EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK. The index laws (
2). QUESTION 1. Use a calculator to find the value of: a 87 b 2.54 c 5291.5.
Logarithmic and Exponential Functions Studen
t Book - Series M-2
y
x
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Logarithmic and exponential functions Student Book - Series M 2 Contents Topics
Date completed
Topic 1 - The index laws
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Topic 2 - Logarithms
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Topic 3 - Change of base
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Topic 4 - The functions y = ax and y = log(ax)
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Topic 5 - The derivative of y = ax
__ /__ /__
Topic 6 - The number e and natural logarithms
__ /__ /__
Topic 7 - The derivative of y = ex
__ /__ /__
Topic 8 - The integral of ex
__ /__ /__
Topic 9 - The derivative of y = ln x
__ /__ /__
Topic 10 - The integral of 1x
__ /__ /__
Topic 11 - Applications of derivatives
__ /__ /__
Topic 12 - Applications of integrals of ex
__ /__ /__
Topic 13 - Applications of integration of 1x
__ /__ /__
Practice Tests Topic 1 - Topic test A
__ /__ /__
Topic 2 - Topic test B
__ /__ /__
Author of The Topics and Topic Tests: AS Kalra Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning
ii
CHAPTER 4
Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 128
Topic 1 - The index laws (1) The index laws (1) QUESTION 1
Use the index laws to simplify:
a
x2 × x5
b
a7 × a
c
4p12 ÷ 2p3
d
x2y3 × xy4
e
(a5)2
f
(3m2n)4
g
a0
h
6y0
i
x9 x3
j
6t8 2t 4
m (2a3b2)3 ÷ 4ab
QUESTION 2 a
23
e
42
1
QUESTION 3
k
n
x2 y5
l
x4 y4
15n9 ÷ 3n5 × 4n
o
2 ab 6 8 a 2b 6
(g4h3)2 × 2(gh2)3
Evaluate: b
104
f
83
2
c
31
d
60
g
25 2
h
320.8
c
4–4
d
10–5
g
16
h
1000
3
Write as fractions (in simplest form):
a
5–1
b
2–3
e
6–2
f
9
–1 2
–1 4
–2 3
95
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 128
Topic 1 - The index laws (2) The index laws (2) QUESTION 1 a
Use a calculator to find the value of:
87
e
2 455 489
QUESTION 2 2x
2.54
c
5291.5
f
5
g
(
371 293
3
0.027
)
4
d
2–7
h
3
(0.027) 4
Simplify:
a
5
÷ 5x
b
73x+4 × 79–3x
c
(3x)2 × (33x)3
d
82x ÷ 26x × 4x
e
93m+1 × 34m–1
f
32n ÷ 82n ÷ 43n
×5
3x
b
QUESTION 3
Solve:
a
k7 = 16 384
b
(25m)2 = 1 048 576
c
(1 – p)5 = 7776
d
93a = 37
e
5x × 253–x = 5
f
43q+5 × 82q–7 = 2
c
x2 × (2x3)3 = 15
QUESTION 4 a
8
x = 12 756
Find the value of x, correct to two decimal places, if: b
2x6 = 12.8
96
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2
Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 129
Topic 2 - Logarithms (1) Logarithms (1) QUESTION 1
Complete:
a
If loga x = c then x =
b
loga xy = loga x +
c
log a x y loga1 =
d
logaa =
f
loga xn =
e
QUESTION 2
Express as an integer:
a
log327
b
log232
c
log55
d
log71
e
log636
f
log10100 000
g
log7343
h
log2256
QUESTION 3
Simplify:
a
log62 + log63
b
log218 – log29
c
log 2 2
d
log520 + log52 – log58
e
log3504 – log37 – log38
f
log a 16 log a 4
c
4 logn3 – logn9
QUESTION 4 a
Express as a single logarithm:
3 loga2 + 2 loga3
b
logm12 + logm4 – logm8
97
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 129
Topic 2 - Logarithms (2) Logarithms (2) QUESTION 1
Complete: If y = ax then x =
QUESTION 2
Use a calculator to find the value, correct to three decimal places, of:
a
QUESTION 3 a
b
log1017
d
log100.35
b
c
loga10 000
logm10
b
logm8
c
logm2.5
d
logm12.5
b
log2x – log27 = 3
a
loga0.01
If logm2 = 0.289 and logm5 = 0.671, evaluate:
a
QUESTION 5
