Calculus Non calc revision
1.
By using an appropriate substitution find
 tan y d y , y

0 .
(Total 6 marks)
2.
Find the gradient of the tangent to the curve x
3 y
2
= cos (πy) at the point (−1, 1).
(Total 12 marks)
3.
A normal to the graph of y = arctan ( x − 1), for x

0, has equation y = −2 x + c , where c

.
Find the value of c .
(Total 6 marks)
4.
Show that

0

6 x sin 2 x d x

8
3


24
.
(Total 6 marks)
6.
A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its volume increases at a rate proportional to its radius.
(a) Show that the radius r cm of the soufflé, at time t minutes after it has been put in the oven, satisfies the differential equation d r d t
 k r
, where k is a constant.
(5)
(b)
Given that the radius of the soufflé is 8 cm when it goes in the oven, and 12 cm when it’s cooked 30 minutes later, find, to the nearest cm, its radius after 15 minutes in the oven.
(8)
(Total 13 marks)
7.
The region bounded by the curve y = ln x through 2

radians about the x -axis.
and the lines x = 1, x = e, y = 0 is rotated
Find the volume of the solid generated.
(Total 12 marks)
9.
The acceleration in m s
–2
of a particle moving in a straight line at time t seconds, t > 0, is given by the formula a =

1
(1
 t )
2
.
When t =1, the velocity is 8 m s
–1
.
(a) Find the velocity when t = 3.
(6)

(b) Find the limit of the velocity as t
 
.
(1)
(c) Find the exact distance travelled between t =1 and t = 3.
(4)
(Total 11 marks)
10.
Solve the differential equation d y d x

1

1
 y x
2
2
, given that y = 3 when x =
3
3
.
Give your answer in the form y = ax
 a
 x a a
where a
 +
.
(Total 6 marks)
11.
Find

0 a arcsin x d x , 0
 a

1 .
(Total 6 marks)
12.
Given that y = e
 x
2
find
(a) d
2 y d x
2
;
(3)
(b) the exact values of the x -coordinates of the points of inflexion on the graph of y = e
 x
2
, justifying that they are points of inflexion.
(3)
(Total 6 marks)
13.
The diagram below shows the shaded region A which is bounded by the axes and part of the curve y
2
= 8 a (2 a − x ), a

0. Find in terms of a the volume of the solid formed when A is rotated through 360

around the x -axis.
14.
(a) Use integration by parts to show that
(Total 6 marks)

 sin x cos x e
 sin x dx
  e
 sin x
(1
 sin x )

C .
(4) d y
Consider the differential equation – y cos x = sin x cos x . d x
(b) Find an integrating factor.
(3)
(c) Solve the differential equation, given that y = − 2 when x = 0. Give your answer in the form y = f ( x ).
(9)
(Total 16 marks)
15.
Car A is travelling on a straight east-west road in a westerly direction at 60 km h
−1
. Car B is travelling on a straight north-south road in a northerly direction at 70 km h
−1
. The roads intersect at the point O. When Car A is x km east of O, and Car B is y km south of O, the distance between the cars is z km.
Find the rate of change of z when Car A is 0.8 km east of O and Car B is 0.6 km south of O.
(Total 6 marks)
16.
Find

0 ln3 e
2 x e x

9
d x , expressing your answer in exact form. (Total 6 marks)
18.
The graph of y = sin (3 x ) for 0
 x

π
4
is is rotated through 2

radians about the x -axis.
Find the exact volume of the solid of revolution formed.
(Total 6 marks)
20.
The volume of a solid is given by
V =
4
3
π r
3  π r
2 h .
At the time when the radius is 3 cm, the volume is 1

cm
3
, the radius is changing at a rate of

2 cm/min and the volume is changing at a rate of 204

cm
3
/min. Find the rate of change of the height at this time.
(Total 6 marks)
21.
The following table shows the values of two functions f and g and their first derivatives when x =1 and x = 0.
x
0
1 f ( x
4
–2
) f ′ (
1
3 x ) g ( x
–4
–1
) g
′ ( x )
5
2
(a) Find the derivative of
3 f ( x ) g ( x )

1
when x = 0.
(b) Find the derivative of f ( g ( x ) + 2 x ) when x =1.
(Total 6 marks)
22.
Solve the differential equation ( x
2
+ 1) d d y x
– xy = 0 where x

