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UAE_Math_Grade_12_Vol_1_SE_718383_ch3
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EXERCISES .1 GO01962-Smith-v1.cls July 4, 2016 13:38

WRITING EXERCISES
Estimate the number of cans that can be sold at (a) 72 and
◦
◦
1. Briefly explain in terms of tangent lines why the approxi- (b) 94 .
mation in example 1.3 gets worse as x gets farther from 8. 11. An animation director enters the position f(t) of a charac-
2. We constructed a variety of linear approximations in this ter’s head after t frames of the movie as given in the table.
section. Some approximations are more useful than oth-
ers. By looking at graphs, explain why the approximation t 200 220 240
sin x ≈ x might be more useful than the approximation f(t) 128 142 136
cos x ≈ 1.
If the computer software uses interpolation to determine
3. In example 1.6, we mentioned that you might think of using the intermediate positions, determine the position of the
a linear approximation instead of Newton’s method. Dis- head at frame numbers (a) 208 and (b) 232.
√
3
cuss the relationship between a linear approximation to 7
√ 12. A sensor measures the position f(t) of a particle t microsec-
3
at x = 8 and a Newton’s method approximation to 7 with onds after a collision as given in the table.
x = 2.
0
4. Explain why Newton’s method fails computationally if t 5 10 15
′
f (x ) = 0. In terms of tangent lines intersecting the x-axis, f(t) 8 14 18
0
′
explain why having f (x ) = 0 is a problem.
0
Estimate the position of the particle at times (a) t = 8 and
In exercises 1–6, find the linear approximation to f(x) at x = x . (b) t = 12.
0
Use the linear approximation to estimate the given number. ............................................................
1. f(x) = √ x, x = 1, √ 1.2 In exercises 13–16, use Newton’s method with the given x to
0
√ 0
3
2. f(x) = (x + 1) 1∕3 , x = 0, 1.2 (a) compute x and x by hand and (b) use a computer or calcu-
2
1
0
3. f(x) = √ 2x + 9, x = 0, √ 8.8 lator to find the root to at least five decimal places of accuracy.
0
2
3
0
4. f(x) = 2∕x, x = 1, 2∕0.99 13. x + 3x − 1 = 0,x = 1
0
14. x + 4x − x − 1 = 0,x =−1
3
2
0
5. f(x) = sin 3x, x = 0, sin(0.3) 15. x − 3x + 1 = 0,x = 1
2
4
0
0
6. f(x) = sin x, x = , sin(3.0) 16. x − 3x + 1 = 0,x =−1
4
2
0
0
............................................................ ............................................................
In exercises 7 and 8, use linear approximations to estimate the
quantity. In exercises 17–24, use Newton’s method to find an approxi-
mate root (accurate to six decimal places). Sketch the graph and
√ √ √
4
4
4
7. (a) 16.04 (b) 16.08 (c) 16.16 explain how you determined your initial guess.
( ) 3 2 4 3 2
8. (a) sin (0.1) (b) sin (1.0) (c) sin 9 17. x + 4x − 3x + 1 = 0 18. x − 4x + x − 1 = 0
4 5 3
............................................................ 19. x + 3x + x − 1 = 0 20. cos x − x = 0
2
2
21. sin x = x − 1 22. cos x = x
Inexercises9–12,uselinearinterpolationtoestimatethedesired 23. e =−x 24. e −x = √ x
x
quantity.
............................................................
9. A company estimates that f(x) thousand software games
can be sold at the price of AEDx as given in the table. In exercises 25–30, use Newton’s method [state the function
f(x) you use] to estimate the given number.
x 20 30 40
√ √ √ √
3
3
f(x) 18 14 12 25. 11 26. 23 27. 11 28. 23
√
√
29. 4.4 24 30. 4.6 24
Estimate the number of games that can be sold at (a) AED24 ............................................................
and (b) AED36.
10. A vending company estimates that f(x) cans of soft drink In exercises 31–36, Newton’s method fails. Explain why the
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can be sold in a day if the temperature is x F as given in the method fails and, if possible, find a root by correcting the
table. problem. Copyright © McGraw-Hill Education
x 60 80 100 31. 4x − 7x + 1 = 0,x = 0
2
3
0
f(x) 84 120 168 32. 4x − 7x + 1 = 0,x = 1
2
3
0
236 | Lesson 4-1 | Linear Approximations and Newton’s Method