Hello everyone and welcome to another add maths tutorial video. Today I am going to discuss form five additional mathematics

chapter 3 integration long question part 1. Before I begin, if you are new to any

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I will be discussing more mathematics tutorial videos in the future.

Alright guys without further delay let’s jump to the question now. okay by the way

guys in this video I’m going to discuss two integration question.

You are now looking at the example one. I’m going to read the question now. The

gradient function of a curve is kx square minus 6x where k is a constant.

Given that the straight line y minus 9x plus 5 is the tangent to the curve at

the point negative 1 negative 9. So the question ask to find the value of k and

the equation of the curve. All right guys I hope you will attempt this question in

the future. If you could manage to solve this question well done guys. But if you

can’t solve this question, no worries I’m going to show you the step-by-step

working and the solutions now. Alright let’s proceed to show the solution now. It is the solution for question part a.

given the gradient function of a curve is dy/dx equals the kx squared minus 6x

and in the question they have already give us the straight line y minus 9x

plus 5 is the tangent to the curve of the point negative 1 negative 9. So as we

learn in the differentiation, the tangent to the curve is also considered a

differentiation guys so how we gonna find the the

differentiation value from Y minus 9x plus 5 so it’s very simple when you

bring negative 9x plus 5 to the right hand side it will become y equals to 9x

minus 5 so these 9 as we learn in the straight

line form in the which is in the y equals the mx plus c, so this is actually

a m. m is actually a gradient of the graph in this case our straight line y

equals to 9x minus 5 is the tangent so the gradient of the tangent is a 9. So

what we going to do now is we’re going to relate this 9 value to the dy/dx.

Sound confusing? No worries guys, I’m going to show you

the step by step working out so please stay with me. Okay for this first step

what we going to do is we’re going to compare the gradient of the graph value.

So which is the gradient of the graph they already give us the kx square minus

6x and then what we’re going to do is we’re gonna compare with the gradient of

the tangent. So it’s given that y minus 9x plus 5 is equal to 0 ok what we’re

doing now is you bring everything to the right hand side so it become y equals to

9x minus 5. As I said earlier, this 9 is the gradient of the tangent value. Okay

what I’m going to do now is I’m going to compare this gradient of tangent value

to the gradient function of a curve by using the coordinate negative 1 negative

9. So please take note guys you only have to use the

x-coordinate value you no need to use the y-coordinate value because we are

calculating with respect to dy/dx, we are not calculating in respect to y. This

is our y value this is the x value which we will be substituting into this dy/dx.

Okay let us continue. So kx squared minus 6x equals to 9 so we’re gonna substitute the x in the equals to

negative 1 so from here we get k in the bracket negative 1 square minus 6 in a

bracket negative 1 it goes to 9 so negative 1 square you get is positive 1.

So k plus 6 negative 6 times negative 1 you get positive 6 okay so plus 6 equals to

9. k plus 6 equals to 9. So what you do is you bring the negative I mean you bring

this positive 6 to the right hand side so will become k equals to 9 minus 6. So

from here you’ll get your k is equals to 3. All right guys we have already solved

our part a solution. Now let us proceed with the part b solution. Okay please hold

on. okay in the part b, since in our part a

we have already find our k value, so what we can do is we can immediately

substitute our value k is equal to three. So now our dy/dx value is equals to 3x

squared minus 6x. So since the question asked to find the equation of the curve,

so how we’re going to find the equation of the curve? Alright guys we are going

to integrate this dy/dx so how you can integrate this dy/dx okay I’m going to

show you now . The first of all, we need to times the whole equation by dx, so this

dx will be in a fraction form will be canceled out and in the right side will

be times by dx. So dy is equals to in a bracket 3x squared minus 6x with respect

to dX. So let us continue. Step one, we are going to integrate y with respect to dx.

