IB Questionbank Mathematical Studies 3rd edition1

1.Consider the function f(x) = x3 +

x

48

, x 0.

(a)Calculate f(2).

(2)

(b)Sketch the graph of the function y = f(x) for 5 x 5 and 200 y 200.

(4)

(c)Find f(x).

(3)

(d)Find f(2).

(2)

(e)Write down the coordinates of the local maximum point on the graph of f.

(2)

(f)Find the range of f.

(3)

(g)Find the gradient of the tangent to the graph of f at x = 1.

(2)

There is a second point on the graph of f at which the tangent is parallel to the tangent at x = 1.

(h)Find the x-coordinate of this point.

(2)

(Total 20 marks)

2.The function f(x) is defined by f(x) = 1.5x + 4 +

x

6

, x 0.

(a)Write down the equation of the vertical asymptote.

(2)

(b)Find f(x).

(3)

(c)Find the gradient of the graph of the function at x = 1.

(2)

(d)Using your answer to part (c), decide whether the function f(x) is increasing or decreasing at x = 1. Justify your answer.

(2)

(e)Sketch the graph of f(x) for 10 x 10 and 20 y 20.

(4)

P1 is the local maximum point and P2 is the local minimum point on the graph of f(x).

(f)Using your graphic display calculator, write down the coordinates of

(i)P1;

(ii)P2.

(4)

(g)Using your sketch from (e), determine the range of the function f(x) for 10 x 10.

(3)

(Total 20 marks)

3.Consider the function f(x) = x3 3x2 24x + 30.

(a)Write down f(0).

(1)

(b)Find f(x).

(3)

(c)Find the gradient of the graph of f(x) at the point where x = 1.

(2)

The graph of f(x) has a local maximum point, M, and a local minimum point, N.

(d)(i)Use f(x) to find the x-coordinate of M and of N.

(ii)Hence or otherwise write down the coordinates of M and of N.

(5)

(e)Sketch the graph of f(x) for 5 x 7 and 60 y 60. Mark clearly M and N on your graph.

(4)

Lines L1 and L2 are parallel, and they are tangents to the graph of f(x) at points A and B respectively. L1 has equation y = 21x + 111.

(f)(i)Find the x-coordinate of A and of B.

(ii)Find the y-coordinate of B.

(6)

(Total 21 marks)

4.(a)Sketch the graph of the function f (x) =

4

3

2

+

+

x

x

, for 10 x 10. Indicating clearly the axis intercepts and any asymptotes.

(6)

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

(2)

(c)On the same diagram sketch the graph of g (x) = x + 0.5.

(2)

(d)Using your graphical display calculator write down the coordinates of one of the points of intersection on the graphs of f and g, giving your answer correct to five decimal places.

(3)

(e)Write down the gradient of the line g (x) = x + 0.5.

(1)

(f)The line L passes through the point with coordinates (2, 3) and is perpendicular to the line g (x). Find the equation of L.

(3)

(Total 17 marks)

5.The following curves are sketches of the graphs of the functions given below, but in a different order. Using your graphic display calculator, match the equations to the curves, writing your answers in the table below.

diagrams not to scale

A

B

C

D

E

F

0

0

0

0

0

0

y

x

y

x

y

x

y

x

y

x

y

x

Function

Graph label

(i)

y = x3 + 1

(ii)

y = x2 + 3

(iii)

y = 4 x2

(iv)

y = 2x + 1

(v)

y = 3x + 1

(vi)

y = 8x 2x2 x3

(Total 6 marks)

6.(a)Sketch the graph of y = 3 +

1

3

-

x

for 10 x 10.

(b)Write down the equations of

(i)the horizontal asymptote;

(ii)the vertical asymptote.

(Total 6 marks)