### Revision Tri 2 Test 1 Calculus No Calculator

```Name____________________________________________________
Calculus Revision: No Calculators Allowed
1.
Let g(x) =
(a)
ln x
x2
, for x > 0.
Use the quotient rule to show that g ʹ′( x) =
1 − 2 ln x
x3
.
(4)
(b)
The graph of g has a maximum point at A. Find the x-coordinate of A.
(3)
(Total 7 marks)
2.
Consider the function f with second derivative f′′(x) = 3x – 1. The graph of f has a minimum
⎛ 4 358 ⎞
point at A(2, 4) and a maximum point at B ⎜ − ,
⎟ .
⎝ 3 27 ⎠
(a)
Use the second derivative to justify that B is a maximum.
(3)
(b)
Given that f′ =
3 2
x – x + p, show that p = –4.
2
(4)
(c)
Find f(x).
(7)
(Total 14 marks)
3.
Let g(x) = 2x sin x.
(a)
Find g′(x).
(4)
(b)
Find the gradient of the graph of g at x = π.
(3)
(Total 7 marks)
IB Questionbank Maths SL
1
4.
⎛ 3 ⎞
The graph of the function y = f(x) passes through the point ⎜ , 4 ⎟ . The gradient function of f is
⎝ 2 ⎠
given as f′(x) = sin (2x – 3). Find f(x).
(Total 6 marks)
5.
Let f(x) =
ax
2
x +1
, –8 ≤ x ≤ 8, a ∈
. The graph of f is shown below.
The region between x = 3 and x = 7 is shaded.
(a)
Show that f(–x) = –f(x).
(2)
IB Questionbank Maths SL
2
(b)
Given that f′′(x) =
2ax( x 2 − 3)
( x 2 + 1) 3
, find the coordinates of all points of inflexion.
(7)
(c)
6.
It is given that
a
∫ f ( x)dx = 2 ln( x
2
+ 1) + C .
(i)
(ii)
Find the value of
8
∫ 2 f ( x − 1)dx .
4
(7)
(Total 16 marks)
1 3
x + 2x2 – 5x. Part of the graph of f is shown below. There is a maximum
3
point at M, and a point of inflexion at N.
Consider f (x) =
(a)
Find f ′(x).
(3)
(b)
Find the x-coordinate of M.
(4)
(c)
Find the x-coordinate of N.
(3)
(d)
The line L is the tangent to the curve of f at (3, 12). Find the equation of L in the form
y = ax + b.
(4)
(Total 14 marks)
IB Questionbank Maths SL
3
7.
(a)
(b)
Find
Find
∫
2
1
1
(3 x 2 − 2) dx.
∫ 2e
(4)
2x
0
IB Questionbank Maths SL
dx .
(3)
(Total 7 marks)
4
8.
On the axes below, sketch a curve y = f (x) which satisfies the following conditions.
x
f (x)
−2 ≤ x < 0
0
–1
0 < x <1
1
1<x≤2
2
f ′ (x)
f ′′ (x)
negative
positive
0
positive
positive
positive
positive
0
positive
negative
(Total 6 marks)
IB Questionbank Maths SL
5
9.
10.
The function f is given by f (x) = 2sin (5x – 3).
(a)
Find f " (x).
(b)
Write down
∫ f ( x ) dx .
(Total 6 marks)
If f ʹ′(x) = cos x, and f ⎛⎜ π ⎞⎟ = – 2, find f (x).
⎝ 2 ⎠
Working:
......................................................................
(Total 4 marks)
IB Questionbank Maths SL
6
11.
The diagram below shows the shaded region R enclosed by the graph of y = 2x 1 + x 2 , the
x-axis, and the vertical line x = k.
y
y = 2x 1+x 2
R
x
k
(a)
Find
dy
.
dx
(3)
(b)
Using the substitution u = 1 + x2 or otherwise, show that
3
∫
2
2 x 1 + x dx = (1 + x2) 2 + c.
3
2
(3)
(c)
Given that the area of R equals 1, find the value of k.
(3)
(Total 9 marks)
12.
x
The diagram shows part of the graph of y = e 2 .
y
x
y = e2
P
ln2
IB Questionbank Maths SL
x
7
(a)
Find the coordinates of the point P, where the graph meets the y-axis.
(2)
The shaded region between the graph and the x-axis, bounded by x = 0
and x = ln 2, is rotated through 360° about the x-axis.
(b)
Write down an integral which represents the volume of the solid obtained.
(4)
(c)
Show that this volume is π.
IB Questionbank Maths SL
(5)
(Total 11 marks)
8
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