M_Bank\YR12-2U\Logarithmic&Exponential.CAT
Logarithmic and Exponential Functions
1)! ¥2U84-1i¥
Express the following as integers.
a. 52´23´10-2.
b. .
c. .†
«® a)2 b)2 c)3 »
2)! ¥2U84-4i¥
Differentiate with respect to x:
a. 7tanx;
b. ;
c. xesinx.†
«® a)7 sec2x b) c) »
3)! ¥2U84-4ii¥
If f(x)=log10x, find f¢(x)and hence f¢(50)correct to 3 significant figures.†
«® f¢(x)=, 8·69´10-3 »
4)! ¥2U84-6i¥
Find the following indefinite integrals:
a. ;
b. ;
c. dx.†
«® a)sin2x+C b)ln(2x2+5x)+C c) »
5)! ¥2U84-10¥
i. Expand e-x(1-e-x).
ii. For the curve y=e-x-e-2x
a. Find where it cuts the axes.
b. Find the coordinates of any stationary points and determine their nature.
c. Determine the values of x for which the curve is monotonic decreasing and hence or otherwise discuss the behaviour of the curve for large values ofx.
d. Sketch the curve.
e. Calculate the area bounded by this curve, the x-axis and the ordinates x=0 and x=log2.†
«® i)e-x-e-2x ii)a)(0, 0) b) is a maximum turning point. c)xln2. As x approaches +¥ the curve asymptotically approaches the xaxis. d) e)units2 »
6)! ¥2U85-3iii¥
Sketch the curves in separate diagrams:
a. y=log(x+2).
b. y=.†
«® a)b) »
7)! ¥2U85-5i¥
Differentiate the following:
a. (5-2x)3;
b. ;
c. .†
«® a)-6(5-2x)2 b) c) »
8)! ¥2U85-5ii¥
Find the primitive function of the following:
a. (x2-2)2;
b. ;
c. .†
«® a)x5-x3+4x+C b)x2-2lnx+C c)ln(x2-2)+C »
9)! ¥2U85-10i¥
The rate at which liquid is flowing into a vessel after t minutes is given by . If (loge2)m3 of liquid flows into the vessel after 3minutes, how much liquid flows in after 8minutes? Give your answer to 3significant figures.†
«® 1·50 m3 »
10)! ¥2U85-10ii¥
Sketch the curve y=xex showing all stationary points and asymptotes. Hence find value(s)ofk for which xex=k has:
a. no solutions;
b. one solution;
c. two solutions.†
«® a)k0·36 b)k³0 and k=-0·36 c)-0·36k0 »
11)! ¥2U86-2i¥
Find the derivatives of the following:
a. (x2+5)tanx;
b. loge(6x2-3);
c. (cosx+sinx)3.†
«® a)(x2+5)sec2x+2xtanx b) c)3(cosx+sinx)2(cosx-sinx)»
12)! ¥2U86-7i¥
Find:
a. ;
b. ;
c. .†
«® a)(3x+5)3 +C b)cos(3x-1)+C c)ln(ex+5)+C »
13)! ¥2U86-10i¥
Given that a2+b2=23ab, express in terms of aandb. Hence show that .†
«® Proof »
14)! ¥2U86-10ii¥
Consider the function y=:
a. For what values of x is this function defined?
b. Describe the behaviour of the function as x:
a. approaches zero.
b. increases indefinitely.
c. Find any stationary points and determine their nature.
d. Sketch the curve of this function.†
«® a)All real numbers, x¹0 b)a)As x approaches zero, the function approaches ±¥ b)Asx approaches infinity, the function approaches ±¥ c)x=-1 is a maximum. d) »
15)! ¥2U87-7i¥
Evaluate:
a. .
b. .†
«® a)ln3 b) »
16)! ¥2U87-7iii¥
Find the volume of the solid formed when the area bounded by the lines x=0 and x=1 and the curve y=ex is rotated about the xaxis. Leave your answer in exact form.†
«® »
17)! ¥2U87-8i¥
Differentiate with respect tox:
a. 7x2-;
b. 4(3x-5)6;
c. 2xcosx;
d. .†
«® a)14x+ b)72(3x-5)5 c)-2xsinx+2cosx d) »
18)! ¥2U87-8ii¥
For the curve y=loge(x-1):
a. write down the domain of this function;
b. use calculus to show that it:
a. has no stationary points;
b. is always increasing;
g. is always concave down.
