All Calculus Review.
2.
Find the gradient of the tangent to the curve x3 y2 = 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 = −2x + c, where c
.
Find the value of c.
(Total 6 marks)
4.
The function f is defined by f (x) = x e2x.
It can be shown that f (n) (x) = (2n x + n 2n−1) e2x for all n
derivative of f (x).
(a)
+
, where f (n) (x) represents the nth
By considering f (n) (x) for n =1 and n = 2, show that there is one minimum point P on the
graph of f, and find the coordinates of P.
(7)
(b)
Show that f has a point of inflexion Q at x = −1.
(5)
(c)
Determine the intervals on the domain of f where f is
(i)
concave up;
(ii)
concave down.
(2)
(d)
Sketch f, clearly showing any intercepts, asymptotes and the points P and Q.
(4)
(e)
Use mathematical induction to prove that f (n) (x) = (2nx + n2n−1) e2x for all n
f (n) (x) represents the nth derivative of f (x).
+
, where
(9)
(Total 27 marks)
5. (Calc)
A television screen, BC, of height one metre, is built into a wall. The bottom of the
ˆ C,
television screen at B is one metre above an observer’s eye level. The angles of elevation ( AO
ˆ B ) from the observer’s eye at O to the top and bottom of the television screen are  and  radians
AO
respectively. The horizontal distance from the observer’s eye to the wall containing the television screen
ˆ C ) is  radians, as shown below.
is x metres. The observer’s angle of vision ( BO
(a)
(i)
Show that  = arctan
2
1
– arctan .
x
x
1
(ii)
Hence, or otherwise, find the exact value of x for which  is a maximum and justify
that this value of x gives the maximum value of .
(iii)
Find the maximum value of .
(17)
(b)
Find where the observer should stand so that the angle of vision is 15.
(5)
(Total 22 marks)
6.
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)
7. (Calc)
For x 
1
, let f (x) = x2 ln (x +1) and g (x) =
2
2 x  1.
(a)
Sketch the graphs of f and g on the grid below.
(b)
Let A be the region completely enclosed by the graphs of f and g.
Find the area of A.
(Total 6 marks)
8.
The following table shows the values of two functions f and g and their first derivatives when x =1
and x = 0.
(a)
x
f (x)
f ′ (x)
g (x)
g′ (x)
0
4
1
–4
5
1
–2
3
–1
2
Find the derivative of
3 f ( x)
when x = 0.
g ( x)  1
2
(b)
Find the derivative of f (g (x) + 2x) when x =1.
(Total 6 marks)
9.
Given that exy − y2 ln x = e for x  1, find
dy
at the point (1, 1).
dx
(Total 6 marks)
10.
The function f is defined by f (x) =
ln x
x3
, x 1.
(a)
Find f ′(x) and f ′′(x), simplifying your answers.
(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.
(6)
(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)
(d)
Show that the area of R is
1
(4 – ln 3).
18
(6)
(Total 26 marks)
11.
Given that 3x+y = x3 + 3y, find
12.
(a)
dy
.
dx
(Total 6 marks)
Write down the term in xr in the expansion of (x + h)n, where 0  r  n, n
+
.
(1)
(b)
Hence differentiate xn, n
+
, from first principles.
(5)
(c)
Starting from the result xn  x–n = 1, deduce the derivative of x–n, n
+
.
(4)
(Total 10 marks)
13. The following diagram shows the points A and B on the circumference of a circle, centre O, and
radius 4 cm, where AÔB = . Points A and B are moving on the circumference so that  is increasing
at a constant rate.
O
A
B
3
Given that the rate of change of the length of the minor arc AB is numerically equal to the rate of
change of the area of the shaded segment, find the acute value of .
(Total 6 marks)
The normal to the curve y = k + ln x2, for x  0, k 
x
3x + 2y = b, where b  . Find the exact value of k.
14.
, at the point where x = 2, has equation
(Total 6 marks)
16. 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
1
 radians and is increasing at
radians per second. Find the speed, in kilometres per hour, at
3
60
which the airplane is moving towards the observer.
Airplane
x
3 km
Observer
(Total 6 marks)
17.
The function f is defined by f (x) =
(a)
(i)
Show that
f (x) =
(ii)
x2
, for x > 0.
2x
2 x – x 2 ln 2
2x
Obtain an expression for f (x), simplifying your answer as far as possible.
(5)
(b)
(i)
Find the exact value of x satisfying the equation f (x) = 0
(ii)
Show that this value gives a maximum value for f (x).
(4)
(c)
Find the x-coordinates of the two points of inflexion on the graph of f.
(3)
(Total 12 marks)
18.
The function f is defined by
f (x) =
(a)
x2 – x 1
x2  x 1
(i)
Find an expression for f (x), simplifying your answer.
(ii)
The tangents to the curve of f (x) at points A and B are parallel to the x-axis. Find the
coordinates of A and of B.
(5)
(b)
(i)
Sketch the graph of y = f (x).
4
(ii)
Find the x-coordinates of the three points of inflexion on the graph of f.
(5)
(c)
Find the range of
(i)
f;
(ii)
the composite function f ° f.
(5)
(Total 15 marks)
19. (Calc)
A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are
2
on the curve y = e – x . The area of this rectangle is denoted by A.
(a)
Write down an expression for A in terms of x.
(b)
Find the maximum value of A.
(Total 6 marks)
20.
A point P(x, x2) lies on the curve y = x2. Calculate the minimum distance from the point
1

A 2, –  to the point P.
2

(Total 3 marks)
21. (Calc)
(a) Let y = a  b sin x , where 0 < a < b.
b  a sin x
(i)
Show that
dy
(b 2  a 2 ) cos x
=
.
dx
(b  a sin x) 2
(4)
(ii)
Find the maximum and minimum values of y.
(4)
(iii)
Show that the graph of y = a  b sin x , 0 < a < b cannot have a vertical asymptote.
b  a sin x
(2)
(b)
For the graph of y  4  5 sin x for 0  x  2,
5  4 sin x
(i)
write down the y-intercept;
(ii)
find the x-intercepts m and n, (where m < n) correct to four significant figures;
(iii)
sketch the graph.
(5)
(c)
The area enclosed by the graph of y  4  5 sin x and the x-axis from x = 0 to x = n is
5  4 sin x
denoted by A. Write down, but do not evaluate, an expression for the area A.
5
(2)
(Total 17 marks)
22. (Calc)
(a) Sketch and label the curves
y = x2 for –2  x  2, and y = – 1 ln x for 0 < x  2.
2
(2)
(b)
Find the x-coordinate of P, the point of intersection of the two curves.
(2)
(c)
If the tangents to the curves at P meet the y-axis at Q and R, calculate the area of the triangle
PQR.
(6)
(d)
Prove that the two tangents at the points where x = a, a > 0, on each curve are always
perpendicular.
(4)
(Total 14 marks)
6