Implicit differentiation

The curve C has equation

Find an expression for in terms of x and y.

The point P(1, 0) lies on C.

Show that the normal to C at P has equation

Prove that the normal to C at P does not meet C again.

You should use algebra for your proof and make your reasoning clear.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation

The point P(–2, 5) lies on the curve.

Find the equation of the normal to the curve at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers to be found.

The curve C has equation

where p and q are constants.

Show that

where a, b and c are integers to be found.

Given that

• the point P (–1, – 4) lies on C

• the normal to C at P has equation 19 x + 26 y + 123 = 0

find the value of p and the value of q.

The curve C has equation

where p and q are constants.

Show that

where a, b and c are integers to be found. Given that

• the point P (–1, – 4) lies on C

• the normal to C at P has equation 19 x + 26 y + 123 = 0

find the value of p and the value of q.

The curve C has equation

Show that

Figure 4 shows a sketch of the curve with equation x² 2xy 3y² 50

The curve is used to model the shape of a cycle track with both x and y measured in km.

The points P and Q represent points that are furthest west and furthest east of the origin O, as shown in Figure 4.

Using part (a),

find the exact coordinates of the point P.

Explain briefly how to find the coordinates of the point that is furthest north of the origin O. (You do not need to carry out this calculation).

The curve C has equation x²y³xxsin(௘y

The point P with coordinates (3, 2) lies on C. The tangent to C at P meets the y-axis at the point Q.

Find the y coordinate of Q, giving your answer in the form where a, b and c are integers to be found.

The curve C has equation

The point P with coordinates (–2, 4) lies on C.

Find the exact value of at the point P. The normal to C at P meets the y-axis at the point A.

Find the y coordinate of A, giving your answer in the form p + q ln2, where p and q are constants to be determined.

The curve C has equation 2x²y + 2x + 4y – cos () = 17

Use implicit differentiation to find in terms of x and y. The point P with coordinates 3, lies on C. The normal to C at P meets the x-axis at the point A.

Find the x coordinate of A, giving your answer in the form , where a, b, c and d are integers to be determined.

The curve C has equation

Find the coordinates of the points on C where = 0 (Solutions based entirely on graphical or numerical methods are not acceptable.)

Figure 3 shows a sketch of part of the curve C with equation

The point P lies on C and has coordinates (2, 9). The line l is a tangent to C at P. The line l cuts the x-axis at the point Q.

Find the exact value of the x coordinate of Q.

The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line l. This region R is rotated through 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated. Give your answer in the form where p and q are exact constants. [You may assume the formula V = r₂h for the volume of a cone.]

A curve C has the equation

Find an equation of the tangent to C at the point (3, –2), giving your answer in the form ax + by + c = 0, where a, b and c are integers.

x² + y² + 10x + 2y – 4xy = 10

Find in terms of x and y, fully simplifying your answer.

Find the values of y for which = 0

The curve C has equation Show that at the point (1, 3) on the curve C can be written in the form where A and yw are integers to be found. (7)

A curve is described by the equation

A point Q lies on the curve. The tangent to the curve at Q is parallel to the y-axis. Given that the x coordinate of Q is negative,

use your answer to part (a) to find the coordinates of Q.

The curve C has equation

Find the coordinates of the points on C where = 0.

Find the gradient of the curve with equation ln y = 2 x ln x, x 0, y 0 at the point on the curve where x = 2. Give your answer as an exact value.

A curve C has equation

Find the exact value of at the point on C with coordinates (3, 2).

The curve C has the equation

The point P lies on C where x

Find the value of y at P.

Find the equation of the tangent to C at P, giving your answer in the form ax + by + cπ = 0, where a, b and c are integers.

A curve C has the equation y² – 3y = x³ + 8.

Hence find the gradient of C at the point where y = 3.

A curve has equation 3x² – y² + xy = 4. The points P and Q lie on the curve. The gradient of the tangent to the curve is at P and at Q.

Use implicit differentiation to show that y – 2 x = 0 at P and at Q.

Find the coordinates of P and Q.

A curve is described by the equation

Find the coordinates of the two points on the curve where x = –8.

Find the gradient of the curve at each of these points.