Solving Past Papers | Differentiation | Class 28 | AS Level P1 9709

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Okay. So, for today's class we're going to do we have completed the topic of differentiation the last class. And we have revised differentiation basically with these examples and now we are going to attempt some past papers from differentiation. And now only integration is left. Okay. So, the question is given in 2021. The function f x is defined by So, they have given you the function f x. It is x squared plus k over x. K over x I can write it down k x power minus one plus two. For look at these values. Sometimes this can be useful. X is greater than zero. So, this means that you can only put in the values of x

which are positive, more than zero. Given that the curve with the function has a stationary point So, they have told us that it has a stationary point when x is two. This means that if your x is equals to two, your first derivative will come out as zero. Because it is a stationary point. So, the value of the derivative should come it multiplies to two x power one. And again the power comes down and multiplies and power reduces by one. And this is a constant so zero is equals to zero when x is two. So, this means now two times two minus k two power minus two is equals to zero. And now we can find the value of

K. So, 4 - K upon 4 is equals to 0. 4 is equals to K upon 4. And so, your K comes out as 16. This would go over there and multiply. So, you have some idea now. >> Yes, sir. >> Then they're asking, "Look, determine the nature of the stationary point." So, for this thing, you always have to find the double derivative. So, and then the first derivative we have found it is It is this, 2X XX Sorry, - KK we have found. K 16 16 X - 2 is your first derivative. And now you will differentiate it again. So, 2X becomes 2. The power comes down and multiplies, so it becomes + 32. X power minus 3 power reduces further.

And now we have to find the nature of this stationary point. So, this So, when your X was 2, look. So, stationary point was when your X is 2. So, if X is 2, so the value of the second derivative at this point will be positive. This will be 42 + 4. I think 32 upon 8 will be 4. So, 2 + 4 is 6. Which is positive, and which means definitely that this point is a minimum point. That's it. Then given that this is the only stationary point of the curve, find the range of F. Given that this is the only stationary point of the curve, find the range of F. We'll have to find this find the range of F. That's a good

question. This is the only stationary point. This is the only stationary point. So, first of all, find the value of Y at the stationary point. So, the function is now the complete function has become x squared plus 16 upon x plus two. So, the turning point was two. X was two. So, F two the value of Y at this point is uh four plus eight plus two. Four plus eight 12 12 plus two is 14. And this is a minimum point. So, this means that the curve is going upwards like this. Isn't it? This is the minimum point like this. And this point is the two {comma} 14. And we know that this is the What is the domain given to us?

The domain was given to us that your x is zero. Oh, sorry. Greater than zero. X is greater than zero. Look at the question. Look. X is greater than zero. So, if it is zero, exactly zero, then your F X will come out as There is There will be no answer. Look. If If your F If you find the value of the function at zero zero squared plus 16 upon zero. So, no solution. This will be asymptote. So, this means the function is coming from above like this. And then coming at this turning point, and then because this is the minimum point, the graph will be going up again. Isn't it? Some getting it or not?

So, what is the range? Rayan? What is the range? Rayan, Rayan, Rayan, what is the range now? >> Greater than like two. >> Greater than greater than two is the Y value, sorry, X value. Range is the Y value. Yes. Yeah, that is your answer. FX is greater than equal to 14. Okay? Getting it or not? >> Yeah. >> Your function Look, you can see if your X is all positive, so your lowest value of Y that you're getting is 14, and the value of Y is increasing on both sides. So, it will always be greater than equal to 14 if your X is greater than zero. That's why this is the answer. Good question. Good question.