Hence sketch the curve.†
«® a)x1 b) »
19)! ¥2U87-8iii¥
For what value ofx is the tangent to the curve y=e3x parallel to the line y=6x.†
«® »
20)! ¥2U88-4i¥
Find the derivatives of the following:
a. e2x+1;
b. ;
c. x2sin3x.†
«® a)2e2x +1 b) c)3x2cos3x+2xsin3x »
21)! ¥2U88-8i¥
Find a primitive function of the following:
a. x2-;
b. 3sec22x;
c. .†
«® a) b)tan2x+C c)ln(x3-2)+C »
22)! ¥2U88-8iii¥
The curve y=logex between the lines x=1 and x=3 is rotated about the yaxis. Find the volume of the solid formed. (Leave your answer in terms of p.)†
«® 4punits3 »
23)! ¥2U88-10ii¥
Consider f(x)=.
a. What is the domain of f(x)?
b. Find f¢(x)and hence determine all stationary points.
c. Sketch the curve of y=f(x)clearly showing all its essential features.†
«® a)x0 b)f¢(x)= c) »
24)! ¥2U89-5a¥
Differentiate:
i. 2x3--4;
ii. xtanx;
iii. .†
«® i)6x2+ ii)xsec2x+tanx iii) »
25)! ¥2U89-10a¥
Simplify:
i. ;
ii. .†
«® i) ii)4 »
26)! ¥2U89-10b¥
A normal is drawn to the curve y=e2x at the point P(loge2,4). The normal cuts the x-axis atQ.
i. Show that the equation of the normal atP is x+8y=32+ loge2.
ii. Find the co-ordinates ofQ.
iii. On a diagram, shade in the region bounded by the curve y=e2x, the normal atP and the co-ordinate axes.
iv. Find the area of the shaded region.†
«® i)Proof ii)Q(32+loge2, 0) iii) iv)Area=65units2 »
27)! ¥2U90-2b¥
Find the co-ordinates of the stationary point on the curve y=.†
«® (0,1)»
28)! ¥2U90-5b¥
Find the exact value of .†
«® loge2 »
29)! ¥2U90-6a¥
Simplify ln27÷ln81.†
«® »
30)! ¥2U90-6b¥
Evaluate log311 correct to 2significant figures.†
«® 2·2 »
31)! ¥2U90-6c¥
Find the exact value of x if: 2(x-1)=7.†
«® log214 »
32)! ¥2U90-8c¥
Find the equation of the tangent drawn to the curve y=ex+3 at the point (2,e5).†
«® y=e5(x-1)»
33)! ¥2U91-3a¥
Differentiate with respect to x:
i. 2x3+4-1;
ii. xe3x;
iii. .†
«® i)6x2+ ii)e3x(3x+1) iii) »
34)! ¥2U91-6c¥
Find:
i. ;
ii. .†
«® i)t3 +2lnt+C ii)e2x+C »
35)! ¥2U92-4c¥
Find the equation of the normal to the curve y=lnx2 at the point x=1.†
«® x+2y-1=0 »
36)! ¥2U92-6a¥
Find the following:
i. ;
ii. .†
«® i)x+4ln x+C ii)tan 3x+C »
37)! ¥2U93-3a¥
Differentiate:
i. 4x3+7;
ii. xe2x;
iii. .†
«® i)12x2 ii)e2x(2x+1) iii) »
38)! ¥2U93-4b¥
Find:
i. ;
ii. .†
«® i)ln(x+5)+C ii)tan3x+C »
39)! ¥2U94-3a¥
Find:
i. ;
ii. .†
«® i)3lnx+C ii) »
40)! ¥2U94-4a¥
Differentiate with respect to x:
i. ln(3x-5);
ii. sin2x;
iii. xe3x.†
«® i) ii)2cos2x iii)e3x(3x+1)»
41)! ¥2U95-3b¥
Differentiate:
i. ln(x3+7);
ii. x2e3x;
iii. .†
«® i) ii)xe3x(3x+2) iii) »
42)! ¥2U95-7b¥
Differentiate y= and hence evaluate .†
«® (ln2)2 »
43)! ¥2U95-8c¥
Find the co-ordinates of the point on the curve y=e3x where the tangent is perpendicular to the line y=x+4.†
«® »
44)! ¥2U96-2a¥
Differentiate:
i. sin(3x-2);
ii. .†
«® i)3cos(3x-2) ii) »
45)! ¥2U96-2c¥
Find
i. ;
ii. ;
iii. .†
«® i)tan3x+C ii) iii)ln3 »
46)! ¥2U96-4a¥
Find the equation of the tangent to the curve y=xlnx at the point (1,0).†
«® y=x-1 »
47)! ¥2U96-9c¥
The diagram shows the area bounded by the graph y=lnx, the co-ordinate axes and the line y=ln3. †
i. Find the shaded area.
