Applications of Differential Calculus Exercise
Application of Differential Calculus
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5A (LHospitals Rule)Date:
Name:________________
1.Evaluate
2.Evaluate the following limits:
(a)
(b)
(c)
3.Evaluate the following limits:
(a)
(b)
(c)
4.Evaluate the following limits:
(a)
(b)
(c)
5.Evaluate the following limits:
(a)
(b)
(c)
Ans:
1.
2.
, ,
3.1, , 14.0, 0,
5.
, ,
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5B(LHospitals Rule)Date:
Name:________________1Evaluate the following limits:
(a)
(b)
2.Evaluate:
(a)
(b)
[HKAL 1994]
3.Evaluate:
(a)
(b)
[HKAL 1998]
Ans:
1.
,
2.
,
3.
,
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5C (Monotonic Functions)Date:
Name:________________1.Show that the function is strictly
increasing for .
2.Show that the function is strictly increasing for .
3.Determine the interval for which the function is
increasing.
4.Determine the interval in for which the function is
decreasing.
Ans:
3.
,
4.
.
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5D(Monotonic Functions)Date:
Name:________________1.Prove for .
2.Prove that for
3.Let .
(a)Show that is strictly increasing on the interval .
(b)Hence, show that if ,
4.Let .
By finding the greatest value of , prove that .
5.(a)Show that for
(i)
(ii)
(b)Let be a positive integer greater than . Deduce from (a)
that
Hence show that
(c)Use the above results to evaluate the limit
(Ans:)
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5E (Maxima and Minima)Date:
Name:________________
1.Find the maximum or minimum points of and .
x
Maximum point=
Minimum point=
2.Find the maximum or minimum points of .
3.Find the maximum or minimum points of .
4.Find the maximum or minimum points of .
5.Find the maximum or minimum points of
Ans:
1.Max. pt ()Min. pt ()
2.Min. pt ()
3.Min. pt ()
4.Max. pt ()Min. pt ()
5.Min. pt ( 0, 0 )
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5F (points of inflexion)Date:
Name:________________1.Find the points of inflexion of the curve
and .
(,)is point of inflexion.
2.Find the points of inflexion of the curve .
3.Find the points of inflexion of the curve .
4.Find the points of inflexion of the curve , ().
5.Find the points of inflexion of the curve .
Ans:
1.
2.
3.no point of inflexion4.
5.
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5G(asymptotes)Date:
Name:________________1.Find the asymptotes to the curve .
2.Find the asymptotes to the curve .
3.Find the asymptotes to the curve .
4.Find the asymptotes to the curve .
Ans:
1.
2.
3.
4.
.
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5H(Curve Sketching)Date:
Name:________________1.Let
(a)
For and , find and
(b)
Show that but both and do not exist.
(c)Show that the graph of has extreme points at and and has
inflexional points at and .
(d)
Sketch the graph of .
Vision Ex 5.10(12)
2.Let .
(a)Show that and do not exist.
(b)Find for and .
(c)Find the range of values of such that
(i)
,
(ii)
(d)Find the maximum, minimum and inflexional points of the graph
of .
(e)Find the asymptote(s) of the graph of .
(f)Sketch the graph of .
Vision Ex 5.10(14)
A-Level Pure Mathematics
Chapter 5Application of Differential Calculus
Exercise 5I(Curve Sketching)Date:
Name:________________1.Let
()
(a)
(i)Evaluate for . Prove that does not exist.
(ii)Determine those values of for which and those values of for
which
.
(iii)Find the relative extreme points of .
(b)
(i)Evaluate for . Hence determine the points of inflexion of
.
(ii)Find the asymptote of the graph of .
(c)
Using the above results, sketch the graph of .
HKAL 94 Paper II2.Let
(a)Find and for .
(b)Determine the range of values of for each of the following
cases:
(i)
,
(ii)
,
(iii)
(iv)
.
(c)Find the relative extreme point(s) and point(s) of inflexion
of .
(d)Find the asymptote(s) of .
(e)Sketch the graph of .
(f)Let
(i)Is differentiable at ? Why?