Okay, so the equation of the curve was given to us Y is equals to 2 plus under root 26 minus X squared. Under root, I'm writing this under root as power half because that is better understanding for working on it. Then, find the coordinates of the point on the curve at which the gradient is 4 upon 3. So, definitely, this would be any curve. Okay, we don't know how is this curve. It can be anything. Most probably, it will be something like this, but I don't know. Okay, we don't know about the curve. But, there can be points on this curve where the tangent is 4 upon 3. So, this means the gradient of the

tangent is always what? The dy by dx. So this means that they have given you the value of the dy by dx 4 upon 3 and they're asking you to find the values of x. Find x when your dy by dx is equal to 4 upon 3. So what do you have to do? You will have to find the derivative. This will become 3 0. 2 is a constant so 0. Power comes down multiply so plus half. The bracket doesn't change. The power reduces by half further and the derivative of the inside which will be minus 2x. 0 minus 2x. And now if you simplify 2 and this 2 cancel out. So your derivative for which the value they have given to us it is 4

upon 3 it is equal to minus x upon 25 minus x square power half. This is your equation that you're that you're forming right now. And now we have to solve the rest solving for the value of x. Look they're asking you for the coordinates. So first we'll find we'll find the values of x when this is happening and then we will we'll find the y coordinate as well. Because it is the complete the coordinates they're asking for the coordinates. Coordinates mean the x and the y both. So how would you solve this? How would you solve this? You can square on both sides. So if you square on both sides because

you're going to get rid of the square root. So this would become 400 is equal to 25 x square. x square will be equal to 24 divided by 25 which will be 16. So your X comes out as plus minus four. So, this means that there are two values of X at which the tangent is the gradient of the tangent is equal to 4 upon 3. And now you want to find the value of We will find the value of Y as well. And not the not just the value of X. So, if your if your X is plus four, so your Y comes out as two plus under root 25 minus four squared which will become 16. 16 minus 25 is nine. Square root of nine is three. So, two

plus nine two plus three is five. This is one of the coordinate. 4, 5. And the other coordinate will be if your X value is equal to minus four. So, which will be the same. You'll get the same answer. Because minus four is squared and plus four squared will be the same thing. So, Y comes out as five. So, the other coordinate is minus five comma minus four comma 25. This is the answer. That's why they called coordinates of the points. So, there are two coordinates. This is the answer. This is the answer. Five marks. Okay. The diagram shows the solid cone which has a slant height of 15 and a vertical height of capital H. Show

that the volume of the cone is given by this. Okay. First of all, what is the formula for the volume of cone? It is 1 upon 3 pi r squared h. And we have the slant height. We don't have the radius. Radius is this one. And this is the capital height that sorry, this straight height. This is the height we We have. So, H the height is is still H. The radius we will have to find the value of the radius in terms of this H by using this Pythagoras theorem. The right angle triangle. So, 15 squared is equals to R squared plus H squared. So, R squared is equals to 15 squared plus H minus H squared. So, R squared is equals to 225 minus H

squared. So, now you can replace this value of R over here. So, your V comes out as 1 upon 3 into pi into 225 minus H squared into H. And then if you expand the bracket or multiply the H outside inside the bracket so, 225H minus H cubed. So, this is the one the same equation. So, two marks you got the two marks forming the equation. Then they have given you that uh given that H can vary. Find the value of H for which V has a stationary point. Determine showing all necessary the nature as well. So, we have to find the turning point. We have to find the value of V, the value of H. And we also have to find the

nature. So, we would have to first of all do the derivative of this equation. V is equals to 1 upon 3 pi 225H minus 1 upon 3 pi H cubed. I have I have expanded the bracket. Now, if you differentiate this will become 1 upon 3 pi into 225 minus the power 3 comes down and multiplies. So, 3 upon 3 pi H squared. And we have to find the stationary value, so it's equals to zero. Now, if you solve this, 225 pi upon 3 minus pi h^2 is equals to zero. So, 225 upon 3 pi is equals to pi h^2. Pi and pi cancel out. And so, your h comes out as 225 upon 3 square root. Plus minus. And we know that h, the height, cannot be negative. Therefore,

the h value positive we will use. So, we have the found the value of h, which is positive square root 225 upon 3. This is the value of h for which we have stationary value. And now, determine whether this is stationary the stationary is minimum maximum. So, we have found the first derivative. Here is the first derivative. And now, you will differentiate it again. So, d v^2 upon d h d^2 h is equals to This is the constant. So, this would become zero. Then, this would become minus 2 pi h. Because three and three this becomes one. So, the two power comes down, 2 pi. And power of h reduces by one, so 2 pi

