Tangents and normals

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

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

The curve crosses the x‑axis at the point A.

Verify that the x coordinate of A is 4

The line l₁ is the tangent to the curve at A.

Use calculus to show that an equation of line l₁ is

12 x + y = 48

The line l₂ has equation y = 8x The region R, shown shaded in Figure 3, is bounded by the curve, the line l₁ and the line l₂

Use algebraic integration to find the exact area of R.

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 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 entirely on calculator technology are not acceptable.

The curve C has equation y = f(x) where x

Given that

• the curve has a stationary point with x coordinate α

• α is small

use the small angle approximation for cosx to estimate the value of α to 3 decimal places.

The point P(0, 3) lies on C

Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

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.

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

The point P(5, –13) lies on C

The line l is the tangent to C at P

Use differentiation to find the equation of l, giving your answer in the form y = mx + c where m and c are integers to be found.

Hence verify that l meets C again on the y-axis.

The finite region R, shown shaded in Figure 2, is bounded by the curve C and the line l.

Use algebraic integration to find the exact area of R.

In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve C with equation

The point P(5, –13) lies on C The line l is the tangent to C at P

Use differentiation to find the equation of l, giving your answer in the form y = mx + c where m and c are integers to be found.

Hence verify that l meets C again on the y-axis. The finite region R, shown shaded in Figure 2, is bounded by the curve C and the line l.

Use algebraic integration to find the exact area of R.

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 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

Figure 2 shows a sketch of part of the curve C with equation y xlnx, x 0

The line l is the normal to C at the point P(e, e)

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

Show that the exact area of R is Ae² B where A and B are rational numbers to be found.

The curve C has equation

Find in its simplest form.

Find an equation of the tangent to C at the point where x = –1 Give your answer in the form ax + by + c = 0, where a, b and c are integers.

A curve with equation y = f(x) passes through the point (4, 9). Given that

find f(x), giving each term in its simplest form.

Point P lies on the curve . The normal to the curve at P is parallel to the line 2 y + x = 0

Find the x coordinate of P.

The curve C has equation y = x² + 8 The line L has equation y = 3 x + k, where k is a positive constant.

Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.

Given that line L is a tangent to C,

find the value of k.

A sketch of part of the curve C with equation

is shown in Figure 3. Point A lies on C and has an x coordinate equal to 2

Show that the equation of the normal to C at A is y = –2 x + 7

The normal to C at A meets C again at the point B, as shown in Figure 3.

Use algebra to find the coordinates of B.

Given that

equal to 10 (4)

Figure 2 shows a sketch of the curve H with equation y = + 4, x

Give the coordinates of the point where H crosses the x-axis.

Give the equations of the asymptotes to H.

Find an equation for the normal to H at the point P (–3, 3).

This normal crosses the x-axis at A and the y-axis at B.

Find the length of the line segment AB. Give your answer as a surd.

The curve C has equation

Find , giving each term in its simplest form. The point P on C has x-coordinate equal to

Find the equation of the tangent to C at the point P, giving your answer in the form y = a x + b, where a and b are constants.

The tangent to C at the point Q is parallel to the line with equation 2x − 3y + 18 = 0

Find the coordinates of Q.

The point P (4, –1) lies on the curve C with equation y = f(x), x > 0, and

Find the equation of the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are integers.

Find f(x).

The curve C has equation

Find an equation of the tangent to C at the point where x = 2

The curve C has equation

The point P has coordinates (2, 7).

Show that P lies on C.

Find the equation of the tangent to C at P, giving your answer in the form y = mx + c, where m and c are constants.

The point Q also lies on C. Given that the tangent to C at Q is perpendicular to the tangent to C at P,

show that the x-coordinate of Q is 2 6.

The curve C has equation

The point P on C has x-coordinate equal to 2.

Show that the equation of the tangent to C at the point P is y = 1 – 2 x.

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

The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B.

Find the area of triangle APB.

Given that the point P (4, 1) lies on C,

find f(x) and simplify your answer.

Find an equation of the normal to C at the point P (4, 1).