Calculus concentrated review

10 

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(1)

1. (a) Differentiate the following function with respect to x: f (x) = 2x – 9 – 25x–1

(b) Calculate the x-coordinates of the points on the curve where the gradient of the tangent to the curve is equal to 6.

(Total 6 marks)

2. (a) Differentiate the function y = x2 + 3x – 2.

(b) At a certain point (x, y) on this curve the gradient is 5. Find the co-ordinates of this point.

(Total 6 marks)

3.

f x( )

A

B

C D

E

F

G H

K L

y = ( )f x

x

Given the graph of f (x) state

(a) the intervals from A to L in which f (x) is increasing.

(1)

(b) the intervals from A to L in which f (x) is decreasing.

(1)

(c) a point that is a maximum value.

(1)

(d) a point that is a minimum value.

(e) the name given to point K where the gradient is zero.

(1) (Total 5 marks)

4. (a) Write 32 in the form 3xa where a∈ .

(2)

(b) Hence differentiate y = 32 giving your answer in the form — where c∈ +.

x xc

b

(Total 6 marks)

5. Consider the function f (x) = 32 + x – 4. x

(a) Calculate the value of f (x) when x = 1.

(2)

(b) Differentiate f (x).

(4)

(c) Find f '(l).

(2)

(d) Explain what f '(l) represents.

(2)

(e) Find the equation of the tangent to the curve f (x) at the point where x = 1.

(3)

(f) Determine the x-coordinate of the point where the gradient of the curve is zero.

(3) (Total 16 marks)

6. The diagram shows a part of the curve y = f(x).

2

1

–1

–2

–3 –2 –1 0 1 2 3 4 5 6 7 8 x

y

(a) For what values of x is f '(x) = 0?

(3)

(b) Solve g(x) = 0.

(2)

(c) (i) Calculate the values of g'(x) when

(a) x = –3;

(b) x = 0.

(ii) Hence state whether the function is increasing or decreasing at

(a) x = –3;

(b) x = 0.

(4) (Total 9 marks)

8. The height (cm) of a daffodil above the ground is given by the function h (w) = 24w – 2.4w2, where w is the time in weeks after the plant has broken through the surface (w≥ 0).

(a) Calculate the height of the daffodil after two weeks.

(2)

(b) (i) Find the rate of growth, .

w h d

d

(2)

(ii) The rate of growth when w = k is 7.2 cm per week. Find k.

(3)

(iii) When will the daffodil reach its maximum height? What height will it reach?

(4)

(c) Once the daffodil has reached its maximum height, it begins to fall back towards the ground. Show that it will touch the ground after 70 days.

(4)

9.

f x( )

A

B

C D

E

F

G H

K L

y = ( )f x

x

Given the graph of f (x) state

(a) the intervals from A to L in which f (x) is increasing.

(1)

(b) the intervals from A to L in which f (x) is decreasing.

(1)

(c) a point that is a maximum value.

(1)

(d) a point that is a minimum value.

(e) the name given to point K where the gradient is zero.

(1) (Total 5 marks)

10. A, B, C, D and E are points on the curve y = f(x) shown in the diagram below.

y

B

D

E

(5)

(b) D has coordinates (a, f(a)), and the x-coordinate at E is a + 4. Write an expression for the gradient of the line segment [DE].

(3) (Total 6 marks)

11. Consider the function f (x) = 2x3 – 3x2 – 12x + 5. (a) (i) Find f ' (x).

(ii) Find the gradient of the curve f (x) when x = 3.

(4)

(b) Find the x-coordinates of the points on the curve where the gradient is equal to –12.

(3)

(c) (i) Calculate the x-coordinates of the local maximum and minimum points.

(ii) Hence find the coordinates of the local minimum.

(6)

(d) For what values of x is the value of f (x) increasing?

(2) (Total 15 marks)

12. The function g is defined as follows

g : xpx2 + qx + c, p, q, c∈ (a) Find g(x)

(1)

(b) If g(x) = 2x + 6, find the values of p and q.

(2)

(c) g(x) has a minimum value of –12 at the point A. Find

(i) the x-coordinate of A;

(ii) the value of c.

(6)

13. A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card as shown in the diagram below. What is left is then folded to make an open box, of length l cm and width w cm.

x x

9 cm

24 cm

(a) Write expressions, in terms of x, for

(i) the length, l;

(ii) the width, w.