h. And now, you will substitute. If your h is positive under root 225 upon 3, you can see the double derivative value will come out as positive. Regardless of the whatever the value that is, we can see that if we put in a positive value over here, and it will multiply with the negative sign, so the overall sign will be negative. So, this means this is a maximum value. Okay? Any questions, anyone? >> An increasing function, they have already defined that it is an increasing function. It's defined for X greater than N. Where N is a integer. Given that the derivative value the derivative function

is given to you. Find the least possible value of N. So the derivative is already given to us. We have to find the least value of N for which X greater than N is increasing. So first of all they have told us that this should be increasing. So this means the derivative first derivative will be positive. So this means your X is squared minus 6X plus 8 should be positive. And how do you solve the quadratic inequality? You first I'm find the roots. So the roots are X is equals to 2 and 4. X is equals to 4. And now with the small sketch because it is this one is 4 this one is 2. Because it is more than 0, so you will

shade the part which is above. So X is more than 4 and X is less than 2. So which one you can see? This means the value of N is 4. The least value of N is 4. X greater than 4 will always be the gradient will always be the dy by dx will always be positive. Okay? Do you get it? Any questions? >> No, sir. This was You can see the logic is still the same. You have to understand what are they telling you, what information they've given you, and how have to you how you have to use that. And further They're just rephrasing the thing so complicatedly. But it's actually the same logic, the increasing decreasing function, you

know. That's what you're doing the practice for, understanding the question what they're asking you, and then the rest of it is simple. The dimension of of the cuboid are given X, 2X, and 4X. Show that the surface area and the volume are connected by this relationship. So, the volume formula is LBW, right? LBH. So, X * 2X * 4X which will become 8X cubed. And the total surface area it is twice length by breadth plus twice breadth by height plus twice length by height. And then you add them up. Right? So, if you find this the total surface area two times length, this let's suppose this length this is breadth, this is height.

So, length by breadth 8X squared plus two breadth by height 4X multiplied by X, this will be 4X squared plus two length by height 2X multiplied by X, this will be 2X squared. And now if you sum all of it 16X squared plus 8 plus 4 16X squared 16 plus 8 + 4 24 and 4 28 28x squared. This is the total surface area. And now they're telling you that S is equals to 7 V 2 upon 3. Let's see. The total surface area we have found it is 28 x squared. Is this the equal same as 7 * 8 x cubed power 2 upon 3. So, how much will this be? 8 power 2 upon 3 The x is already I can see it is same. 7 multiplied by 8 power 2 upon 3

Sorry. 7 multiplied by 8 power Uh look 28. So, 28 x squared is the same thing. Left Left hand is equals to right hand. So, therefore shown. Shown. When the volume of the cuboid is 1,000 the surface area is increasing at Mhm. So, volume is 1,000. The surface area the total surface area is increasing at 2 cm squared per second per second. Find the rate of increase of the volume. They're asking you to find DV by DT. DT. Using this we can find the DS by DT. Sorry, DS by DV. If you differentiate you will get 7 * 2 upon 3 V power minus 1 upon 3. 14 upon 3 V negative 1 upon 3. And there are they've given you that V

is 1,000. So, if V is 1,000, so the value of the dS by dV it is equals to 14 upon 3 and 2,000. 1 upon 3 negative. So, 14 upon 3,000 14 upon 10, I think. 30 30 30. 7 upon 15. Okay, so they've simplified it so 7 upon 15. And now using the rates of change formula, we have dS over dV. It will be equal to dS over dT divided by dV upon dT. Now, we have found the value of dS upon dV. dS by dT this one already given the two divided by they're asking you to find dV by dT. So, this would go over there and multiply. Right? And the other part will go over there and divide. So, dV by dT it is equals to two

upon 7 upon 15. Which will come out as 30 upon 7. This is your answer. See it? This is your answer. 4.29 So what I got is Water is poured into a tank at a constant rate of look it is volume per second. So DV by DT is given to you. 500 The depth of the water in the tank is H. Depth is H. When the depth of water in the tank is H, the volume V is given by So V is equals to 4 upon 3 25 minus plus H whole square sorry whole cube minus 62500 upon 3. Okay, this is the formula for the volume they've given it to us. Find the rate at which H is increasing when at the instant when H is 10. So first if you differentiate