(ii)Sketch the graph of .
HKAL 02 Paper II5
_1128106360.unknown
_1128108756.unknown
_1128109574.unknown
_1128173939.unknown
_1128174052.unknown
_1128174202.unknown
_1128174258.unknown
_1128174359.unknown
_1128713931.unknown
_1128174389.unknown
_1128174286.unknown
_1128174252.unknown
_1128174088.unknown
_1128174113.unknown
_1128174083.unknown
_1128174001.unknown
_1128174033.unknown
_1128174046.unknown
_1128174009.unknown
_1128173966.unknown
_1128173991.unknown
_1128173949.unknown
_1128109819.unknown
_1128110697.unknown
_1128173452.unknown
_1128173612.unknown
_1128173852.unknown
_1128173930.unknown
_1128173622.unknown
_1128173679.unknown
_1128173575.unknown
_1128173587.unknown
_1128173527.unknown
_1128110740.unknown
_1128110760.unknown
_1128110726.unknown
_1128110539.unknown
_1128110553.unknown
_1128110554.unknown
_1128110552.unknown
_1128110436.unknown
_1128110447.unknown
_1128110474.unknown
_1128109855.unknown
_1128109776.unknown
_1128109789.unknown
_1128109688.unknown
_1128109037.unknown
_1128109171.unknown
_1128109447.unknown
_1128109560.unknown
_1128109182.unknown
_1128109120.unknown
_1128109130.unknown
_1128109096.unknown
_1128108794.unknown
_1128108935.unknown
_1128109023.unknown
_1128108912.unknown
_1128108779.unknown
_1128108783.unknown
_1128108776.unknown
_1128107582.unknown
_1128108126.unknown
_1128108280.unknown
_1128108331.unknown
_1128108726.unknown
_1128108322.unknown
_1128108203.unknown
_1128108229.unknown
_1128108192.unknown
_1128107763.unknown
_1128107997.unknown
_1128108078.unknown
_1128107883.unknown
_1128107761.unknown
_1128107762.unknown
_1128107759.unknown
_1128107760.unknown
_1128107658.unknown
_1128106824.unknown
_1128106880.unknown
_1128106964.unknown
_1128107019.unknown
_1128106936.unknown
_1128106845.unknown
_1128106859.unknown
_1128106835.unknown
_1128106573.unknown
_1128106644.unknown
_1128106814.unknown
_1128106620.unknown
_1128106630.unknown
_1128106600.unknown
_1128106538.unknown
_1128106560.unknown
_1128106522.unknown
_1128101990.unknown
_1128105791.unknown
_1128105959.unknown
_1128106217.unknown
_1128106295.unknown
_1128106199.unknown
_1128105935.unknown
_1128105944.unknown
_1128105816.unknown
_1128105618.unknown
_1128105690.unknown
_1128105738.unknown
_1128105671.unknown
_1128102074.unknown
_1128105596.unknown
_1128102012.unknown
_1128102057.unknown
_1128101999.unknown
_1128101639.unknown
_1128101868.unknown
_1128101904.unknown
_1128101979.unknown
_1128101889.unknown
_1128101828.unknown
_1128101860.unknown
_1128101799.unknown
_1128100959.unknown
_1128101143.unknown
_1128101634.unknown
_1128101636.unknown
_1128101637.unknown
_1128101250.unknown
_1128101352.unknown
_1128101420.unknown
_1128101163.unknown
_1128101027.unknown
_1128101087.unknown
_1128100965.unknown
_1128100577.unknown
_1128100755.unknown
_1128100840.unknown
_1128100855.unknown
_1128100828.unknown
_1128100703.unknown
_1128100730.unknown
_1128090496.unknown
_1128099279.unknown
_1128099548.unknown
_1128099553.unknown
_1128100023.unknown
_1128098808.unknown
_1128098834.unknown
_1128098894.unknown
_1128098965.unknown
_1128098873.unknown
_1128098819.unknown
_1128098578.unknown
_1128098791.unknown
_1128098500.unknown
_1003691584.unknown
_1128090488.unknown
_1003691593.unknown
_1003690388.unknown
