IB SL calculus exam practice

1.

2

 x

Let f  Let f ( x)  x) = cos( x ) and g  and g ( x)  x) = e , for –1.5 ≤ x ≤ 0.5. Find the area of the region enclosed by the graphs of  f and g . (Total 6 marks)

2.

The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) (counterclockwise) direction.

diagram not to scale

The diameter of the wheel is 8 metres. The centre of the wheel, A, A, is 2 metres above the water level. After t  seconds,  seconds, the height of the bucket above the water level is given by h = a sin bt     2.

(a)

!how that a = ". (2)

The wheel turns at a rate of one rotation every #$ seconds. 

(b)

!how that b = 15 . (2)

%n the first rotation, there are two values of t  when  when the bucket is descending at a rate of   –1 $.& m s . (c) (c)

'ind 'ind the these valu value es of t . (6)

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(d)

eter etermi mine ne whet whether her the the bucke buckett is under underwa water ter at the the secon second d value value of t . (4) (Total 14 marks)

d y

3.

 A gradient gradient function is given by d x when  x = *.

= 10e 2 x − 5

. hen x hen x = $, y  =  = 8. 'ind the value of y 

(Total 8 marks)

4.

+et f  +et f ( x   x ) = x  = x  cos x   cos x , for $ ≤ x ≤ !. (a)

'ind f "(  x ). ). "( x  (3)

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2

 

(b)

n the grid below, sketch the gra-h of y = f "( x ).

(4) (Total 7 marks)

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3

5.

 x 

+et f ( x ) = e sin 2 x   *$, for $ ≤ x ≤ #. $art of the graph of  f is given below.

There is an x interce-t at the -oint A, a local ma/imum -oint at 0, where x = p and a local minimum -oint at 1, where x = q.

(a)

rite down the x coordinate of A. (1)

(b)

'ind the value of  (i)

 p

(ii)

q. (2)

q

(c)

'ind

∫  f ( x)d x . 3/-lain why this is not the area of the shaded region.  p

(3) (Total 6 marks)

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6.

2

4onsider  f ( x ) = x ln(" – x  ), for  –2 % x  5 2. The gra-h of  f is given below.

(a)

+et 6 and 7 be -oints on the curve of  f where the tangent to the gra-h of  f is -arallel to the x a/is. (i)

'ind the x coordinate of 6 and of 7.

(ii)

4onsider f    ( x ) = k . rite down all values of k  for which there are e/actly two solutions. (5)

#

2

+et g ( x ) = x   ln("  – x  ), for –2 % x  5 2.

(b)

!how that g "( x ) =

− 2 x # + & x 2 ln(# − x 2 ) 2 # −  x . (4)

(c)

!ketch the gra-h of g ". (2)

(d)

4onsider g "( x ) = w . rite down all values of w  for which there are e/actly two solutions. (3) (Total 14 marks)

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7.

 –1 The velocity v  m s  of an obect after t  seconds is given by v (t ) = 15 t  − &t  , for $ ≤ t ≤ 25.

(a)

n the grid below, sketch the gra-h of v , clearly indicating the ma/imum -oint.

(3)

+et   be the distance travelled in the first nine seconds. (b)

(i)

rite down an e/-ression for  .

(ii)

9ence, write down the value of  . (4) (Total 7 marks)

8.

#

2

+et f "( x ) = –2# x    : x    # x   *. (a)

There are two -oints of infle/ion on the gra-h of f . rite down the x coordinates of these -oints. (3)

(b)

+et g ( x ) = f '( x ). 3/-lain why the gra-h of  g has no -oints of infle/ion. (2) (Total 5 marks)

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9.

2

+et f ( x ) = x  ln("  – x  ), for –2 % x  5 2. The gra-h of  f is shown below.

The gra-h of  f crosses the x a/is at x = a, x = $ and x = b.

(a)

'ind the value of a and of b. (3)

The gra-h of  f has a ma/imum value when x = " . (b)

'ind the value of " . (2)

(c)

The region under the gra-h of  f from x = $ to x = "  is rotated #;$ abot the x a/is. 'ind the volume of the solid formed. (3)

(d)

+et #  be the region enclosed by the curve, the x a/is and the line x = " , between  x = a and x = " . 'ind the area of # . (4) (Total 12 marks)

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10.

2

+et f ( x ) = x ( x  – 5) , for $ ≤ x ≤ !. *he follo+ing diagra sho+s the graph of f .

+et #  be the region enclosed by the x a/is and the curve of f .

(a)

'ind the area of # . (3)

(b)

'ind the volume of the solid formed when #  is rotated through #;$  abot the x  a/is. (4)

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(c)

The diagram below shows a -art of the gra-h of a