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
Ā (x , y) is y/x
* Lets solve the problem
āµ y = -5 sin(x)
āµ m = dy/dx , where m is the slope of the tangent to the curve
- The differentiation of sin(x) is cos(x)
ā“ m = -5 cos(x)
āµ The tangent passes through the origin and a point on the curve
- The general point on the curve is (x , -5 sin(x))
āµ m = y/x
ā“ m = -5 sin(x)/x
āµ The slopes are equal
ā“ -5 sin(x)/x = -5 cos(x)
- Multiply both sides by x
ā“ -5 sin(x) = -5x cos(x)
- Divide both sides by -5 cox(x)
# Remember that sin(x)/cos(x) = tan(x)
ā“ tan(x) = x
ā“ tan(x) - x = 0
* Lets use Newton's method
- In the Newton's method x(n+1) = x(n) - y/y'
āµ y = tan(x) - x
ā“ y' = secĀ²(x) - 1
āµ secĀ²(x) - 1 = tanĀ²(x)
ā“ x(n+1) = x(n) - [tan(x) - x]/tanĀ²(x)
- Let x(1) = 4.5
ā“ x(2) = 4.5 - [tan(4.5) - 4.5]/tanĀ²(4.5) = 4.493614
ā“ x(3) = 4.493409
ā“ x(4) = 4.493409
ā“ x(5) = 4.493409
ā“ The value of x = 4.493409 is the largest value
āµ m = -5 cos(x)
ā“ m = -5 cos(4.493409) = 1.086168
ā“ The slope is 1.086168
