# Maths Task-Differential Calculus, Anti differentiation, Problem Solving

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PART 1-Differential Calculus
SEND: Work for Submission
TECHNOLOGY FREE
1. The figure at right shows part of the
hyperbola with equation y = 4 andP is the
y
x
point at which x=2.
(a) A1 is a point on the curve whose
x coordinate is 2.5. Find its y coordinate
and hence the gradient of the chord P A1 .
places.
(b) Repeat for point A2 whose x coordinate is 2.1.
y= 4
x
P
P
2
O
2
x
(c) Repeat for point A3 whose x coordinate is 2.01.
(d) Repeat for point A4 whose x coordinate is 2.001
(e) What do results of (a), (b), (c) and (d) suggest for the value of the gradient of the function
4
y = at point P?
x
36 – x 2
5 x3 – 5
2.
Find (a) lim
(b) lim
x ?1 x – 1
x ?6 6 – x
3.
For the function f : R ? R, f ( x) = 7 – x 2 , find:
(a)
the value of f ?(x) from the first principles,
(b)
the value of f ?(x) ,using rules of differentiation,
f ? ( -1) . Show all working.
(c)
4.
Find the derivatives of the following functionsby rule. Do not use first principles.
(i)
f(x) = 3×10 – 8 x5 + 2 x3 + 6 x – 9
(ii)
f(x) = x ( x + 1)( 2 x -1)
(iii)
f(x) = 6 x – ( 2 x – 3) – 2
(iv)
f(x) =3
(v)
f(x) = x 7
2
6
7
(vi)
3
4 x4 – 5x
x
For the curve with equation y = x 2 + 4 x + 3 , find:
(vii)
5.
f(x) = ( x + 2 )
f(x) =
(a)
(b)
the gradient of the curve where it cuts the x-axis,
the coordinates of the points on the curve where the tangent is:
(i) parallel to the x-axis,
(ii) parallel to the line 6 x + 3 y = 7 .
CAS Calculators
6.
Use your CAS calculator to demonstrate the limits for SEND Question 2. Use
connectivity software to include a screen capture of your answers. Alternatively, draw
by hand a picture of your screen.
7.
Find the following derivatives.
(a)
f ( x ) = x3 cos ( 2 x )
(b)
f ( x ) = 1 + tan 2 ( x )
(c)
f ( x ) = ecos( 2 x )
(d)
f ( x ) = log ( sin ( 3x ) )
Enter as f ( x ) = 1 + (tan x)2
PART 2-Differential Calculus
SEND: Work for Submission
TECHNOLOGY FREE
1.
Find the equations of
i. the tangent
ii. the normal
to the curve y = 2 x3 – x 2 + 4 x – 7 at the point wherex = 1.
2.
Show that f ( x ) = -2 x3 + 3×2 – 3x + 2 has no stationary points.
3.
(a) Sketch a neat graph of f ( x ) = 14 – 5x – 8×2 – x3 , showing the coordinates of any
axis intersections and stationary points. (Note you may use your calculator for
arithmetical purposes, but not for graphing).
(b) Verify the stationary points (explain why they are points of maximum or
minimum).
4.
Sketch the graph f : R ? R wheref(x) = -2(x-1)(x-2)(x+3), giving the coordinates of
any stationary points and axis intersections. What is the range of f(x)?
5.
Use differentiation to find the minimum value off(x) = 3×2- 5x + 10 and the value(s) of
x for which this minimum occurs.
6.
A piece of wire of length 60 centimetres is cut into two sections. Each section is then
bent into the shape of a square. Find the smallest possible value of the sum of the areas
of the two squares.
Hints for Q6:
i. Let x represent the length of the first square and y,the length of the second square.
ii. Find an equation connecting x and y, and then write y in terms of x.
iii. Establish the equation for the sum of the areas of the two squares. Write your
equation so that it is in terms of x.
iv. Using calculus, find the minimum value for this area.
CAS Calculators
7
Use your calculator to sketch the gradient graph for each of the following
functions. (Note that it is not necessary to find the equation of the gradient
function).
Include a copy of your screen shots with your work. For each of the parts i – iii,
your screen should show the graph of the function as well as the gradient
function.
i.
f ( x) = x 2 – 2
ii.
f ( x) = 4 x 3 – 3x 2 – 7 x + 1
iii.
f ( x) = x 4 + x 2 + 1
PART 3-Anti differentiation
SEND: Work for Submission
TECHNOLOGY FREE
1. Find the following (show all working out):
(a) ? ( x5 + 2 x – 7 ) dx
(b) ? x ( 3 x – 4 ) dx
(c) ? ( x + 4 )( x – 1) dx
(d) ? ( x – 2 ) dx
3
2
? x4
?
(e) ? ? – 3x ? dx
? 2
?
2. Find the definite integral of the following (show all working out). (Note you may
use a calculator when evaluating, but do not use the calculator to integrate.
(a)
(b)
? ( 3x
5
-1
?
