DIFFERENTIATION
1. Show that the points (0, 0) and ( 2 , 2 ) on the curve e(x + y) = cos (xy) have a common tangent.
(Total 7 marks)
2.
1 (9 + 8 x 2 x 4 ) The curve C has equation y = 8 . dy Find the coordinates of the points on C at which dx = 0.
(4)
(a)
(b)
The tangent to C at the point P(1, 2) cuts the x-axis at the point T. Determine the coordinates of T.
(4)
(c)
The normal to C at the point P cuts the y-axis at the point N. Find the area of triangle PTN.
(7) (Total 15 marks)
3.
x The function f is defined by f(x) = e
2 x 1.5
.
(2)
(a)
Find f(x).
(b)
f ( x) You are given that y = x 1 has a local minimum at x = a, a > 1. Find the value of a.
(6) (Total 8 marks)
4.
Find the equation of the normal to the curve x3y3 xy = 0 at the point (1, 1).
(Total 7 marks)
IB Mathematics Higher Level Worksheet Dileep
5.
Let f be a function defined by f(x) = x arctan x, x (a) Find f(1) and f( 3 ).
(2)
(b)
Show that f(x) = f(x), for x
.
(2)
(c)
< f ( x) < x + 2 , for x Show that x 2
.
(2)
(d)
Find expressions for f(x) and f(x). Hence describe the behaviour of the graph of f at the origin and justify your answer.
(8)
(e)
Sketch a graph of f, showing clearly the asymptotes.
(3)
(f)
Justify that the inverse of f is defined for all x
and sketch its graph.
(3) (Total 20 marks)
6.
Let f be a function with domain
that satisfies the conditions,
f(x + y) = f(x) f(y), for all x and y and f (0) 0.
(a)
Show that f (0) = 1.
(3)
(b)
Prove that f(x) 0, for all x
.
(3)
IB Mathematics Higher Level Worksheet Dileep
(c)
Assuming that f(x) exists for all x , use the definition of derivative to show that f(x) satisfies the differential equation f(x) = k f(x), where k = f(0).
(4)
(d)
Solve the differential equation to find an expression for f(x).
(4) (Total 14 marks)
7.
Consider the part of the curve 4x2 + y2 = 4 shown in the diagram below.
(a)
dy Find an expression for dx in terms of x and y.
(3)
(b)
2 2 , 5 5 . Find the gradient of the tangent at the point
(1)
IB Mathematics Higher Level Worksheet Dileep
(c)
A bowl is formed by rotating this curve through 2 radians about the xaxis. Calculate the volume of this bowl.
(4) (Total 8 marks)
8.
1 x Let f(x) = 1 + x and g(x) =
x + 1 , x > 1.
(Total 7 marks)
Find the set of values of x for which f (x) f(x) g(x).
9.
(a) (b)
2 Differentiate f(x) = arcsin x + 2 1 x , x [1, 1].
(3)
Find the coordinates of the point on the graph of y = f(x) in [1, 1], where the gradient of the tangent to the curve is zero.
(3) (Total 6 marks)
10.
The cubic curve y = 8x3 + bx2 + cx + d has two distinct points P and Q, where the gradient is zero. (a) Show that b2 > 24c.
(4)
(b)
1 3 , 12 and , 20 2 , respectively, find the Given that the coordinates of P and Q are 2 values of b, c and d.
(4) (Total 8 marks)
IB Mathematics Higher Level Worksheet Dileep
11.
5 Consider the graphs y = ex and y = ex sin 4x, for 0 x 4 . 5 On the same set of axes draw, on graph paper, the graphs, for 0 x 4 . Use a scale of 1 cm to 8 on your x-axis and 5 cm to 1 unit on your y-axis.
(3)
(a)
(b)
n Show that the x-intercepts of the graph y = e sin 4x are 4 , n = 0, 1, 2, 3, 4, 5.
x
(3)
(c)
Find the x-coordinates of the points at which the graph of y = ex sin 4x meets the graph of y = ex. Give your answers in terms of .
(3)
(d)
(i)
Show that when the graph of y = ex sin 4x meets the graph of y = ex, their gradients are equal.
(ii)
Hence explain why these three meeting points are not local maxima of the graph y = ex sin 4x.
(6)
(e)
(i)
Determine the y-coordinates, y1, y2 and y3, where y1 > y2 > y3, of the local maxima
5 of y = ex sin 4x for 0 x 4 . You do not need to show that they are maximum values, but the values should be simplified.
(ii)
Show that y1, y2 and y3 form a geometric sequence and determine the common ratio r.
(7) (Total 22 marks)
IB Mathematics Higher Level Worksheet Dileep
12.
Find the gradient of the curve exy + ln(y2) + ey = 1 + e at the point (0, 1).
(Total 7 marks)
13.
Consider the curve with equation x2 + xy + y2 = 3. (a) Find in terms of k, the gradient of the curve at the point (1, k).
(5)
(b)
Given that the tangent to the curve is parallel to the x-axis at this point, find the value of k.
(1) (Total 6 marks)
14.
Andr wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = 2 km and PY = 2 km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.
When Andr swims he covers 1 km in 5 5 minutes. When he runs he covers 1 km in 5 minutes.
(a)
If PQ = x km, 0 x 2, find an expression for the time T minutes taken by Andr to reach point Y.
(4)
(b)
Show that
5 5x dT = 5. dx x2 + 4
(3)
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(c)
(i)
dT = 0. Solve dx
(ii)
Use the value of x found in part (c) (i) to determine the time, T minutes, taken for Andr to reach point Y.
(iii)
d 2T 20 5 = 2 3 dx 2 2 x + 4 Show that and hence show that the time found in part (c) (ii) is a minimum.
(11) (Total 18 marks)
15.
Find the gradient of the tangent to the curve x3 y2 = cos (y) at the point (1, 1).
(Total 6 marks)
16.
Find the equation of the normal to the curve 5xy2 2x2 =18 at the point (1, 2).
(Total 7 marks)
17.
(a)
Use the derivatives of sin x and cos x to show that the derivative of tan x is sec2 x.
(3)
(b)
dy 1 = dx dx 1 dy , show that the derivative of arctan x is 1 + x 2 . Hence by using
(4) (Total 7 marks)
IB Mathematics Higher Level Worksheet Dileep
18.
1 dy 2 ln (1 + e 2 x ) = , show that dx 3 (ey 3). If y = 3
(Total 7 marks)
19.
5x 2. The function f is given by f(x) = 2sin
(a) Write down f(x).
(2)
(b)
f Given that 2 = 1, find f(x).
(4) (Total 6 marks)
20.
Find the gradient of the normal to the curve 3x2y + 2xy2 = 2 at the point (1, 2).
(Total 6 marks)
21.
A curve C is defined implicitly by xey = x2 + y2. Find the equation of the tangent to C at the point (1, 0).
(Total 7 marks)
IB Mathematics Higher Level Worksheet Dileep