ii. Hence find the exact value of .†
«® i)2units2 ii)3ln3-2 »
48)! ¥2U97-1a¥
Find the value of log39.†
«® 2 »
49)! ¥2U97-3a¥
Differentiate:
i. ;
ii. (ex-e-x)2;
iii. .†
«® i) ii)2e2x-2e-2x iii) »
50)! ¥2U97-3c¥
Evaluate .†
«® 0·5(e2-1)»
51)! ¥2U97-8a¥
For the curve y=ln(x-2):
i. write down its domain;
ii. sketch the curve.†
«® i)x2 ii) »
52)! ¥2U97-9a¥
Using loga2=0·387 and loga3=0·613, find the value of loga12.†
«® 1·387 »
53)! ¥2U98-3a¥
Differentiate with respect to x:
i. (4-3x)6;
ii. x2e2x;
iii. .†
«® i)18(3x–4)5 ii)2x(x+1)e2x iii) »
54)! ¥2U98-3c¥
Find the equation of the normal to the curve y=lnx at the point where x=1.†
«® y=-x+1 »
55)! ¥2U98-8b¥
Evaluate .†
«® 0·5ln5 »
56)! ¥2U99-2a¥
Differentiate the following functions:
i.
ii. (x+1)lnx
iii. .
«® i) ii) iii) »
57)! ¥2U99-2b¥
Find:
i.
ii. .†
«® i) ii) »
58)! ¥2U99-6a¥
The diagram shows part of the hyperbola and the line y=4–x. The hyperbola and line intersect at the points (0,4) and (3,1). Calculate the exact area of the shaded region.
«® units2 »
59)! ¥2U99-8b¥
The region bounded by the curve y=ex+e–x, the xaxis and the line x=0 and x=2 is rotated about the xaxis. Find the volume of the solid formed. (Leave your answer in terms ofe.)†
«® »
60)! ¥2U00-3a¥
Differentiate the following expressions with respect tox:
i. (3x2+2)3
ii. 3xcos2x
iii. †
«® i)18x(3x2+2)2 ii)–6xsin2x+3cos2x iii) »
61)! ¥2U00-3b¥
Evaluate the following definite intervals:
i.
ii. †
«® i) ii) »
62)! ¥2U00-3c¥
Find .†
«® »
63)! ¥2U00-6a¥
Find all the values of xfor which lnx=2lnx.†
«® x=1 »
64)! ¥2U01-3a¥
Find:
i.
ii. .†
«® i) ii) »
65)! ¥2U01-3b¥
Evaluate .†
«® loge4 »
66)! ¥2U01-7c¥
i. Without using calculus, sketch the graph of y=ex–3.
ii. On the same sketch, find graphically the number of solutions of the equation ex–3=-x2. †
«® i) ii)2solutions »
67)! ¥2U01-8a¥
Differentiate y=log2x.†
«® »
68)! ¥2U02-1e¥
- Solve 4x=32.
- Hence, or otherwise, write down the value of log432.†
«® i)x=2×5 ii)2×5 »
69)! 2¥2U02-2a¥
Differentiate with respect tox:
i.
ii.
iii. †
«® i) ii) iii) »
70)! 2¥2U02-3a¥
†
«® »
71)! 2¥2U02-3c¥
Find the equation of the normal to the curve at the point (e,e).†
«® x+2y–3e=0 »
72)! 2¥2U02-6b¥
Calculate the exact volume generated when the region enclosed by the curve y=1+2e–x for 0£x£1, is rotated about the x axis.†
«® p(7–4e–1–2e–2)cubic units »
73)! ¥2U03-1f¥
Sketch the curve y=2e–x, clearly showing where the curve cuts the y-axis.†
«® »
74)! ¥2U03-3a¥
Differentiate with respect to x:
i.
