REVIEW - For Integration Test (2012)

8/2/2019 REVIEW - For Integration Test (2012)

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1

1. A particle moves along a straight line so that its velocity, v ms1

at time tseconds is given by v =

6e3t

+ 4. When t= 0, the displacement, s, of the particle is 7 metres. Find an expression for s in

terms oft.(Total 7 marks)

2. (Calc) Consider the functionf(x) = e(2x1) + ,x .

(a) Sketch the curve offfor 2 x 2, including any asymptotes.(3)

(b) (i) Write down the equation of the vertical asymptote off.

(ii) Write down which one of the following expressions does not represent an area

between the curve offand thex-axis.

f(x)dx

f(x)dx

(iii) Justify your answer.(3)

(c) The region between the curve and thex-axis betweenx = 1 andx = 1.5 is rotated through

360

about thex-axis. Let Vbe the volume formed.

(i) Write down an expression to represent V.

(ii) Hence write down the value ofV.(4)

(d) Findf (x).(4)

(e) (i) Write down the value ofx at the minimum point on the curve off.

(ii) The equationf(x) = khas no solutions forpkq. Write down the value ofp and of

q.(3)

(Total 17 marks)

4. (calc) Letf(x) = , wherep, q+.

Part of the graph off, including the asymptotes, is shown below.

125

x

21

2

1

2

0

22

3

qx

xp


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2

(a) The equations of the asymptotes arex =1,x= 1,y = 2. Write down the value of

(i) p;

(ii) q.(2)

(b) LetR be the region bounded by the graph off, thex-axis, and they-axis.

(i) Find the negativex-intercept off.

(ii) Hence find the volume obtained whenR is revolved through 360 about thex-axis.(7)

(c) (i) Show thatf (x) = .

(ii) Hence, show that there are no maximum or minimum points on the graph off.(8)

(d) Let g (x) =f (x). LetA be the area of the region enclosed by the graph of g and thex-axis,

betweenx = 0 andx = a, where a 0. Given thatA = 2, find the value ofa.(7)

(Total 24 marks)

5. (calc) The functionf(x) is defined asf(x) = 3 + ,x .

(a) Sketch the curve offfor 5 x 5, showing the asymptotes.(3)

(b) Using your sketch, write down

(i) the equation of each asymptote;

(ii) the value of thex-intercept;

(iii) the value of the y-intercept.

22

2

1

13

x

x

52

1

x

2

5


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3

(4)

(c) The region enclosed by the curve off, thex-axis, and the linesx = 3 andx = a, is revolved

through 360 about thex-axis. Let Vbe the volume of the solid formed.

(i) Find dx.

(ii) Hence, given that V= , find the value ofa.

(10)

(Total 17 marks)

8. The velocity v of a particle at time tis given by v = e2t

+ 12t. The displacement of the particle at

time tis s. Given that s = 2 when t= 0, express s in terms oft.(Total 6 marks)

11. Consider the functionsfand g wheref(x) = 3x5 and g (x) =x2.

(a) Find the inverse function,f1

.(3)

(b) Given that g1

(x) =x + 2, find (g1

f) (x).(2)

(c) Given also that (f1

g) (x) , solve (f1 g) (x) = (g

1f) (x).

(2)

Let h (x) = ,x 2.

(d) (i) Sketch the graph ofhfor 3 x 7 and 2 y 8, including any asymptotes.

(ii) Write down the equations of the asymptotes.(5)

(e) The expression may also be written as 3 + . Use this to answer the following.

(i) Find dx.

(ii) Hence, calculate the exact value of h (x)dx.

(5)

(f) On your sketch, shade the region whose area is represented by h (x)dx.

(1)

(Total 18 marks)

2

52

1

52

69

xx

3ln3

3

28

3

3x

)()(

xgxf

3

53

x

x

2

1

x

)(xh

5

3

5

3


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12. Letf(x) = x2

+x + 4.

(a) (i) Write downf (x).

(ii) Find the equation of the normal to the curve offat (2, 3).

(iii) This normal intersects the curve offat (2, 3) and at one other point P.

Find thex-coordinate of P.(9)

Part of the graph offis given below.

