may work, you cannot please him.
9. In spite of his illness, he attends office regularly.
10. The apples that I bought yesterday were good.

DIRECT TO INDIRECT SPEECH: GENERAL RULES

Direct speech: The boy said, ‘I’m happy with my results.’
Indirect speech: The boy said that he was happy with his results.

General rules for changing direct speech into indirect speech

Omit all inverted commas or quotation marks. End the sentence with a full stop.
If the verb inside the inverted commas/quotation marks is in the present tense, change it into the corresponding past tense. If it is in the simple past tense, change it into the past perfect tense.
Direct speech: The girl said, ‘I like singing.’
Indirect speech: The girl said that she liked singing.
Direct speech: Rahul said, ‘I will have to reach home by 8.30.’
Indirect speech: Rahul said that he would have to reach home by 8.30.
Direct speech: Alina said, ‘I met James yesterday.’
Indirect speech: Alina said that she had met James yesterday.
When the verb inside the quotation marks expresses a universal truth, we do not normally change it into the past tense.
He said, ‘All people have equal rights.’
He said that all people have equal rights. (More natural than ‘He said that all people had equal rights.’)
Use pronouns appropriately.
Study the examples given below.
Direct speech: The boy told the girl, ‘I told you that we were not going on a holiday.’
Indirect speech: The boy told the girl that he had told her that theywere not going on a holiday.

Notes

When the reporting verb is in a present or future tense, we do not change the tense of the verb inside the quotation marks.
Direct speech: She says, ‘I will come.’
Indirect speech: She says that she will come.