- x3e–3x
- .†
«® i) ii)3x2e–3x(1–x) iii) »
75)! ¥2U03-3b¥
Find .†
«® »
76)! ¥2U03-4d¥
Solve forx the equation .†
«® »
77)! ¥2U03-5c¥
Given that y=3e–2x, show that .†
«® Proof »
78)! ¥2U03-7a¥
Find given that .†
«® »
79)! ¥2U03-10a¥
i. Show that .
ii. Hence, or otherwise, find
iii. The graph shows the curve y=1nx2, (x0) which meets the line x=5 at Q. Using your answers from (i) and (ii), or otherwise, find the area of the shaded region.
†
«® i)Proof ii)2(xlnx–x)+constant iii)8units2 »
80)! ¥2U03-10b¥
Consider the function f(x)=e–xcosxfor0£x£2p.
- Find the x values where the stationary points occur.
- Determine the nature of the stationary points.
- Sketch the curve showing the coordinates of the stationary points in exact form and the intercepts with the axes.
- Find the number of solutions to the equation in the domain 0£x£2p. Justify your answer. †
«® i) ii)Minimum turning point at . Maximum turning point at . iii) iv)One »
81)! 2U¥04-2¥a
Write down the derivatives of:
i. (3x+4)7
ii. x3ex
iii. .†
«® i) 21(3x+4)6 ii) x2ex(x+3) iii) »
82)! 2U¥04-2b¥
i. Write down the primitive function of .
ii. Find the exact value of .
iii. Given that =2x–sinx and y=2 when x=0, findy in terms ofx.†
«® i) ii) iii) y=x2+cosx+1 »
83)! 2U¥04-7b¥
i. Sketch the graph of f(x)=ex for all values ofx in the domain and state its range.
ii. The curve f(x)=ex is rotated about the y-axis to give a solid. Show that the volume Vy of the solid formed, from y=3 to y=5, is given by
iii. Use Simpson’s rule with 5function values to find the volume of this solid, correct to 2significant figures.†
«® i) ii) Proof iii) 12units3 »
84)! ¥2U05-1e¥
Sketch the curve y=ex. State its range.†
«® , Range: y0 »
85)! ¥2U05-3b¥
Differentiate the following functions:
i. sinxlogex.
ii. .†
«® i) ii) »
86)! ¥2U05-3c¥
Find:
i. .
ii. leaving your answer in exact form.†
«® i) cos(e–x)+c ii) ln2 »
87)! ¥2U05-3d¥
Find the equation of the normal to the curve y=e4x–1 at the point on the curve where x=0.†
«® y=–x »
88)! ¥2U05-6a¥
i. Factorise the expression 2a2–7a+3.
ii. Hence, solve the following equation for x: 2(log2x)2–7(log2x)+3=0.†
«® i) (2a–1)(a–3) ii) x= or x=8 »
89)! ¥2U05-10a¥
i. Simplify logee2ax
ii. Hence evaluate †
«® i) 2ax ii) a3 »
90)! ¥2U06-3b¥
Differentiate with respect tox the following expressions:
i. 3x1ogex.
ii. sin2x.†
«® i) 3+3logex ii) 2sinxcosx »
91)! ¥2U06-3c¥
Find:
i. .
ii. . (Leave your answer in exact form).†
«® i) ii) »
92)! ¥2U06-6b¥
The shaded region bounded by the graph , the line y=5 and the y- axis is rotated about the y- axis to form a solid of revolution.
i. Show that the volume of the solid is given by .
- Copy and complete the following table into your writing booklet. Give all answers correct to three decimal places.
y / 1 / 2 / 3 / 4 / 5
loge y / 0 / 0×693 / 1×099 / 1×609
iii. Use Simpson’s Rule with five function values to approximate the volume of the solid of revolutionVy, correct to three decimal places.†
«® i) Proof ii) 1∙386 iii) 12∙695cubic units »
93)! ¥2U06-7a¥
Solve the following equation: log2x+log2(x+7)=3, forx0.†
«® x=1 »
†©CSSA of NSW 1984 - 2006
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