this This is a question of rates of change. Whenever you have time and variable two variable two other variables volume and height and time so definitely it's rates of change. So first derivative 4 upon 3 multiplied by 3 25 plus H power square and the derivative of the inside will be one minus zero And if value of H is given to us it is 10. So you can find the value of DV by DH. 3 and 3 cancel out. 4 into 25 plus 10 squared 4 into 35 squared This is the value of DV by DH. Now the the formula of the rates of change that you can form right now is DV divided by DH divided by DT divided by DT. So we have this value 4900.

DV by DT it is given to us 500 divided by DH by DT. This would go over there and multiply. And 4900 will come and divide. 5 upon 49 this is the answer. So yeah. >> G sir. >> At another instant the rate at which H is increasing is 0.075. Find the value of V at this moment. At this instant, okay. So they have given given that to us that DV by DT it is equals to 500. And they've also now given that DH by DT it is equals to 0.075. So now using these two and using the formula we can find the value of DV by DH. So DV by DH is is it is equals to 500 DV by DT divided by DH by DT. Which will be how much?

2000 20,000 upon 3. And this is equals to the derivative function that we have found. Look. And now we can find the value of h. And then using the value of h, we will find the value of v as well. So now divided by 4, this would become 5,000 upon 3. 25 plus h whole squared. Square root and you will get two answers. And then minus 25. So the positive answer for h will be considered. Minus 25 plus under root 5,000 upon 3. This is 15. 15.82 5 meters. And if we take negative, it is a negative answer. So we won't consider the negative one. This is what we have found and now we find the value of v using the original

formula. That's what they were asking you. So using this answer, this is your answer. 4 upon 3 into 25 plus answer cubed minus 62500 upon 3. This is your answer. Correct to three significant figures if you round it off. This is the value of the volume. Okay? >> Okay, then Sanjeev. >> Yes, sir. >> Good question. So, second part was interesting. And then the third part of this one. The equation of a curve is given to you. Find the derivative. Okay, we will find it. Find the x-coordinate of the stationary point in terms of k. You will Okay, find them. Then given that k is 1.5 10.5 Find the equation of the normal to the

curve at the point where the tangent to the curve makes an angle of Okay, this is which air? Which air is it? Have they given you the air? >> No, air needed. >> This is something which is very very few times that is only tested. I'll give you the information. If you have a straight line and it has a gradient m to the angle that the straight line makes with the horizontal theta it is equals to tan inverse of the gradient. This is the logic behind it. Why? tan inverse of gradient? Because what is gradient? If I move the 10 on the other side, so 10 theta it is equals to gradient is what? Change in Y upon

change in X. So, let's suppose this is the change in X and this is the change in Y. So, can you see? 10 is equals to opposite upon adjacent. Enough? Do you see this? Understand? >> Yeah. >> Mhm. This is the logic behind it. So, now when they have given you the 10 inverse 2 So, the theta they have given you the value of the theta. Theta is equals to 10 inverse of 2. So, this means the gradient M is equals to 10 theta. If I move the 10 back on the towards the theta, so this means that your gradient is coming out as 2, which is the value of your dy by dx. So, when they have given you the value of dy by dx

this is the gradient of the tangent. And so therefore, the gradient of normal will be negative 1 upon 2. So, we have the gradient of normal. And uh what more do we need? We need the point. We need the point. How will you find the point? They have given us that the value of K is 10.05. So, you already have the dy by dx value was which is equals to 2, and you put in the value of K as well. So, you can find the value of X. And when you have X, you can find Y as well using original function. >> So you have the X and the Y and the gradient of the normal. You have everything. And then you would use Y

minus Y1 is equals to M X minus X1 to find the equation of the normal. So yeah. I would ask you people to try this again. Okay, you you should try this question again. Ryan, you as well. This is a good question. >> Thanks. >> Okay, I'm stopping the recording now. Good. Good. Good. Do more questions. Bring in questions over here. We'll discuss.