So this is the dy form. so when you change it this into y form, you need to

integrate the right hand side of the equation. So y is equals to integrate of

3x squared minus 6x with respect to dx. Let us proceed to the next step. Okay

when you integrate this, you should be getting your answer Y is equal to 3x

cubed over 3 because you’re adding your power X to the power of 2 you add plus 1

so it becomes X cube and these 2 plus 1 you have to divide as well so 3x cubed

divided by 3 minus 6x squared over 2 so this x was in a power of 1 so when you

integrate you need to add 1 so it becomes 6x square so that 2 you need to

divide as well, 6x squared divided by 2 and don’t forget to add a c guys. c is

a constant since we are dealing with the indefinite integral okay. So you have y

equals to x cubed minus 3x square plus C but this is not the final answer guys, we

still need to find the value of C the constant value. so how are you going to

find the value of C? All right guys I hope you remember that in the earlier

of the question they have already gave us the coordinate

x and y. So what we’re going to do do now is we’re going to use those respective x

and y coordinate values and we’re going to substitute in side this equation and

then to find the respective c value let us continue so we’re going to use the

point negative 1 negative 9. So negative 1 is your x coordinate and negative 9 is

your y coordinate so negative 9 equals to negative 1 cube minus 3 negative 1

square plus with C. So from here let us continue. So negative 1 cube, you should

be getting negative 1. Negative 1 square you should be getting positive 1 but

then you times with negative 3 you still get negative 3. So negative 1 minus

negative 3 you should be getting negative 4. So negative 9 equals to

negative 1 minus negative 3 you should get negative 4 and then your c,

should be getting your answer is negative 5. All right guys we have

almost close to our final answer therefore the equation of our curve is y

is equals to x cube minus 3x squared minus 5. So this is your final answer

guys. All right guys I’ve already complete my first question

of integration now I’m going to proceed with the second question. Okay please

hold on for a moment Okay

let’s read the example two question together. The gradient function of a

curve is 3x square plus x minus 2. Given that the curve passes through the

point (2,5), the question ask to find a) the equation of the curve. b) the coordinates

of the turning point of the curve and determine whether each turning point is

a maximum or minimum point. All right Guys let us proceed further to the

solution now. Okay the first question part a), they asked to find us the

equation of the curve and they’ve already gave us the gradient of the

curve which is our the dy/dx. By the way guys the gradient function of the curve

is also known as dy/dx please take a note of that .So they already give us dy/dx

equals to 3x squared plus x minus 2. Okay what we going to do to find the equation

of the curve?Yes guys you’re correct! we need to integrate this value. So we bring

the dx over here or you can time the whole equation by dx, it is up to you. dy is

equals to excuse me guys in a bracket 3x squared plus X minus 2

with respect to dx. So we’re going to integrate y with respect to dx. So what

we do is we integrate 3x squared plus x minus 2 with respect to dx. Let us

proceed So from here, y is equals to 3x cube

over 3 plus x squared over 2 minus 2x plus c, so don’t forget to add a c guys

since we are dealing with the indefinite integral over here so these 3 and 3 will

cancel off. Okay let us proceed but this is not the final answer guys as I said

in the example 1 we still have the C value so we’re going to use the

respective point to calculate the c value. So how are you going to do that ? Okay let me show you the working now. So 2 is the x-coordinate, 5 is the

y-coordinate so I’m going to substitute the 5 as a y and 2 in the x.5 equals to 2

cubed which is 8, plus 2 square over 2 which is 4 divided by 2 which we get

back is 2 and then minus with 4 . 2 times 2 is 4 plus c. So 5 equals to 8 plus

2 minus 4 plus c, so you should be getting your c is equals to negative 1.

So 8 plus 2 you get is 10. 10 minus 4 is 6 .So 5 minus with 6 you get back is

negative 1. So from here you get the equation of the curve is y equals to x

cubed plus x squared over 2 minus 2x and minus 1 so don’t forget to rewrite the

whole equation guys. If you don’t write this you might lose your marks. Okay we

have already solved the part a) question let us proceed to the part b) now. Okay for the part b, they ask us to find

the coordinate of the turning point and then they ask us to determine whether

each coordinate of the turning point is a maximum point or minimum point. so what is the turning point guys? so this is the basic explanation I get to show to you.

