Calculus -Differentiation [304 marks]

1. [Maximum mark: 16] SPM.1.SL.TZ0.8

Let f (x) =
1

3
x
3
+ x
2
− 15x + 17 .

The graph of f has horizontal tangents at the points where x = a and x = b, a < b.

(a) Find f ′
(x) . [2]

(b) Find the value of a and the value of b. [3]

(c.i) Sketch the graph of y = f

′
(x) . [1]
(c.ii) Hence explain why the graph of f has a local maximum point at x .
= a [1]

(d.i) Find f ′′
(b) . [3]
(d.ii) Hence, use your answer to part (d)(i) to show that the graph of f has a
local minimum point at x = b. [1]

(e) The normal to the graph of f at x = a and the tangent to the graph of f at
x = b intersect at the point ( p, q ) .

Find the value of p and the value of q. [5]

2. [Maximum mark: 7] SPM.2.SL.TZ0.6
The displacement, in centimetres, of a particle from an origin, O, at time t seconds, is given by
s(t) = t 2 cos t + 2t sin t, 0 ≤ t ≤ 5.

(a) Find the maximum distance of the particle from O. [3]

(b) Find the acceleration of the particle at the instant it first changes direction. [4]
3. [Maximum mark: 15] EXN.1.SL.TZ0.9
The following diagram shows a ball attached to the end of a spring, which is suspended
from a ceiling.

The height, h metres, of the ball above the ground at time t seconds after being released can
be modelled by the function h(t)= 0. 4 cos(πt)+1. 8 where t ≥ 0.

(a) Find the height of the ball above the ground when it is released. [2]

(b) Find the minimum height of the ball above the ground. [2]
(c) Show that the ball takes 2 seconds to return to its initial height above the
ground for the first time. [2]

(d) For the first 2 seconds of its motion, determine the amount of time that the
ball is less than 1. 8 + 0. 2√2 metres above the ground. [5]
(e) Find the rate of change of the ball’s height above the ground when t = 1

. Give your answer in the form pπ√q ms

−1
where p ∈ Q and q ∈ Z
+
. [4]
4. [Maximum mark: 7] EXN.2.AHL.TZ0.6
The curve C has equation e 2y
= x
3
+ y .

(a) Show that

dy
=
3x
2

. [3]
2y
dx 2e −1

(b) The tangent to C at the point Ρ is parallel to the y -axis.

Find the x-coordinate of Ρ. [4]

5. [Maximum mark: 4] 22N.1.SL.TZ0.2
The function g is defined by g(x)= , where x .
2
x +1
e ∈ R

Find g'(−1). [4]

6. [Maximum mark: 15] 22N.1.SL.TZ0.9
The function f is defined by f (x)= cos
2
x − 3 sin
2
x, 0 ≤ x ≤ π .

(a) Find the roots of the equation f (x)= 0. [5]

(b.i) Find f '(x). [2]

(b.ii) Hence find the coordinates of the points on the graph of y = f (x) where
f '(x)= 0 . [5]
(c) Sketch the graph ofy = f (x), clearly showing the coordinates of any
points where f '(x)= 0 and any points where the graph meets the
coordinate axes. [3]
7. [Maximum mark: 7] 22N.1.AHL.TZ0.7
Consider the curve with equation (x 2 2
+ y )y
2
= 4x
2
where x ≥ 0 and
−2 < y < 2 .

Show that the curve has no local maximum or local minimum points for x > 0 . [7]
8. [Maximum mark: 20] 22N.1.AHL.TZ0.10
The function f is defined by f (x)= cos
2
x − 3 sin
2
x, 0 ≤ x ≤ π .

(a) Find the roots of the equation f (x)= 0. [5]

(b.i) Find f '(x). [2]

(b.ii) Hence find the coordinates of the points on the graph of y = f (x) where
f '(x)= 0 . [5]
(c) Sketch the graph of y =|f (x)|, clearly showing the coordinates of any
points where f '(x) = 0 and any points where the graph meets the
coordinate axes. [4]
(d) Hence or otherwise, solve the inequality |f (x)|> 1. [4]
9. [Maximum mark: 6] 22N.2.AHL.TZ0.8
The following diagram shows liquid in a round-bottomed glass flask, which is
made of a sphere and a cylindrical neck.

Initially, the flask is empty. Liquid is poured into the flask at a rate of 2 3
cm s
−1
. You
may assume that the liquid does not reach the cylindrical neck.

The volume V cm
3
and the height h cm of the liquid in the flask satisfy the
equation

V = 5πh
2
−
1

3
3
πh .

Find the rate of change of the height of the liquid in the flask at the instant when
the volume of the liquid is 200 cm
3
. [6]
10. [Maximum mark: 14] 22M.1.SL.TZ1.7
A function, f , has its derivative given by f '(x) = 3x
2
− 12x + p , where p ∈ R . The
following diagram shows part of the graph of f '.

The graph of f ' has an axis of symmetry x = q .

The vertex of the graph of f ' lies on the x-axis.

The graph of f has a point of inflexion at x = a .