log101.65
If a2.37 = 10, find:
loga10
QUESTION 4
c
log10205
Find the value of x if:
log315 + log3x = log35
98
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4
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EXCEL HSC MATHEMATICS page 130
Change of base (1) Topic 3 - Change of base (1) QUESTION 1
Complete:
QUESTION 2
Simplify:
a
log927
d
2log816
QUESTION 3
log m a = log m b
b
log84
e
c
log432
log49 + log23
Find the value, correct to four decimal places, of:
a
log310
b
log715
c
log29
d
log411
e
log135
f
log689
g
log30.6
h
log20.75
i
log90.08
99
CHAPTER 4 – Logarithmic and exponential functions
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5
Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 130
Topic 3 - Change of base (2) Change of base (2) QUESTION 1 a
Express as a logarithm to the given base:
log411 (base 2)
QUESTION 2
b
log256 (base 5)
c
log2732 (base 3)
Find the value of x, correct to three decimal places:
a
3x = 17
b
2x = 75
c
5x = 0.275
d
2(6x) = 45
e
7x–1 = 16
f
3 – 2x = 0.37
100
EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS
Topic 4 - The functions y = a and y = log The functions y = a x and y = logaxa x
page 130
x
QUESTION 1 a
y=2
Sketch the graph of:
x
b
y = 7x
y
c y
x
QUESTION 2 a
y = 3–x y
x
x
Sketch the graph of: b
y = log10x y
c
y = log5x y
x
y = log2x y
x
x
QUESTION 3 a
On the same diagram sketch the graph of y = 3x and y = log3x
y
x
b
Complete: The graph of y = 3x and y = log3x are reflections of each other in the line 101
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS
xx Topic 5 - The derivative of y = a
page 131
The derivative of y = a
Fill in the blanks in the derivation from first principles of y = 10x
QUESTION 1
f ( x + h )– f ( x ) dy = lim dx h → 0 h
= lim
h
h→0
= lim
x
10 (
)
h→0
= 10 x lim
h→0
QUESTION 2 a
)
Use a calculator to find, to two decimal places, the approximate value of:
10 h – 1 lim h h→0
QUESTION 3 a
(
h
b
2h – 1 lim h h→0
c
3h – 1 lim h h→0
c
ln 3
Find, to two decimal places: b
ln 10
ln 2
QUESTION 4
ah – Use a calculator to find the value of a, to two decimal places, for which lim h h→0
QUESTION 5
e h – 1 Using a calculator, find lim h h→0
QUESTION 6
Write down the derivative of:
a
y = 5x
b
y = 7x
c
y = 4x
d
y = 11x
e
y = 6x
f
y = 9x
g
y = 8x
h
y = 15x
102
1 =1
EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK
Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning
8
Logarithmic andand exponential functionsfunctions Logarithmic exponential Topic 6 - The number e and natural logarithms The number e and natural logarithms QUESTION 1
Write down the exact value of:
a
e0
b
ln 1
c
ln e
d
ln e2
e
eln2
f
7 ln e
g
ln e7
h
eln5
QUESTION 2
Find the value, correct to four decimal places, of:
a
e2
b
e4
c
2e5
d
e–1
e
ln 1.25
f
ln 7.8
g
loge3.6
h
ln 0.237
i
6 loge4
j
4e3 + 1
k
e
l
4 ln 3 – 1
QUESTION 3
Find the value of k, correct to three decimal places, if:
a
ek = 1.6
b
ln k = 1.9
c
3ek = 5.87
d
7e2k = 6
e
5e–4k = 3
f
10e3k+1 = 0.456
103
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS
xx
page 131
Topic 7 - The derivative of y = e (1) The derivative of y = e (1) QUESTION 1
Find the derivative of:
a
y = ex
b
y = 3ex
c
f(x) = e2x
d
y = 4ex + 3
e
y = 2e5x
f
y = e–x
g
y = x – ex
h
y = 6e2x+5
i
f(x) = 4e–8x
j
y = 6x3 – 3e3x
k
y = ex – e–x
l
f(x) = 6 – 7e–9x
QUESTION 2
Use the product rule to differentiate:
a
y = xex
b
y = x2e2x
c
y = (3x – 4)e–x
d
y = 5e7x(x2 – 9x + 2)
104
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EXCEL HSC MATHEMATICS
