Parametric differentiation

The curve C has parametric equations

The point P with x coordinate –3 lies on C.

Find the y coordinate of P.

Find a Cartesian equation for C, giving the answer in the form y = f (x)

Hence, or otherwise, find the gradient of C at the point P.

Figure 4 shows a sketch of the curve C with parametric equations

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

Using parametric differentiation, show that the tangent to C at P has equation

The curve C is used to model the profile of a slide at a water park.

Units are in metres, with y being the height of the slide above water level.

Find, according to the model, the greatest height of the slide above water level.

The curve C has parametric equations

Show that a Cartesian equation for C is

where A and B are integers to be found. The curve C cuts the y-axis at the point P

Show that the equation of the tangent to C at P can be written in the form ax + by = c ln 5 where a, b and c are integers to be found.

Figure 6 shows a sketch of the curve C with parametric equations

The line l is the normal to C at the point P where t =

Using parametric differentiation, show that an equation for l is

Show that all points on C satisfy the equation

The straight line with equation

intersects C at two distinct points.

Find the range of possible values for k.

The curve C has parametric equations

Find an expression for in terms of θ

Hence find the exact value of the gradient of the tangent to C at the point where y = 8

The curve C has parametric equations

Find an expression for in terms of θ

Hence find the exact value of the gradient of the tangent to C at the point where y = 8

The curve C has parametric equations

where θ is a parameter.

Find an expression for in terms of . The curve C cuts the y-axis at the point A. The line l is the normal to C at the point A.

Show that an equation for l is

The line l intersects the curve C again at the point B.

are rational constants. (Solutions based entirely on graphical or numerical methods are not acceptable.)

The curve C has parametric equations

The point P lies on C where t =

Find the equation of the tangent to C at the point P. Give your answer in the form y = px + q, where p and q are integers to be determined.

Show that the cartesian equation for C can be written in the form

where a and b are integers to be determined.

Figure 2 shows a sketch of the curve C with parametric equations

The point P lies on C and has coordinates 4 .

Find the exact value of at the point P. Give your answer as a simplified surd.

The point Q lies on the curve C, where 0

Find the exact coordinates of the point Q.

A curve C has parametric equations

Find the value of at the point on C where t = 2, giving your answer as a fraction in its simplest form.

Show that the cartesian equation of the curve C can be written in the form

where a and b are integers to be determined.

Figure 4 shows a sketch of part of the curve C with parametric equations

The point P lies on C and has coordinates (3, 2). The line l is the normal to C at P. The normal cuts the x-axis at the point Q.

Find the x coordinate of the point Q.

The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid of revolution, giving your answer in the form + q², where p and q are rational numbers to be determined. [You may use the formula V πr²h for the volume of a cone.]

The curve shown in Figure 3 has parametric equations

The point A, with coordinates (k, 1), lies on the curve. Given that k > 0

find the exact value of k,

find the gradient of the curve at the point A.

There is one point on the curve where the gradient is equal to −

Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal places. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

A curve C has parametric equations

Find at the point where t =

Find a cartesian equation for C in the form

stating the value of the constant k.

Write down the range of f(x).

Figure 2 showsa sketch of the curve C with parametric equations

Show that the cartesian equation of C may be written in the form

stating the values of a and b. (3)

shown shaded in Figure 3. This region is rotated through 27 radians about the x-axis to form a solid of revolution.

Use calculus to find the exact value of the volume of the solid of revolution. (5)

Figure 2 shows a sketch of part of the curve C with parametric equations

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

Show that A has coordinates (0, 3).

Find the x coordinate of the point B.

Find an equation of the normal to C at the point A.

The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the x-axis.

Use integration to find the exact area of R.

Figure 2 shows a sketch of the curve C with parametric equations

Show that = kt, where k is a constant to be determined.

Find an equation of the tangent to C at the point where t = Give your answer in the form y = ax + b, where a and b are constants.

Find a cartesian equation of C.

Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates ˲ 3, 3.

Find the value of at the point P.

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

Show that Q has coordinates k ˲ 3, 0, giving the value of the constant k.

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x ˲3 and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the volume of the solid of revolution, giving your answer in the form p ¥ 3 + q ², where p and q are constants.

The curve C has parametric equations

Find

a cartesian equation of C.

The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x ln 2 and the line x ln 4. The area R is rotated through360 about the x-axis.

Use calculus to find the exact volume of the solid generated.

A curve C has parametric equations

Find the x-coordinate of P.

The curve C shown in Figure 3 has parametric equations

where t is a parameter. Given that the point A has parameter t = –1,

find the coordinates of A.

The line l is the tangent to C at A.

Show that an equation for l is 2 x – 5 y – 9 = 0.

The line l also intersects the curve at the point B.

Find the coordinates of B.

Figure 3 shows the curve C with parametric equations

The point P lies on C and has coordinates (4, 2√3).

Find the value of t at the point P.

The line l is a normal to C at P.

Show that an equation for l is y = –x√3 + 6√3.

The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3.

Show that the area of R is given by the integral.²64 sin² t cost dt

Use this integral to find the area of R, giving your answer in the form a + b√3, where a and b are constants to be determined.