(a) Find the value of q. [2]

(b.i) Write down the value of the discriminant of f '. [1]

(b.ii) Hence or otherwise, find the value of p. [3]

(c) Find the value of the gradient of the graph of f ' at x = 0 . [3]

(d) Sketch the graph of f '', the second derivative of f . Indicate clearly the x-
intercept and the y -intercept. [2]
(e.i) Write down the value of a. [1]

(e.ii) Find the values of x for which the graph of f is concave-down. Justify your
answer. [2]
11. [Maximum mark: 15] 22M.1.SL.TZ2.7
The following diagram shows part of the graph of a quadratic function f .

The graph of f has its vertex at (3, 4) , and it passes through point Q as shown.

The function can be written in the form f (x) = .

2
a(x − h) + k

The line L is tangent to the graph of f at Q.

Now consider another function y = g(x) . The derivative of g is given by g'(x) = f (x) − d

, where d ∈ R.

(a) Write down the equation of the axis of symmetry. [1]

(b.i) Write down the values of h and k. [2]

(b.ii) Point Q has coordinates (5, 12) . Find the value of a. [2]

(c) Find the equation of L. [4]

(d) Find the values of d for which g is an increasing function. [3]

(e) Find the values of x for which the graph of g is concave-up. [3]
12. [Maximum mark: 25] SPM.3.AHL.TZ0.2
This question asks you to investigate some properties of the sequence of functions of
the form f (x) = cos (n arccos x), −1 ≤ x ≤ 1 and n ∈ Z .
n
+

Important: When sketching graphs in this question, you are not required to find
the coordinates of any axes intercepts or the coordinates of any stationary points unless
requested.

(a) On the same set of axes, sketch the graphs of y = f 1 (x) and y = f 3 (x) for
−1 ≤ x ≤ 1. [2]

For odd values of n > 2, use your graphic display calculator to systematically vary the value
of n. Hence suggest an expression for odd values of n describing, in terms of n, the number
of
(b.i) local maximum points; [3]

(b.ii) local minimum points; [1]

(c) On a new set of axes, sketch the graphs of y = f 2 (x) and y = f 4 (x) for −1
≤ x ≤ 1. [2]

For even values of n > 2, use your graphic display calculator to systematically vary the value
of n. Hence suggest an expression for even values of ndescribing, in terms of n, the number
of

(d.i) local maximum points; [3]

(d.ii) local minimum points. [1]

(e) Solve the equation f n

'
(x) = 0 and hence show that the stationary points
on the graph of y = f n (x) occur at x = cos
kπ

n
where k ∈ Z
+
and 0 < k <
n. [4]
The sequence of functions, f n (x) , defined above can be expressed as a sequence
of polynomials of degree n.

(f ) Use an appropriate trigonometric identity to show that f 2 (x) = 2x

2
− 1 . [2]

Consider f n+1 (x) = cos ((n + 1) arccos x) .

(g) Use an appropriate trigonometric identity to show that
f n+1 (x) = cos (n arccos x) cos (arccos x) − sin (n arccos x) sin (arccos x)

. [2]

(h.i) Hence show that f n+1 (x) ,

+ f n−1 (x) = 2xf n (x) n ∈ Z
+
. [3]

(h.ii) Hence express f 3 (x) as a cubic polynomial. [2]

13. [Maximum mark: 5] EXN.1.AHL.TZ0.6
2
2x cos ( x )
Use l’Hôpital’s rule to determine the value of lim( ) .
x→0
5 tan x [5]
14. [Maximum mark: 22] EXN.2.AHL.TZ0.12
Consider the differential equation

dy y
= f( ), x > 0
dx x

(a) Use the substitution y = vx to show that ∫ dv

= ln x + C where C
f ( v ) −v

is an arbitrary constant. [3]

The curve y = f (x) for x > 0 has a gradient function given by

2 2
dy y +3xy+2x

dx
= 2
.
x

The curve passes through the point (1, −1) .

(b) By using the result from part (a) or otherwise, solve the differential equation
and hence show that the curve has equation y = x(tan (ln x)−1). [9]
(c)
π π

The curve has a point of inflexion at (x , y ) where e

−
1 1
2 < x1 < e 2

. Determine the coordinates of this point of inflexion. [6]

(d) Use the differential equation
dy
=
2
y +3xy+2x
2

to show that the points of

dx x2

zero gradient on the curve lie on two straight lines of the form y = mx

where the values of m are to be determined. [4]

15. [Maximum mark: 7] 22N.1.AHL.TZ0.7
Consider the curve with equation (x 2 2
+ y )y
2
= 4x
2
where x ≥ 0 and
−2 < y < 2 .

Show that the curve has no local maximum or local minimum points for x > 0 . [7]
16. [Maximum mark: 20] 22M.2.AHL.TZ1.10
Consider the function f (x)= √ x2 − 1 , where 1 ≤ x ≤ 2 .

(a) Sketch the curve y = f (x) , clearly indicating the coordinates of the
endpoints. [2]

(b.i) Show that the inverse function of f is given by f −1 2

(x)= √ x + 1 . [3]
(b.ii) State the domain and range of f −1
. [2]

The curve y = f (x) is rotated 2π about the y -axis to form a solid of revolution that is used
to model a water container.