4
0
(
2
– 2 x – 1) dx
3x –
1
2
) dx
2
3. Find the equation of the curve which goes through the point (3,-4) and has the
dy
= 2x – 5
derivative
dx
4. If f(x) = (x+1)2 and F(1) = 2, find the antiderivative F(x)
5. Suppose that a point moves along some unknown curve y = f(x) in such a way that
at each point (x, y) on the curve the tangent line has a slope
x2
2
. Find an equation
for the curve, given that it passes through (1, 1).
6. The rate of change of height of a baseball is given by
dh
= 15 – 10t where h is the
dt
height above the ground in metres after t seconds.
(a) If h = 0.5 when t = 0, express h as a function of t.
(b) Find the height of the baseball after 2 seconds.
(c) How long does it take the baseball to reach the ground? Give your
answer correct to 3 decimal places.
7. Find the area under the curve f ( x ) = 2 x3 + 3×2 – 2 x bounded by the x-axis and the
lines x = 2 and x = 0.
CAS Calculators
8. Sketch the curve f ( x ) = 2 cos
( x ) , -2? ? x ? 2? .
1
4
Copy the CAS calculator screen output onto your page.
On your graph, shade in the section which represents the area under the curve
bounded by the x-axis and the lines x = 0 and x = 2?
Use your CAS calculator to determine this area.
9. An object moves in a straight line so that its velocity, v m/s, at any time t seconds is
given byv(t) = 0.5t2- 2.5t + 2
(a) Find the equation which represents the acceleration at any time t.
(b) Find the equation representing the position, x metres, at any timet, given that
the object is 2.5 metres to the right of the originwhen t = 0 seconds.
(c) Find the position and acceleration when velocity is equal to zero. Give your
answers correct to 2 decimal places.
(d) Find the average acceleration between t = 0 and t = 1.
PART 4-Problem Solving
You must submit for assessment your complete solutions for problems 1 to 3clearly showing
all working. Include, where appropriate, the CAS calculator screens, either copied out by
hand or pasted electronically into your work.
PROBLEM 1.
Icebergs
In order to provide more water for a particular desert country with a very hot climate, the
government plans to tow icebergs from the Antarctic and to convert them to fresh water on
arrival. As a preliminary to such an expensive venture, a number of possible models are
evaluated.
Model 1
(3 + 1 + 1 + 1 + 2 = 8 marks)
Assume that the iceberg is a cube of 1 331 000 m3 and that it melts uniformly so that a top
layer of 1 m is lost each day.
(i)
Find an expression for V, the volume of the iceberg after n days. Sketch the graph
of V against n.
(ii)
After how many days would the volume of the iceberg be halved?
(iii) After how many days would the ice have melted completely?
(iv) If it takes 20 days for the trip, how much of the iceberg is left on arrival?
(v)
The rate of change of volume V (m3) of ice with respect to time (days) is given by
dV
.
dn
Find this and comment.
Model 2
(3 + 1 + 1 + 1 + 2 = 8 marks)
Assume the iceberg is a cube of 1 331 000 m3 and that it melts so that the face which is
totally exposed to the air melts twice as quickly, i.e. 2 m melts away each day. Assume that
the iceberg retains a cuboid shape. Make the same calculations as those in parts (i) to (v)
above for this new model. (You may need to use numerical methods to solve some of the
resulting equations.)
Model 3
(3 + 1 + 1 + 1 + 2+ 1 + 1 + 2 = 12 marks)
Assume that the iceberg is a sphere of volume 1 331 000 m3 which melts uniformly so that
dr
= -1, where r metres is the radius of the sphere. Repeat the calculations for parts (i) to
dn
(v).
Use the information gained for each of these possible models to answer the following:
(vi) What are the faults for each of the models?
(vii) Is there a best shape for an iceberg when used for this purpose?
(viii) Can you suggest a better model?
PROBLEM 2.
Water Slide
(2 + 2 + 2 + 2 = 8 marks)
The slope of a water slide at a theme park is given by
dy
1
= – ( x 2 – 16 x + 60) for x ? [0, 18]
24
dx
where the origin is taken to be at the ground level beneath the highest point of the slide,
which is 18 m above the ground. All units are in metres.
(a) Find the equation of the curve which describes the slide.
(b) Sketch the curve of the slide, labelling stationary points.
(c) Does the slope of the slide ever exceed 45º? (Give reasons for your answer.)
If so, when (i.e. for what values of x) does this occur? (Give answers correct to two
decimal places.)
(d) What is the maximum value of the slope of the slide? (Give an answer correct to the
nearest degree.)
PROBLEM 3.
NJ Constructions(1 + 2 + 2 + 3 = 8 marks)
The diagram below shows a property where NJ Constructions has decided to build a caravan
park.Fencing forms the boundary on three sides of the property:
Side 1: the y axis
Side 2: the x axis
Side 3: the line HK, which cuts the x-axis and y-axisat H and K respectively.
There is a creek flowing through the property modelled by the curvey = a×x (5  x).
A(3, 3) is a point on thecurve.
y
K
(a) Findthe value of the coefficient a .
(b) Find the equation of the fence HK, if
the line HK is tangent to the curve at
A.
(c) Find the area of OHK, where O is
the origin.
(d) Find the ratio of the area P to the
area Q.
H

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