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CEGEP CHAMPLAIN - ST. LAWRENCE
201-103-RE: Differential Calculus
Patrice Camiré
Problem Sheet #21
Linear Approximation
1. (a) Give the equation of the tangent line to the curve y = f (x) at the point xo .
(b) State the main principle behind the method of linear approximation.
2. Use the method of linear approximation to estimate the following numbers by hand.
√
√
3
5
1
(a) (1.0004)12
(g) 131
(j) 249
(d)
5.01
√
√
4
√
6
√
(b) 66
(h) 78
60
(k)
(e) 120
√
4
2/3
√
√
(c) (8.05)
(i) 632
3
(f) 62
(l) 80
3. Use the method of linear approximation to estimate the following numbers by hand.
√
√
(a) (0.997)6
(k) 144.1
(f) 99.9
√
√
3
(b) (2.0003)7
(g) 65
(l) 139.97
√
√
5
(c) 99.99
(h) (2.01)10
(m) 0.99
1
√
(i) ln(1.02)
5
(d)
(n)
1.1
1002
0.03
√
√
(j) e
(o) 82
(e) 25.1
Answers
1. (a) Txo (x) = f (xo ) + f 0 (xo )(x − xo )
(b) Near the point of tangency, the tangent line is very close to the curve. Mathematically, this
can be stated as
f (x) ≈ f (xo ) + f 0 (xo )(x − xo ) , if x ≈ xo .
2. (a) f (1.0004) = (1.0004)12 ≈ 1.0048
f (x) = x12 and xo = 1
√
65
(b) f (66) = 66 ≈
= 8.125
8
√
f (x) = x and xo = 64
(c) f (8.05) = (8.05)2/3 ≈ 4.016̄
f (x) = x2/3 and xo = 8
(d) f (5.01) =
1
≈ 0.1996
5.01
1
f (x) = and xo = 5
x
√
241
(e) f (120) = 120 ≈
= 10.954
22
√
f (x) = x and xo = 121
√
95
3
(f) f (62) = 62 ≈
= 3.9583̄
24
√
f (x) = 3 x and xo = 64
3. (a) (0.997)6 ≈ 0.982
(b) (2.0003)7 ≈ 128.1344
√
(c) 99.99 ≈ 9.9995
(d)
(e)
(f)
(g)
1
1002
√
√
≈ 0.000998
25.1 ≈ 5.01
99.9 ≈ 9.995
√
3
65 ≈
193
= 4.02083̄
48
√
127
3
(g) f (131) = 131 ≈
= 5.08
25
√
f (x) = 3 x and xo = 125
√
107
4
(h) f (78) = 78 ≈
= 2.972̄
36
√
f (x) = 4 x and xo = 81
√
2507
4
(i) f (632) = 632 ≈
= 5.014
500
√
f (x) = 4 x and xo = 625
√
407
5
(j) f (249) = 249 ≈
= 3.0148
135
√
f (x) = 5 x and xo = 243
√
95
6
= 1.97916̄
(k) f (60) = 60 ≈
48
√
f (x) = 6 x and xo = 64
√
(l) f (80) = 80 ≈ 8.94̄
√
f (x) = x and xo = 81
(h) (2.01)10 ≈ 1075.2
(i) ln(1.02) ≈ 0.02
(j) e0.03 ≈ 1.03
√
(k) 144.1 ≈ 12.00416̄
√
(l) 139.97 ≈ 11.832083̄
√
5
(m) 0.99 ≈ 0.998
√
5
(n) 1.1 ≈ 1.02
√
(o) 82 ≈ 9.05̄
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