(b) LetR be the region under the curve offfromx= 1 tox = 2.

(i) Write down an expression for the area ofR.

(ii) Calculate this area.

(iii) The regionR is revolved through 360 about thex-axis. Write down an expression

for the volume of the solid formed.(6)

(c) Find giving your answer in terms ofk.

(6)

(Total 21 marks)

14. The graph ofy = sin 2x from 0x is shown below.

The area of the shaded region is 0.85. Find the value ofk.

(Total 6 marks)

4

3

k

xxf1

,d)(


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15. A particle moves with a velocity v m s1

given by v = 25 4t2

where t 0.

(a) The displacement, s metres, is 10 when tis 3. Find an expression for s in terms oft.(6)

(b) Find twhen s reaches its maximum value.(3)

(c) The particle has a positive displacement for mtn. Find the value ofm and the value

ofn.(3)

(Total 12 marks)

17. The curvey =f(x) passes through the point (2, 6).

Given that = 3x25, findy in terms ofx.

(Total 6 marks

18. Letf(x) = .

(a) Write down the equation of the horizontal asymptote of the graph off.(1)

(b) Findf (x).(3)

(c) The second derivative is given byf(x) = .

Let A be the point on the curve offwhere the gradient of the tangent is a maximum. Find

thex-coordinate of A.(4)

(d) LetR be the region under the graph off, betweenx = andx = ,

as shaded in the diagram below

Write down the definite integral which represents the area ofR.(2)

(Total 10 marks)

x

y

d

d

21

1

x

32

2

)(1

26

x

x

2

1

2

1

R

1

2

1

1

112

1

2

x

y


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19. (calc) Let h (x) = (x2) sin (x1) for5 x 5. The curve ofh (x) is shown below. There isa minimum point at R and a maximum point at S. The curve intersects thex-axis at the points (a, 0) (1, 0)

(2, 0) and (b, 0).

(a) Find the exact value of

(i) a;

(ii) b.(2)

The regions between the curve and thex-axis are shaded for ax 2 as shown.

(b) (i) Write down an expression which represents the total area of the shaded regions.

(ii) Calculate this total area.(5)

(c) (i) They-coordinate of R is0.240. Find they-coordinate of S.

(ii) Hence or otherwise, find the range of values ofkfor which the equation

(x2) sin (x1) = khas four distinct solutions.(4)

(Total 11 marks)

20. (calc) The diagram below shows a sketch of the graph of the functiony = sin (ex

) where1 x

2, andx is in radians. The graph cuts they-axis at A, and thex-axis at C and D. It has a

maximum point at B.

5 4 3 2 1 1 2 3 4 5 x

y

S

( , 0)b( , 0)a

R

4

3

2

1

1

2

3

4

5

6

7


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(a) Find the coordinates of A.(2)

(b) The coordinates of C may be written as (ln k, 0). Find the exact value ofk.(2)

(c) (i) Write down they-coordinate of B.

(ii) Find .

(iii) Hence, show that at B,x = ln .

(6)

(d) (i) Write down the integral which represents the shaded area.

(ii) Evaluate this integral.(5)

(e) (i) Copy the above diagram into your answer booklet. (There is no need to copy the

shading.) On your diagram, sketch the graph ofy =x3.

(ii) The two graphs intersect at the point P. Find thex-coordinate of P.(3)

(Total 18 marks)

22. Consider functions of the formy = ekx

(a) Show that = (1ek).

(3)

(b) Let k = 0.5

(i) Sketch the graph ofy = e0.5x

, for1 x 3, indicating the coordinates of they-intercept.

(ii) Shade the region enclosed by this graph, thex-axis,y-axis and the linex = 1.

(iii) Find the area of this region.(5)

(c) (i) Find in terms ofk, wherey = ekx

.

1 0 1 2

A

B

C D

y

x

x

y

d

d

2

1

0

de x

kx

k

1

x

y

d

d


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The point P(1, 0.8) lies on the graph of the functiony = ekx

.

(ii) Find the value ofkin this case.

(iii) Find the gradient of the tangent to the curve at P.(5)

(Total 13 marks)

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REVIEW - For Integration Test (2012)