(c.i) Show that the volume, V m

3
, of water in the container when it is filled to a
height of h metres is given by V .
1 3
= π(
3
h + h)
[3]
(c.ii) Hence, determine the maximum volume of the container. [2]

At t = 0, the container is empty. Water is then added to the container at a constant rate of
0. 4 m
3
s
−1
.

(d) Find the time it takes to fill the container to its maximum volume. [2]
(e) Find the rate of change of the height of the water when the container is
filled to half its maximum volume. [6]
17. [Maximum mark: 8] 22M.2.AHL.TZ2.7
arctan ( cos x ) −k
Consider lim x2
, where k ∈ R .
x→0

(a) Show that a finite limit only exists for k = π

4
. [2]

(b) Using l’Hôpital’s rule, show algebraically that the value of the limit is − .
1
[6]
4
18. [Maximum mark: 28] 22M.3.AHL.TZ2.1
This question asks you to explore properties of a family of curves of the type
y
2
= x
3
+ ax + b for various values of a and b, where a, b ∈ N .

On the same set of axes, sketch the following curves for −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2

, clearly indicating any points of intersection with the coordinate axes.

(a.i) y
2 3
= x , x ≥ 0 [2]

(a.ii) y
2
= x
3
+ 1, x ≥ −1 [2]
(b.i) Write down the coordinates of the two points of inflexion on the curve
y
2
= x
3
+ 1 . [1]

(b.ii) By considering each curve from part (a), identify two key features that
would distinguish one curve from the other. [1]

Now, consider curves of the form y 2

= x
3
, for x
+ b
3
≥ −√b , where b ∈ Z
+
.

(c) By varying the value of b, suggest two key features common to these
curves. [2]
Next, consider the curve y 2
= x
3
+ x, x ≥ 0 .

(d.i) Show that

dy 2
3x +1
, for x .
dx
= ±
3
2√x +x
> 0
[3]

(d.ii) Hence deduce that the curve y 2

= x
3
+ x has no local minimum or
maximum points. [1]

The curve y = x + x has two points of inflexion. Due to the symmetry of the curve these
2 3

points have the same x-coordinate.

(e) Find the value of this x-coordinate, giving your answer in the form
p√3+q
x = √
r
, where p, q, r ∈ Z . [7]

P(x, y) is defined to be a rational point on a curve if x and y are rational numbers.

The tangent to the curve y 2

= x
3
+ ax + b at a rational point P intersects the curve at
another rational point Q.

Let C be the curve y 2

= x
3
+ 2 , for x ≥ −√2
3
. The rational point P(−1, − 1) lies on C.

(f.i) Find the equation of the tangent to C at P. [2]

(f.ii) Hence, find the coordinates of the rational point Q where this
tangent intersects C, expressing each coordinate as a fraction. [2]

(g) The point S(−1 , 1) also lies on C. The line [QS] intersects C at a further
point. Determine the coordinates of this point. [5]
19. [Maximum mark: 15] 20N.1.AHL.TZ0.H_11
Consider the curve C defined by y 2
= sin (xy) , y ≠ 0 .

(a) Show that

dy y cos ( xy )
.
dx
=
2y−x cos ( xy )
[5]

(b) Prove that, when

dy
= 0 , y = ±1 . [5]
dx
(c) Hence find the coordinates of all points on C, for 0 < x < 4π , where
dy

dx
= 0 . [5]
20. [Maximum mark: 6] 20N.1.AHL.TZ0.F_1
Use l’Hôpital’s rule to determine the value of

.
2 sin x−sin 2x
lim
x→0
x3 [6]
21. [Maximum mark: 7] 20N.3.AHL.TZ0.Hca_1
Use l’Hôpital’s rule to find
2
cos ( x −1 ) −1
lim .
x→1
ex−1
−x [7]
22. [Maximum mark: 9] 19M.1.AHL.TZ1.H_7
Find the coordinates of the points on the curve y 3
+ 3xy
2
− x
3
= 27 at which
dy

dx
= 0 . [9]
23. [Maximum mark: 4] 19M.2.AHL.TZ1.H_1
Let l be the tangent to the curve y = xe
2x
at the point (1, e ).
2

Find the coordinates of the point where l meets the x-axis. [4]
24. [Maximum mark: 9] 19M.3.AHL.TZ0.Hca_4
Using L’Hôpital’s rule, find lim( .
tan 3x−3 tan x
)
x→0
sin 3x−3 sin x [9]
25. [Maximum mark: 5] 18N.2.AHL.TZ0.H_5
Differentiate from first principles the function f (x) = 3x
3
− x . [5]
26. [Maximum mark: 8] 18N.3.AHL.TZ0.Hca_2
(a) Use L’Hôpital’s rule to determine the value of
2
−3x
⎛e + 3 cos (2x) − 4 ⎞
lim
2
x→0 ⎝ 3x ⎠
[5]

(b) ⎛ ∫
x

0
−3t
(e
2

+3 cos(2t)−4) dt ⎞
Hence find lim⎜ x ⎟ .
x→0 ∫ 3t2 dt
⎝ 0
⎠
[3]
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