Explain why the following four statements ask for the same information: i) Find

Question

Explain why the following four statements ask for the same information: i) Find the roots of f(x) = x^3 - 3x - 1. ii) Find the x-coordinates of the intersections of the curve y = x^3 with the line y = 3x + 1. iii) Find the x-coordinates of the points where the curve y = x^3 - 3x crosses the horizontal line y = 1. iv) Find the values of x where the derivative of g(x) = (1/4) x^4 - (3/2)x^2 - x + 5 equals zero. Locating a planet To calculate a planet's space coordinates, we have to solve equations like x = 1 + 0.5 sin x. Graphing the function f(x) = x - 1 - 0.5 sin x suggests that the function has a root near x = 1.5. Use one application of Newton's method to improve this estimate. That is, start with x_0 = 1.5 and find x_1. (The value of the root is 1.49870 to five decimal places.) Remember to use radians.

Explanation / Answer

11.a .f(x) = x^3 -3x-1

Finding the roots means solving for x

ii. Intersection of the curve means both have same coordinates that is

x^3 =3x+1

x^3-3x-1=0

iii. Crosses the horizontal line means

x^3-3x=1

x^3 -3x-1=0

iv. derivative is

4(1/4)x^3 -2(3/2)x -1=0

x^3 -3x-1=0

so all four equations are same

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