0, y

0, given that y =1 when x = 1. (Total 6 marks)
23.
Given that e xy
− y
2
ln x = e for x
 d y
1, find at the point (1, 1).
d x
(Total 6 marks)
24.
Let f ( x ) = x ln x − x , x

0.
(a) Find f
′ ( x ).
(b) Using integration by parts find

(ln
2 x ) d x .
(Total 6 marks)
25.
The function f is defined by f ( x ) = ln x
3 x
, x

1.
(a) Find f
′( x ) and f
′′( x ), simplifying your answers.
(6)
(b) (i) Find the exact value of the x -coordinate of the maximum point and justify that this is a maximum.
(ii) Solve f
′′( x ) = 0, and show that at this value of x , there is a point of inflexion on the graph of f .
(iii) Sketch the graph of f , indicating the maximum point and the point of inflexion.
(11)

The region enclosed by the x -axis, the graph of f and the line x = 3 is denoted by R .
(c) Find the volume of the solid of revolution obtained when R is rotated through 360
 about the x -axis.
(3)
1
(d) Show that the area of R is (4 – ln 3).
18
(6)
(Total 26 marks)
26.
Solve the differential equation
( x + 2)
2 d y
= 4 xy ( x d x

–2) given that y =1 when x = −1.
(Total 6 marks)

1
27.
Let f be the function defined for x > by f ( x ) = ln (3 x + 1).
3
(a) Find f
′( x ).
(b) Find the equation of the normal to the curve y = f ( x ) at the point where x = 2.
Give your answer in the form y = ax + b where a , b

.
(Total 6 marks)
28.
Let f ( x ) = cos
3
(4 x + 1), 0
 x

1.
(a) Find f ′ ( x ).
(b) Find the exact values of the three roots of f ′ ( x ) = 0.
(Total 6 marks)
29.
Find
 e 2 x
sin x d x .
(Total 6 marks)
30.
Let f ( x ) = 3 x
2
– x + 4. Find the values of m for which the line y = mx + 1 is a tangent to the graph of f .
(Total 6 marks)
31.
Find
 e x
cos x d x .
(Total 6 marks)

x
3
32.
The curve y = – x
3
2
– 3 x + 4 has a local maximum point at P and a local minimum point at Q. Determine the equation of the straight line passing through P and Q, in the form ax + by + c =
0, where a , b , c

.
(Total 6 marks)
33.
The function f is defined by f ( x ) = e px
( x + 1), here p

.
(a) (i) Show that f

( x ) = e px
( p ( x + 1) + 1).
(ii) Let f
( n )
( x ) denote the result of differentiating f ( x ) with respect to x , n times.
Use mathematical induction to prove that f
( n )
( x ) = p n
–1 e px
( p ( x + 1) + n ), n
 +
.
(7)
(b) When p = 3 , there is a minimum point and a point of inflexion on the graph of f .
Find the exact value of the x -coordinate of
(i) the minimum point;
(ii) the point of inflexion. (4)
(c) Let p =
1
2
. Let R be the region enclosed by the curve, the x -axis and the lines x = –2 and x = 2. Find the area of R .
(2) (Total 13 marks)
34.
The temperature T °C of an object in a room, after t minutes, satisfies the differential equation d T
= k ( T – 22), where k is a constant. d t
(a) Solve this equation to show that T = A e kt
+ 22, where A is a constant. (3)
(b) When t = 0, T = 100, and when t = 15, T = 70.
(i) Use this information to find the value of A and of k .
(ii) Hence find the value of t when T = 40.
(7)
(Total 10 marks)
35.
Find
 ln x x d x . (Total 6 marks)
36.
An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the

angle

is
1
3

radians and is increasing at
1
60
radians per second. Find the speed, in kilometres per hour, at which the airplane is moving towards the observer.
Airplane x
3 km
Observer
(Total 6 marks)
37.
Using the substitution y = 2 – x , or otherwise, find




2 x
– x



2 d x.
38.
The diagram below shows the graph of y
1
= f ( x ), 0

x

4.
y
(Total 6 marks)
0 1 2 3 4 x
On the axes below, sketch the graph of y
2
=

0 x f ( t ) d t , marking clearly the points of inflexion. y
0 1 2 3 4 x
(Total 6 marks)