Topic 2 Part 5
[140 marks]
Let
f(x) = ln(x + 5) + ln 2 , for
x > −5 .
1a. Find
[4 marks]
1b. Let
[3 marks]
f −1 (x) .
g(x) = ex .
Find
(g ∘ f)(x) , giving your answer in the form
ax + b , where
a,b ∈ Z .
Let
f(x) = 3(x + 1)2 − 12 .
2a. Show that
[2 marks]
2b. For the graph of f
[8 marks]
f(x) = 3x2 + 6x − 9 .
(i)
write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the y-intercept;
(iv) find both x-intercepts.
2c. Hence sketch the graph of f .
[2 marks]
2d. Let
[3 marks]
g(x) = x2 . The graph of f may be obtained from the graph of g by the two transformations:
a stretch of scale factor t in the y-direction
followed by a translation of
p
( ).
q
Find
p
( ) and the value of t.
q
Let
f(x) = 4tan2 x − 4 sin x ,
− π3 ≤ x ≤ π3 .
3a. On the grid below, sketch the graph of
[3 marks]
3b. Solve the equation
[3 marks]
y = f(x) .
f(x) = 1 .
The following diagram shows the graphs of
f(x) = ln(3x − 2) + 1 and
g(x) = −4 cos(0.5x) + 2 , for
1 ≤ x ≤ 10 .
4a. Let A be the area of the region enclosed by the curves of f and g.
(i)
[6 marks]
Find an expression for A.
(ii) Calculate the value of A.
4b.
(i) Find
f ′ (x) .
(ii) Find
g′ (x) .
[4 marks]
4c. There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.
[4 marks]
The following diagram shows part of the graph of f , where
f(x) = x2 − x − 2 .
5a. Find both x-intercepts.
5b.
Find the x-coordinate of the vertex.
[4 marks]
[2 marks]
Part of the graph of a function f is shown in the diagram below.
6a. On the same diagram sketch the graph of
y = −f(x) .
[2 marks]
6b.
Let
g(x) = f(x + 3) .
[4 marks]
(i) Find
g(−3) .
(ii) Describe fully the transformation that maps the graph of f to the graph of g.
Let
f(x) = ex (1 − x2 ) .
7a. Show that
[3 marks]
f ′ (x) = ex (1 − 2x − x2 ) .
Part of the graph of
y = f(x), for
−6 ≤ x ≤ 2 , is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.
7b. Write down the equation of the horizontal asymptote.
7c.
7d.
7e.
[1 mark]
Write down the value of r and of s.
[4 marks]
Let L be the normal to the curve of f at
P(0, 1) . Show that L has equation
x+y = 1 .
[4 marks]
Let R be the region enclosed by the curve
y = f(x) and the line L.
[5 marks]
(i)
Find an expression for the area of R.
(ii) Calculate the area of R.
A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were
280 taxis in the city. After n years the number of taxis, T, in the city is given by
T = 280 × 1.12n .
8a.
(i)
Find the number of taxis in the city at the end of 2005.
(ii) Find the year in which the number of taxis is double the number of taxis there were at the end of 2000.
[6 marks]
8b.
At the end of 2000 there were
25600 people in the city who used taxis.
[6 marks]
After n years the number of people, P, in the city who used taxis is given by
P=
(i)
2560000
.
10 + 90e−0.1n
Find the value of P at the end of 2005, giving your answer to the nearest whole number.
(ii) After seven complete years, will the value of P be double its value at the end of 2000? Justify your answer.
8c.
Let R be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis [5 marks]
if
R < 70 .
(i)
Find the value of R at the end of 2000.
(ii) After how many complete years will the city first reduce the number of taxis?
Let
f(x) = 3(x + 1)2 − 12 .
9a.
9b.
Show that
f(x) = 3x2 + 6x − 9 .
[2 marks]
For the graph of f
[7 marks]
(i)
write down the coordinates of the vertex;
(ii) write down the y-intercept;
(iii) find both x-intercepts.
9c. Hence sketch the graph of f .
9d.
Let
g(x) = x2 . The graph of f may be obtained from the graph of g by the following two transformations
[3 marks]
[3 marks]
a stretch of scale factor t in the y-direction,
followed by a translation of
p
( ).
q
Write down
p
( ) and the value of t .
q
Let
f(x) = 4x − ex−2 − 3 , for
0≤x≤5.
10a.
Find the x-intercepts of the graph of f .
[3 marks]
10b.
10c.
On the grid below, sketch the graph of f .
Write down the gradient of the graph of f at
x=3.
[3 marks]
[1 mark]
Let
h(x) = 2x−1
,
x+1
x ≠ −1 .
11a. Find
[4 marks]
h−1 (x) .
(i) Sketch the graph of h for
−4 ≤ x ≤ 4 and
−5 ≤ y ≤ 8 , including any asymptotes.
11b.
[7 marks]
(ii) Write down the equations of the asymptotes.
(iii) Write down the x-intercept of the graph of h .
11c.
Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line
x = 3.
(i)
Find the area of R.
(ii) Write down an expression for the volume obtained when R is revolved through
360∘ about the x-axis.
[5 marks]
A rock falls off the top of a cliff. Let
h be its height above ground in metres, after
t seconds.
The table below gives values of
h and
t.
12a.
Jane thinks that the function
f(t) = −0.25t3 − 2.32t2 + 1.93t + 106 is a suitable model for the data. Use Jane’s model to
(i)
[5 marks]
write down the height of the cliff;
(ii) find the height of the rock after 4.5 seconds;
(iii) find after how many seconds the height of the rock is
30 m.
12b. Kevin thinks that the function
[3 marks]
12c. (i)
[6 marks]
g(t) = −5.2t2 + 9.5t + 100 is a better model for the data. Use Kevin’s model to find when the rock hits the ground.
On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, plot the data given in the table.
(ii) By comparing the graphs of f and g with the plotted data, explain which function is a better model for the height of the falling rock.
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