As you can see that this is a curve and this is your tangent just assume this is

a tangent this is your curve. So when your curve and at this is your tangent

and when your tangent is in the horizontal form you can assume this is

your dy/dx is equal to 0 because make sense guys because this straight line

has no gradient and your gradient is 0 it’s a flat line so this is your dy/dx equals

is 0 when your dy DX is 0 it’s belief that this is your turning point a

turning point meets when your dy/dx equals to 0. All right guys this is the

basic explanation ok for step 1 to find dy/dx I mean to find the turning

point we need to set our dy/dx equals to 0 because from this diagram the turning

point is the point exact point when your dy/dx is exactly equals to 0, so for that

we need to set our dy/dx is equal to 0. In the question is given that dy/dx is

equal to 3x square plus x minus 2 so when it set it equals 0. Now if you

notice here, you’ve you’re already form a quadratic equation. Let us proceed to the

next step. So when you factor this out you should be getting 3x minus 2 and x

plus 1 equals to 0. From here you get two x values guys the first x value is x

equals to 2 over 3 and x equals to negative 1.

Okay guys, we have already successfully do our step 1. Let us proceed to the step

2. Okay once you’ve already find a turning

point and you set your dy/dx equals 0 and you’re finding the respect and you

found the respective x value, so what you’re going to do now? So step 2 is you

need to do the double differentiation. Okay double

differentiation why we are doing this is to find that whether the particular

x-coordinate I mean the point is a maximum point or minimum point guys. okay but in this case our dy/dx is 3x squared plus x minus 2. So when you do double

differentiation which is d square y over dx square, you should be getting 6x plus

1, so you bring the negative I mean the two times three times and and then the

power you minus by 1, so this x will become 1, so this minus 2 since this is a

constant so it will be removed away. So d square Y over the x value get is 6x plus

1. So what we’re going to do now is we’re going to substitute each x value

and we’re going to determine whether the x value gives us a maximum value or

minimum value. Okay substitute the respective x values, so we’re gonna start

with d square y over dx square we gonna substitute 2 over 3, so when x equals to

2 over 3 so 6 times 2 over 3 plus 1 so where you get your d square Y/dx square

is equals to 5. All right guys I’m going to tell you a small explanation. When you

get your d square y over dx square into a positive value that means that

particular turning point indicates that it’s a minimum point

so since give us a positive value remember this positive value always give

you the minimum point so in this case d square y over dx square is equals to 5. This

coordinate gives us a minimum point. Let us proceed for the next x value. So when

x equals to negative 1, d square y over dx squared equals to 6 in a bracket

negative 1 plus 1. So from here you get your d square y over dx square equals to negative 5. All right guys, since this is in a negative value, so this point where your x

equals to negative 1 give you a maximum point.

Alright guys you already know this is the two over three gives you a minimum

point and negative one gives you a maximum point, so now we need to find the

respective y value. So what we’re going to do now is we’re gonna use this x

coordinate and then we need to substitute back into the original

equation of the curve to find the respective y coordinates. So let us

proceed to the next step Substitute back the x value to the

equation of the curve to find the y value, so that is our step three guys

respectively. So when x equals to two over three

so y is equals to two over three square I mean 2 over 3 cube plus 2 over 3

square over 2 minus 2 times with 2 over 3 minus 1. From here you get your y is

equals to negative 49 over 27 so therefore you should know that 2 over 3

comma negative 49 over 27 gives you a minimum point. Okay let us proceed for the

second x value. So when x equals to negative 1,

you substitute back the negative 1 value y equals the negative 1 cube plus

negative 1 square over 2 minus 2 times with negative 1 minus 1. So from here, you

get your y value is 1 over 2. So therefore negative 1 comma 1 over 2 give

us the maximum point. All right guys, that is all for my step-by-step working and

solution for this question. Any questions or feedback feel free to put in the

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