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Unformatted text preview: Homework 3/31
Part 1:
1. Use diﬀerentials to approximate (2.002)5 .
2. Use diﬀerentials to approximate (1.997)5 .
3. Use diﬀerentials to approximate (2 + h)5 .
4. Use diﬀerentials to approximate (x)5 near x = 2.
NOTE: Your answer to this problem is called the linearization of x5 at x = 2.
5. Graph y = x5 and the linearization at x = 2 on the same axes. Your graph must extend at
least from x = 0 to x = 3.
6. Use your linearization to solve x5 = 35.
7. Use your calculator to solve x5 = 35.
Part 2: Warning!. Work this problem in radians!
1. Use diﬀerentials to approximate tan 46◦ .
[Hint: Convert ∆x = 1◦ to radians.]
2. Use diﬀerentials to approximate tan 44◦ .
3. Use diﬀerentials to approximate tan(π/4 + h).
4. Use diﬀerentials to approximate tan x near x = π/4. Simplify your answer.
NOTE: Your answer to this problem is called the linearization of tan x at x = π/4.
5. Graph y = tan x and your answer to part (4) on the same axes. Your graph must extend at
least from x = 0 to x = π/2.
6. Use your linearization to solve tan x = 5/6.
7. Use your calculator to solve tan x = 5/6.
Part 3:
Suppose that f (x) = x + sin x − 0.2e0.1x .
1. Find the linearization of f (x) at x = 1.
2. Use your linearization to ﬁnd a root of f . (That is, solve f (x) = 0.)
1 3. Find the linearization of f (x) at x = 0.
4. Use your linearization to ﬁnd a root of f .
5. If possible, use your calculator to solve f (x) = 0. Answers
Part 1:
1) 32.16; 2) 31.76; 3) 32 + 80h; 4) 32 + 80(x − 2); 6) x = 2 + 3/80 ≈ 2.0375; 7) x ≈ 2.0362.
Part 2:
1) 1 + π/90; 2) 1 − π/90; 3) 1 + 2h; 4) 1 + 2(x − π/4);
6) x = π/4 − 1/12 ≈ 0.7021; 7) x = arctan(5/6) ≈ 0.6947.
Part 3:
1) f (x) ≈ (1 + sin 1 − 0.2e0.1 ) + (1 + cos 1 − 0.02e0.1 )(x − 1) ≈ 1.620 + 1.518(x − 1).
1 + sin 1 − 0.2e0.1
≈ −0.0673.
2) x ≈ 1 −
1 + cos 1 − 0.02e0.1
3) f (x) ≈ −0.2 + 1.98x.
.2
4) x ≈
≈ 0.101.
1.98
5) x ≈ 0.1011.
2 ...
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This note was uploaded on 10/12/2011 for the course MATH 170 taught by Professor Staff during the Spring '08 term at Boise State.
 Spring '08
 STAFF
 Calculus

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