x x (2) Topic 7 - The derivative of y = e
page 131
The derivative of y = e (2) QUESTION 1 a
y = (ex + 5)4
QUESTION 2 a
y=
b
f (x) = (4x – ex)3
b
the quotient rule
b
y=
Differentiate y = x , using: ex
the product rule
QUESTION 3 a
Find the derivative of:
Differentiate:
ex x+1
3e x x2 – 5
105
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS
x x Topic 8 - The integral of e
page 132
The integral of e QUESTION 1 a
∫e
d
∫e
g
∫
Find: b
∫ 5e
dx
e
∫
( e x + 2 x ) dx
h
∫
x
dx
2 x +3
QUESTION 2 a
d
g
∫
2
∫
–1
∫
2
0
–3
1
c
∫e
4 e – x dx
f
∫e
e 4 x dx 2
i
∫ (x
6 e x dx
c
1 e –2 x dx 2
f
x
dx
3x
dx
3–2 x
2
dx
– 8 x – 6 e –2 x ) dx
Find the exact value of:
e x dx
b
e 2 x+7 dx
e
( e x – e – x ) dx
h
∫
1
∫
2
∫
ln2
0
0
0
7 e x dx
106
i
∫
3
∫
3
∫
2
0
1
1
e 4 x dx
e 4– x dx
( e 3 x + x ) dx
EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 133
Topic 9 - The derivative of y = ln x (1) The derivative of y = ln x (1) QUESTION 1
Differentiate:
a
y = loge x
b
y = loge 2x
c
f(x) = ln 6x
d
y = loge(7x + 5)
e
y = ln (1 – 2x)
f
y = ln (5x + 3)
g
y = ln x2
h
f(x) = ln x5
i
y = ln x9
j
f(x) = ln (x2 + 5)
k
y = ln (3x2 – 4)
l
y = loge (x3 – 7x2)
QUESTION 2 a
Find the derivative of:
y = (ln x)2
b
f(x) = loge (3x – 1)2
107
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 133
Topic 9 - The derivative of y = ln x (2) The derivative of y = ln x (2) QUESTION 1 a
c
b
f(x) = ln x
QUESTION 2 a
Find the exact value of f′(e) if:
y = x ln x
ln x x2
c
f(x) = loge (2x – 1)
f(x) = 3 ln (x2 + 1)
Differentiate: b
d
y = x4 loge x
x+1 ln x
108
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Logarithmic and exponential functions
Logarithmic and exponential functions 1
EXCEL HSC MATHEMATICS page 134
1 The integral of Topic 10 - The integral of x
x
QUESTION 1
Find:
a
∫
dx x
b
d
∫
2 x dx x +5
g
∫
8 x dx x2 – 3
2
QUESTION 2 a
c
∫
5
∫
3
2
0
∫
6 dx x
c
∫
e
∫
3 x 2 dx x3 – 2
f
∫ 3 x – 7 dx
h
∫
dx 4x – 1
i
∫ 1 – 2 x dx
3 dx x+2
3
7
Find the exact value of:
dx x–1
b
2 x dx x2 + 3
d
∫
e
∫
4
1
2
dx 2x
3x2 + 1 x3 + x
dx
109
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS page 135
Topic 11 - Applications of derivatives (1) Applications of derivatives (1) QUESTION 1
Find the equation of the tangent to the curve y = 2 ln x at the point where x = e
QUESTION 2
Find the equation of the normal to the curve y = 2e–x at the point where x = 1
QUESTION 3
The tangent to the curve y = ex at the point P meets the x-axis at an angle of 45°. Find the coordinates of P.
QUESTION 4
Find the maximum value of
ln x x
110
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EXCEL HSC MATHEMATICS page 135
Topic 11 - Applications of derivatives (2) Applications of derivatives (2) QUESTION 1
Consider f(x) = ex(1 – x)
a
Where does the curve y = f(x) cross the x-axis?
b
Find any stationary points and determine their nature.
c
Find any points of inflexion.
d
Complete: i
e
as x → ∞, y →
ii as x → –∞, y →
Sketch the curve y = f(x) y
x
111
CHAPTER 4 – Logarithmic and exponential functions
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Logarithmic andand exponential functionsfunctions Logarithmic exponential
EXCEL HSC MATHEMATICS
Topic 12 - Applications of integrals of exx
pages 135–136
Applications of integrals of e QUESTION 1 a
Find the area bounded by the curve y = ex, the x-axis, x = 0 and x = ln 3
y y = ex
3
b
0
Hence find the shaded area.
ln 3
x
QUESTION 2
A curve y = f(x) has a turning point at (0, 4). If f ″(x) = ex + e–x find the equation of the curve.