(2)

(b) Show that the volume (B m3) of the box is given by B = 4x3 – 66x2 + 216x.

(1)

(c) Find . x B d d

(1)

(d) (i) Find the value of x which gives the maximum volume of the box.

(ii) Calculate the maximum volume of the box.

(4) (Total 8 marks)

14. The velocity, vms–1, of a kite, after t seconds, is given by v = t3 – 4t2 + 4t, 0≤t≤ 4. (a) What is the velocity of the kite after

(i) one second?

(ii) half a second?

(7)

(c) (i) Find in terms of t. Find the value of t at the local maximum and minimum t

v d d

values of the function.

(ii) Explain what is happening to the function at its local maximum point. Write down the gradient of the tangent to its curve at this point.

(8)

(d) On graph paper, draw the graph of the function v = t3 – 4t2 + 4t, 0t≤ 4.

Use a scale of 2 cm to represent 1 second on the horizontal axis and 2 cm to represent 2 ms–1 on the vertical axis.

(5)

(e) Describe the motion of the kite at different times during the first 4 seconds. Write down the intervals corresponding to changes in motion.

(3)

The acceleration, a, of a second kite is given by the equation a = 4t – 3, 0≤t≤ 4.

(f) After 1 second, the velocity, u, of this kite, is 3 ms–1. Write the equation for u in terms of t.

(4) (Total 24 marks)

15. The diagram below is a part of the graph of the function y = 2x2. P is the point (1, 2) on the curve and Q is a neighbouring point.

y

x

O M N

(1, 2)P R Q

y = 2x2

(a) Copy and complete the following table

PR 0.1 0.01 0.001

QN

QR

Gradient of PQ

(8)

(b) Show how you can use your table to

(i) predict the gradient of the curve y = 2x2 at the point (1,2);

(1)

(ii) deduce the gradient of the curve y = 2x2 + 3 at the point (1,5).

(3) (Total 8 marks)

16. A farmer wishes to enclose a rectangular field using an existing fence for one of the four sides. Existing fence

x

y

(a) Write an expression in terms of x and y that shows the total length of the new fence.

(1)

(b) The farmer has enough materials for 2500 metres of new fence. Show that

y = 2500 – 2x

(1)

(c) A(x) represents the area of the field in terms of x.

(i) Show that

A(x) = 2500x – 2x2

(2)

(ii) Find A'(x).

(1)

(iii) Hence or otherwise find the value of x that produces the maximum area of the field. (3)

(iv) Find the maximum area of the field.

(9)

17. A cylinder is cut from a solid wooden sphere of radius 8 cm as shown in the diagram. The height of the cylinder is 2h cm.

D C

O

A B

E

h h

8cm

8 cm

(a) Find AE (the radius of the cylinder), in terms of h.

(2)

(b) Show that the volume (V) of the cylinder may be written as

V= 2πh(64 –h2) cm3.

(2)

(c) (i) Determine, correct to three significant figures, the height of the cylinder with the greatest volume that can be produced in this way.

(5)

(ii) Calculate this greatest volume, giving your answer correct to the nearest cm3.

(3) (Total 12 marks)

18. The letters A to E are placed at particular points on the curve y = f(x).

y

x

A B

C

D

E

(10)

(a) What is the gradient of the curve y =f(x) at the point marked C?

(1)

(b) In passing from point B, through point C, to point D what is happening to ? Is it

x y d d

decreasing or increasing?

(2) (Total 3 marks)

19. The function f (x) is given by f (x) = x3 – 3x2 + 3x, for –1 ≤x ≤ 3. (a) Differentiate f (x) with respect to x.

(2)

(b) Copy and complete the table below.

x –1 0 1 2 3

f (x) 0 1 2 9

f'(x) 12 0 12

(3)

(c) Use the information in your table to sketch the graph of f (x).

(2)

(d) Write down the gradient of the tangent to the curve at the point (3, 9).

(1) (Total 8 marks)

20. The perimeter of a rectangle is 24 metres.

(a) The table shows some of the possible dimensions of the rectangle. Find the values of a, b, c, d and e.

Length (m) Width (m) Area (m2)

1 11 11

a 10 b

Figure

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References

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