Answer:
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
