# Section 4.5

```BC Calculus – FDW 3rd Edition – Section 4.5 Even Answers
Quick Review 4.5
x
f(x)
g(x)
2. 1 cos x ( x 21)sin x 4. -.322 6. y  2ex  e  1 8. 0.7 -1.457 -1.7 10.
( x 1)
0.8 -1.688 -1.8
0.9 -1.871 -1.9
Section 4.5
1.0 -2.000 -2.0
2.(a) L( x )   45 x  95 (b) error &lt; 10-3
1.1 -2.069 -2.1
4.(a) L( x)  x (b) error &lt; 10-2
1.2 -2.072 -2.2
-3

6.(a) L( x )   x  2 (b) error &lt; 10
1.3 -2.003 -2.3
8.(a)  1.2, 1.002100  1.2  101 (b)  1.003,
3
1.009  1.003  105
10. Method 1/ Use  a  b   a n  na n 1b1  ... then
n
(a)  4  3x    4  
1
3

(b) 2  x
2

1
3
1
2
1
3
 23
 4   3x 
1
 2 2  12  2 
 12
(c) 1  21 x  3  13  23  1
2
 
1
1
3
4  3
1
x2
 21 x 
 13
2
1
2
1
42
1
2 2
x
1
43
34
 4 3 1  4x 
1

2
x 2  2 1  x4

 1  23  21 x   1  623 x
Method 2/ Factor out (divide out) to make the first term (or ‘a’) equal to 1, then use: 1 + nx
A problem like (c) is already to go! For (a) and (b) factoring out 41/3 or 21/2 may be tricky!
1
1
1
2
1
2

(a)  4  3x  3  4 3 1  34x

(b) 2  x
2


 2 1
x2
2

1
3

1
2
(c) No factoring here! 1 


 4 3 1  13  34x   4 3 1  4x 
1
    2 1  
 2 1  12
1
2 x

2
3
1
x2
2
x2
4
 1  23  21 x  1  623 x
So? Well… if our binomial is of the form (1 + b)n vs (a + b)n, then method 2 is best!
1 x  27  3  y  2.963
12. y  27


1 x  81  9  y  8.949
14. y  18


16. x  1.452627, 1.164035 (Why six decimal places? Why?!?)
18. 1.189207
20.(a) dy 
22.(a) dy 
2 2 x 2
1 x2 
2
1 2 x 2


1
1 x 2 2
dx (b) dy  .024
dx (b) dy  .2
24.(a) dy  csc 1  3x  cot 1  3x  dx (b) dy  .205525
26.(a) dy 
  dx (b) dy  .0375
2 x y
2 x
BC Calculus – Section 4.5 Even Answers (continued)
28. dy  5e5 x  5x 4 dx

30. dy  8

x

ln8  8 x 7 dx
32.(a) .231 (b) .2 (c) .031
34.(a) .04060401 (b) .04 (c) .00060401
36. A  8 a dr  4 cm2
38. S  12a dx  6 cm2
40. A  2 r dh  .1 r cm2
42. dV  4 r 2dr  4 82 .3  241in2
44. A  s 4 3  dA  2 s4 3 ds  23  20 .5  8.7 cm2
46.(a) .08  .2513 (b) 2%
48. 3%
50.(a) within .5% (b) within 5%
52. Measure the height to within 1/3 % of ‘h’.
54. A  4 r 2 , we have dA  8 r dr  8 r  161   2r which is the surface area error in in2.
2
1
3
56.(a) dT   L2 g 2 dg (b) T decreases and the clock speeds up. dT &amp; dg have opposite
signs (c) dg  .9765, so g  979.0235
58. False. Recall the product rule: d(uv) = udv + vdu 60. A 62. A
64. If x1  h, then f '( x1 ) 
1
1
2h 2
and x2  h  
66.(a) b0  f ( a ), b1  f '( a ), b2 
1
h 2
 11

 2h 2
f ''( a )
2




  h . If x1   h, x2  h .
(b) 1  x  x 2
(c) They appear identical. (d) 1  ( x  1)  ( x  1)2 , the same
2
(e) 1  2x  x8 , they appear the same (f) f: 1 + x, g: 1 – (x – 1), h: 1  2x
68. Formulas for differentials? Just multiply the derivative formulae by ‘dx’.
70. The error is zero at x = a. The error is negligible when compared with (x – a).
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