QUESTION 3
Show that the volume of the solid of revolution formed by rotating the curve y = ex, between x = 0 and x = 5 about the x-axis is given by π ( e 10 – 1) units3. 2 y y = ex
0
112
5
x
EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK
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18
Logarithmic and exponential functions
EXCEL HSC MATHEMATICS
Logarithmic and exponential functions 1 Applications of integration of
pages 135–136
Topic 13 - Applications of integration of 1xx
QUESTION 1
Find the exact area bounded by the curve y = 4 , the x-axis and the ordinates x = 2 and x = 4 x y
x
QUESTION 2
The gradient function of a curve is given by 6 x – passes through the point (1, 7).
QUESTION 3
2 . Find the equation of the curve if it 2x – 1
Find the area shaded in the diagram.
y
y = 2 x
2 1 0
x
113
CHAPTER 4 – Logarithmic and exponential functions
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19
Logarithmic and exponential functions
Logarithmic TOPIC TEST Topic Test
and exponential functions PART A
Time allowed: 1 hour Instructions
Total marks = 100
This part consists of 12 multiple-choice questions Each question is worth 1 mark
SECTION I Multiple-choice questions Calculators may be used
12 marks
Instructions This section consists of 12 multiple-choice questions Fill in only ONE CIRCLE for each question Each question is worth 1 mark Fill in only ONE CIRCLE Time allowed: 30 minutes Calculators may be used
Total marks = 12
1 23 × 2 2 = ? A 25
B
26
C
45
D
46
B
18
C
84
D
86
B
2
C
7
D
8
2 88 ÷ 8 2 = ? A 14
3 7m0 + 70 = ? A 1
4 p–3 = ? A
5
3
p
B
p3
C
1 p3
D
none of these
xn
C
xm xn
D
none of these
1
C
m2 m3
D
m2 3 m
m
xn =? A
6 m
–2 3
n
xm
B
m
=? 1
A
m3
B
3
m2
7 log42 = ? A
1 2
B
1
C
2
D
4
B
loga4.5
C
2 loga1.5
D
cannot be simplified
6.581
D
7.389
8 2 loga3 – loga2 = ? A loga7
9 The value of e2 correct to three decimal places is? A 0.301
B
0.693
C
114
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Logarithmic and exponential functions Topic Test 10 10
PART A
d (e 2 x ) = ? dx d (e 2 x ) = ? 2x A dx e
C
2e2x
11 The could be aBsketch e2x 2ex of the graph of: C A diagram
2e2x
B
2ex
1 e2x 2 1 e2x 2
D y
x
y = 2 could be a sketch of the graph of: A diagram 11 The
D
y = f(x)
y
B y = 2x–x A
y = f(x)
1
–x x C y = 2 log B 2
2 ln2xx D y = log C D y = 2 ln x
01
1
2
x
0
1
2
x
12 log27 = ? ln 7 12 log A 27 = ? ln 2 ln 7 A ln 2
B B
ln 2 ln 7 ln 2 ln 7
C
2 ln 7
D
7 ln 2
C
2 ln 7
D
7 ln 2
Total marks achieved for PART A 12 88 marks
SECTION II
Topic Test SECTION II
PART 88 marksB
Show all necessary working
Show all necessary working 13 Simplify: Instructions Show all necessary working Time allowed: 30 minutes 2–x
b
log645 + log620 – log625
8x+1 × 25x ÷ 42–x
b
log645 + log620 – log625
x+1
a 8 ×2 13 Simplify: a
5x
÷4
14 Find x if: b
(1 – x)3 = 0.512
c
logx16 = 4
x8 = 1 679 616
b
(1 – x)3 = 0.512
c
logx16 = 4
15 Find, correct to three decimal places: a log b 15 Find, correct 102.9 to three decimal places: a
log102.9
1 mark each
1 mark each
a x8 = 1 679 616 14 Find x if: a
1 mark each Total marks = 88
b
1 mark each
1 mark each 5
9.3187
c
log211
5
9.3187
c
log211
CHAPTER 4 – Logarithmic and exponential functions
