Find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval [0, 2π]. What can you conclude about the slope of the sine function sin ax at the origin? y = sin 3x

Solution:

The slope of the Sin x curve at any point lying on it is given by the derivative of the function at that point and that is also the slope of the tangent to the Sine curve at that point.

The derivative of Sin x is given as:

d/dx (Sin x)  =  Cos x 

As evident from the diagram below the slope of the tangent line to the sine function is given by the cosine function. The slope of the sine function at the origin is 1 as evident by point B on the cosine curve.

Slope of the Sine Curve at origin  = d/dx(Sin x)  =  Cos x

At origin, x = 0

Cos 0 = 1

Hence slope of the Sine curve at origin = 1.

At π/2 Slope of the sine curve = Cos (π/2) =0

At  π Slope of the sine curve = Cos (π) = -1

At 3π/2Slope of the sine curve = Cos (3π/2) = 0

The slope of the function Sinax is given as

d/dx Sin (ax) =  aCos ax

d/dx Sin (3x) = 3Cos 3x

Due to the addition of a or 3, the number of cycles of Sin3x from 0 to 2π  becomes 3, and the slope at the origin becomes thrice the original slope of 1 at the origin. Both these inferences are concluded by the graph below:

Find the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete cycles in the interval [0, 2π]. What can you conclude about the slope of the sine function sin ax at the origin? y = sin 3x

Summary:

The slope of the tangent line to the Sin x function at the origin is 1. The slope of the sine function at the origin is 1 and that of y = sin 3x, the slope of the sine function becomes thrice the original slope of 1 at the origin.