121616949-math.144 - Figure 6.18 y = tan x and y = 2 x...

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130 Chapter 6 Applications of the Derivative x 5 = 1 . 165561636. So we guess that the first three places are correct, but that is not the same as saying 1 . 165 is correct to three decimal places—1 . 166 might be the correct, rounded approximation. How can we tell? We can substitute 1 . 165, 1 . 1655 and 1 . 166 into the original function tan x - 2 x ; this gives - 0 . 002483652, - 0 . 000271247, 0 . 001948654. Since the first two are negative and the third is positive, tan x - 2 x crosses the x axis between 1 . 1655 and 1 . 166, so the correct value to three places is 1 . 166. 0 5 10 15 1 1 . 5 ......................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....................................................................................................................................................................................
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Unformatted text preview: Figure 6.18 y = tan x and y = 2 x . Exercises 1. Approximate the fifth root of 7, using x = 1 . 5 as a first guess. Use Newton’s method to find x 3 as your approximation. ⇒ 2. Use Newton’s Method to approximate the cube root of 10 to two decimal places. ⇒ 3. The function f ( x ) = x 3-3 x 2-3 x + 6 has a root between 3 and 4, because f (3) =-3 and f (4) = 10. Approximate the root to two decimal places. ⇒ 4. A rectangular piece of cardboard of dimensions 8 × 17 is used to make an open-top box by cutting out a small square of side x from each corner and bending up the sides. (See exercise 6.1 . 20 .) If x = 2, then the volume of the box is 2 · 4 · 13 = 104. Use Newton’s method to find a value of x for which the box has volume 100, accurate to 3 significant figures. ⇒...
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