Logarithmic and exponential functions CHAPTER 4 – Logarithmic and exponentialMathletics functions Instant Workbooks – Series M 2 Copyright © 3P Learning
1 mark each
115 11521
Logarithmic and exponential functions Topic Test
PART B
16 If loga3 = 0.565 and loga2 = 0.356 find:
1 mark each
a
b
loga6
c
loga9
loga1.5
17 Find the value of x, correct to three decimal places, if: a
5x = 424
b
2 marks each
1 – 3x = 0.57
c
6e2x+1 = 192
18 Write down the exact value of: a
1 mark each b
9 ln e
ln e4
c
eln8
19 Sketch the graph of: a
2 marks each
y = 8x
b
y = log8x
y
y
x
x
Logarithmic and exponential functions
116
22
EXCEL HSC MATHEMATICS REVISION & EXAM WORKBOOK Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning
Logarithmic and exponential functions Topic Test
PART B
20 Differentiate: a
y=7
d
g
x
1 mark each b
y=e
y = 3e–2x
e
y = 5e7x–4
h
x
c
y = ln x
y = ln (5x – 4)
f
y = ln (x2 + 6x)
y = 4 loge(6 – 3x)
i
y = ex
21 Find the derivative of: a
y = x3e2x
c
e 6x – 1
6x
2
3 marks each b
y = 2x logex
d
ln x 4x + 1
Logarithmic and exponential functions CHAPTER 4 – Logarithmic and exponential functions Instant Workbooks – Series M 2 Copyright © 3P Learning Mathletics
117 23
Logarithmic and exponential functions Topic Test
PART B
22 Find:
2 marks each
a
∫
3 dx x
b
∫e
d
∫
e –2 x dx 2
e
∫ 2 x + 1 dx
8x
dx
3
c
∫ 2x
f
∫
23 Find the exact value of: a
c
e
∫
e
∫
5
∫
3
1
1
1
4 x dx 2 –3
1 e 5–3 x dx 4
3 marks each
1 dx x
b
dx ex
d
4 dx 7 – 2x
f
∫
1
∫
4
∫
2
0
1
0
e x dx 2
2 x + 5 dx x2 + 5x
e 3 x–4 dx
Logarithmic and exponential functions
118
24
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Logarithmic and exponential functions
24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0
Topic Test
3 marks
at the point where x = 0
3 marks PART B
24 Find the equation of the tangent to the curve y = 2ex+1 at the point where x = 0
3 marks
25 Find the coordinates of the stationary point of the curve y = x ln x
4 marks
25 Find the coordinates of the stationary point of the curve y = x ln x
4 marks
25 Find the coordinates of the stationary point of the curve y = x ln x
4 marks
26 Find the area bounded by the curve y = 1 , the x-axis and the lines x = 1 and x = 5 x 26 Find the area bounded by the curve y = 1 , the x-axis and the lines x = 1 yand x = 5 x y 1 26 Find the area bounded by the curve y = , the x-axis and the lines x = 1 and x = 5 x
3 marks
24 Find the equation of the tangent to the curve y = 2e
x+1
3 marks 3 marks
y
x x –x
27 Find the volume of the solid of revolution formed when that portion of the curve y = e between x = –1 x and xthe = 1volume is rotated about 3 xmarks 27 Find of the solidthe of x-axis. revolution formed when that portion of the curve y = e–x between = –1 and x = 1 is rotated about the x-axis. 3 marks 27 Find the volume of the solid of revolution formed when that portion of the curve y y= e–x between x = –1 and x = 1 is rotated about the x-axis. 3 marks y = e–x y y = e–x
y
y = e–x
CHAPTER 4 – Logarithmic and exponential functions
–1
0
1
x
–1
0
1
x
–1
0
1
x
Total marks achieved for PART B
119
88
119
CHAPTER 4 – Logarithmic and exponential functions
Logarithmic and exponential functions
CHAPTER 4 – Logarithmic and exponential functions Mathletics Instant Workbooks – Series M 2 Copyright © 3P Learning
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