Of the infinitely many lines that are tangent to the curve y = −5 sin(x) and pass through the origin, there is one that has the largest slope. use newton's method to find the slope of that line correct to six decimal places.

The slope is 1.086168

Step-by-step explanation:

* Lets explain how to solve the problem

- The slope of any curve y = f(x) is dy/dx

- The slope of the line which passes through the origin and the point

 (x , y) is y/x

* Lets solve the problem

∵ y = -5 sin(x)

∵ m = dy/dx , where m is the slope of the tangent to the curve

- The differentiation of sin(x) is cos(x)

∴ m = -5 cos(x)

∵ The tangent passes through the origin and a point on the curve

- The general point on the curve is (x , -5 sin(x))

∵ m = y/x

∴ m = -5 sin(x)/x

∵ The slopes are equal

∴ -5 sin(x)/x = -5 cos(x)

- Multiply both sides by x

∴ -5 sin(x) = -5x cos(x)

- Divide both sides by -5 cox(x)

# Remember that sin(x)/cos(x) = tan(x)

∴ tan(x) = x

∴ tan(x) - x = 0

* Lets use Newton's method

- In the Newton's method x(n+1) = x(n) - y/y'

∵ y = tan(x) - x

∴ y' = sec²(x) - 1

∵ sec²(x) - 1 = tan²(x)

∴ x(n+1) = x(n) - [tan(x) - x]/tan²(x)

- Let x(1) = 4.5

∴ x(2) = 4.5 - [tan(4.5) - 4.5]/tan²(4.5) = 4.493614

∴ x(3) = 4.493409

∴ x(4) = 4.493409

∴ x(5) = 4.493409

∴ The value of x = 4.493409 is the largest value

∵ m = -5 cos(x)

∴ m = -5 cos(4.493409) = 1.086168

∴ The slope is 1.086168

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step-by-step explanation:

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step-by-step explanation: