8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives

2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).

1. Find a normal vector at the point \(( x , y , z )\) on the surface.
2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.

2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).

2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).

2 You are given that \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } + 2 x z + 2 y z + 4 z - 3\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). The surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), and \(\mathrm { P } ( - 2 , - 1,1 )\) is a point on \(S\).
2. Find an equation for the normal line to the surface \(S\) at the point P .
3. A point Q lies on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find the constant \(c\) such that \(\mathrm { PQ } \approx c | h |\).
4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
5. Given that \(x + y + z = k\) is a tangent plane to the surface \(S\), find the two possible values of \(k\).

2 A surface has equation \(z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y\).

1. For a point on the surface at which \(\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }\), show that either \(y = x\) or \(y = 1 - x\).
2. Show that there are exactly two stationary points on the surface, and find their coordinates.
3. The point \(\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)\) is on the surface, and \(\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)\) is a point on the surface close to P . Find an approximate expression for \(h\) in terms of \(w\).
4. Find the four points on the surface at which the normal line is parallel to the vector \(24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }\).

2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Find the equation of the normal line to the surface \(S\) at the point P .
3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.

2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the three stationary points on the surface.
3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).

2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).

1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).

1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).

8 The motion of two remote controlled helicopters \(P\) and \(Q\) is modelled as two points moving along straight lines. Helicopter \(P\) moves on the line \(\mathbf { r } = \left( \begin{array} { r } 2 + 4 p \\ - 3 + p \\ 1 + 3 p \end{array} \right)\) and helicopter \(Q\) moves on the line \(\mathbf { r } = \left( \begin{array} { l } 5 + 8 q \\ 2 + q \\ 5 + 4 q \end{array} \right)\).
The function \(z\) denotes \(( P Q ) ^ { 2 }\), the square of the distance between \(P\) and \(Q\).

1. Show that \(z = 26 p ^ { 2 } + 81 q ^ { 2 } - 90 p q - 58 p + 90 q + 50\).
2. Use partial differentiation to find the values of \(p\) and \(q\) for which \(z\) has a stationary point.
3. With the aid of a diagram, explain why this stationary point must be a minimum point, rather than a maximum point or a saddle point.
4. Hence find the shortest possible distance between the two helicopters. The model is now refined by modelling each helicopter as a sphere of radius 0.5 units.
5. Explain how this will change your answer to part (d). \section*{END OF QUESTION PAPER}

2 The surface \(S\) has equation \(z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y\).
It is given that \(S\) has two stationary points; one at the origin, \(O\), and the other at the point \(A\).
Determine the coordinates of \(A\).

5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1). \begin{figure}[h] \end{figure} The student's model uses the surface with equation \(\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }\), a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. - Describe two problems with this model.
   Explain what has happened.

3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.

2 An open-topped rectangular box is to be manufactured with a fixed volume of \(1000 \mathrm {~cm} ^ { 3 }\). The dimensions of the base of the box are \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The surface area of the box is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(\mathrm { A } = \mathrm { xy } + 2000 \left( \frac { 1 } { \mathrm { x } } + \frac { 1 } { \mathrm { y } } \right)\).
2. 1. Use partial differentiation to determine, in exact form, the values of \(x\) and \(y\) for which \(A\) has a stationary value.
   2. Find the stationary value of \(A\).

5 Let \(\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1\). The surface \(S\) has equation \(z = \mathrm { f } ( x , y )\).

1. (a) Find \(f _ { x }\).
   (b) Find \(\mathrm { f } _ { y }\).
   (c) Show that \(S\) has a stationary point at ( \(0,0,1\) ).
   (d) Find the coordinates of the second stationary point of \(S\).
2. The section \(z = \mathrm { f } ( a , y )\), where \(a\) is a constant, has exactly one stationary point. Determine the equation of the section. A customer takes out a loan of \(\pounds P\) from a bank at an annual interest rate of \(4.9 \%\). Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of \(\pounds M\) in order to reduce the outstanding amount of the loan. Let \(L _ { n }\) denote the outstanding amount of the loan at the end of month \(n\) after the fixed payment has been made, with \(L _ { 0 } = P\).
3. Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation $$L _ { n + 1 } = 1.004 L _ { n } - M$$ with \(L _ { 0 } = P , n \geq 0\).
4. Solve, in terms of \(n , M\) and \(P\), the first order recurrence relation given in part (i).
5. The loan amount is \(\pounds 100000\) and will be fully repaid after 10 years. Find, to the nearest pound, the value of the monthly repayment.
6. The bank's procedures only allow for calculations using integer amounts of pounds. When each monthly amount of the outstanding \(\operatorname { debt } \left( L _ { n } \right)\) is calculated it is always rounded up to the nearest pound before the monthly repayment ( \(M\) ) is subtracted.
   Rewrite (*) to take this into account.
7. Let \(N = 10 a + b\) and \(M = a - 5 b\) where \(a\) and \(b\) are integers such that \(a \geq 1\) and \(0 \leq b \leq 9\). \(N\) is to be tested for divisibility by 17 .
   (a) Prove that \(17 \mid N\) if and only if \(17 \mid M\).
   (b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that \(17 \mid 4097\).
8. (a) Show that, for \(n \geq 2\), any number of the form \(1001 _ { n }\) is composite.
   (b) Given that \(n\) is a positive even number, provide a counter-example to show that the statement "any number of the form \(10001 _ { n }\) is prime" is false. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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1 The surface \(E\) has equation \(z = \sqrt { 500 - 3 x ^ { 2 } - 2 y ^ { 2 } }\).

1. Determine the values of \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\) at the point \(P\) on \(E\) with coordinates \(( 11 , - 8,3 )\).
2. Find the equation of the tangent plane to \(E\) at \(P\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.

9 For all real values of \(x\) and \(y\) the surface \(S\) has equation \(z = 4 x ^ { 2 } + 4 x y + y ^ { 2 } + 6 x + 3 y + k\), where \(k\) is a constant and an integer.

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Determine the smallest value of the integer \(k\) for which the whole of \(S\) lies above the \(x - y\) plane.

2 A surface \(S\) has equation \(\mathrm { z } = 4 \mathrm { x } \sqrt { \mathrm { y } } - \mathrm { y } \sqrt { \mathrm { x } } + \mathrm { y } ^ { 2 }\) for \(x , y \geqslant 0\). Determine the equation of the tangent plane to \(S\) at the point (1,4,20). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.

2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).

1. (a) Find the following.

1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).

1. Find

2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).

1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
   (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
   (B) Deduce that the stationary point A is a local minimum.
   (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .

3 The surface \(S\) has equation \(z = f ( x , y )\), where \(f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\) for all real values of \(x\) and \(y\). You are given that \(S\) has a stationary point at the origin, \(O\), and a second stationary point at the point \(P ( a , b , c )\), where \(\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )\). \begin{enumerate}[label=(\alph*)] \item Determine the values of \(a , b\) and \(c\). \item Throughout this part, take the values of \(a\) and \(b\) to be those found in part (a).

1. Evaluate \(\mathrm { f } _ { x }\) at the points \(\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )\) and \(\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )\).
2. Evaluate \(\mathrm { f } _ { y }\) at the points \(\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )\) and \(\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )\).
3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of \(S\), given by

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. 1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
   2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
2. Write down the equation of any one contour of \(S\) which does not include the origin.

In this question you must show detailed reasoning. A surface \(S\) is defined by \(z = \mathrm{f}(x, y)\) where \(\mathrm{f}(x, y) = x^3 + x^2 y - 2y^2\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
2. Find the stationary points on \(S\). [7]