2024年5月1日发(作者：如何深度清理c盘)

C1 Differentiation

2010 qu. 6

Find the gradient of the curve y = 2x +

6

x

at the point where x = 4.

2010 qu. 10

(i) Find the coordinates of the stationary points of the curve y = 2x

3

+ 5x

2

– 4x.

(ii) State the set of values for x for which 2x

3

+ 5x

2

– 4x is a decreasing function.

(iii) Show that the equation of the tangent to the curve at the point where x =

1

2

is

10x – 4y – 7 = 0.

(iv) Hence, with the aid of a sketch, show that the equation 2x

3

+ 5x

2

– 4x =

57

2

x−

4

has two

distinct real roots.

3.

Jan 2010 qu. 3

Find the equation of the normal to the curve

y = x

3

– 4x

2

+ 7 at the point (2, –1), giving your

answer in the form ax + by + c = 0, where a, b and c are integers.

2010 qu. 6

Not to scale

The diagram shows part of the curve y = x

2

+ 5. The point A has coordinates (1, 6). The point B

has coordinates (a, a

2

+ 5), where a is a constant greater than 1. The point C is on the curve

between A and B.

(i) Find by differentiation the value of the gradient of the curve at the point A.

(ii) The line segment joining the points A and B has gradient 2.3. Find the value of a.

(iii) State a possible value for the gradient of the line segment joining the points A and C.

2010 qu. 9

Given that f(

x) =

1

x

− x + 3

,

(i) find f′(x),

(ii) find f′′(4).

2009 qu. 1

Given that y = x

5

+

1

x

2

, find

(i)

dy

dx

,

(ii)

d

2

y

dx

2

.

2009 qu. 10

(i) Solve the equation 9x

2

+ 18x – 7 = 0.

(ii) Find the coordinates of the stationary point on the curve y = 9x

2

+ 18x – 7.

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(iii) Sketch the curve y = 9x

2

+ 18x – 7, giving the coordinates of all intercepts with the axes.

(iv) For what values of x does 9x

2

+ 18x – 7 increase as x increases?

2009 qu. 11

The point P on the curve y = k

x

has x-coordinate 4. The normal to the curve at P is parallel to

the line 2x + 3y = 0.

(i) Find the value of k.

(ii) This normal meets the x-axis at the point Q. Calculate the area of the triangle OPQ, where

O is the point (0, 0).

2009 qu. 5

Find

dx

dy

in each of the following cases:

(i) y = 10x

–5

,

(ii) y =

4

x

,

(iii) y = x(x + 3)(1 – 5x).

2009 qu. 9

The curve y = x

3

+ px

2

+ 2 has a stationary point when x = 4. Find the value of the constant p

and determine whether the stationary point is a maximum or minimum point.

2009 qu. 10

A curve has equation y = x

2

+ x.

(i) Find the gradient of the curve at the point for which x = 2.

(ii) Find the equation of the normal to the curve at the point for which x = 2, giving your

answer in the form ax + by + c = 0, where a, b and c are integers.

(iii) Find the values of k for which the line y = kx – 4 is a tangent to the curve.

2008 qu. 5

Find the gradient of the curve

y=8x+x

at the point whose x-coordinate is 9.

13.

Jan 2008 qu. 8

(i) Find the coordinates of the stationary points on the curve

y = x

3

+ x

2

− x + 3.

(ii) Determine whether each stationary point is a maximum point or a minimum point.

(iii) For what values of x does x

3

+ x

2

− x + 3 decrease as x increases?

2007 qu. 5

x metres

The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20

metres, is used to form the remaining three sides. The width of the enclosure is x metres.

(i) Show that the enclosed area, A m

2

, is given by

A = 20x – 2x

2

.

(ii) Use differentiation to find the maximum value of A.

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2007 qu. 7

Find

dy

dx

in each of the following cases.

(i) y = 5x + 3

(ii) y =

2

x

2

(iii) y = (2x + 1)(5x − 7)

16.

June 2006 qu. 1

The points A (1, 3) and B (4, 21) lie on the curve y = x

2

+ x + 1.

(i) Find the gradient of the line AB.

(ii) Find the gradient of the curve y = x

2

+ x + 1 at the point where x = 3.

206 qu. 8

A cuboid has a volume of 8 m

3

. The base of the cuboid is square with sides of length x metres.

The surface area of the cuboid is A m

2

.

(i) Show that A = 2x

2

+

32

x

.

(ii) Find

dA

dx

.

(iii) Find the value of x which gives the smallest surface area of the cuboid, justifying your

answer.

18.

Jan 2006 qu. 6

(i) Find the coordinates of the stationary points on the curve y = x

3

– 3x

2

+ 4.

(ii) Determine whether each stationary point is a maximum point or a minimum point.

(iii) For what values of x does x

3

– 3x

2

+ 4 increase as x increases?

2005 qu. 10

(i) Given that y =

1

dy

3

x

3

– 9x, find

dx

.

(ii) Find the coordinates of the stationary points on the curve y =

1

3

x

3

– 9x.

(iii) Determine whether each stationary point is a maximum point or a minimum point.

(iv) Given that 24x + 3y + 2 = 0 is the equation of the tangent to the curve at the point (p, q),

find p and q.

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