Estimate the slope of the curve y = cosx at x=π\2
Comment on the slope as x gets closer to π\2
To estimate the slope of the curve y = cos(x) at x = π/2, we can use the concept of the derivative.
The derivative of y = cos(x) is given by dy/dx = -sin(x).
Substituting x = π/2 into -sin(x), we have dy/dx = -sin(π/2).
The sine of π/2 is equal to 1, so the slope at x = π/2 is -1.
As x gets closer to π/2, the slope becomes steeper and approaches -1.
To estimate the slope of the curve y = cosx at x = π/2, we can use the derivative of the function. The derivative of the cosine function is equal to the negative sine function: dy/dx = -sinx.
At x = π/2, the cosine function reaches its maximum value of 1. Therefore, the slope at this point can be estimated by substituting π/2 into the derivative:
dy/dx = -sin(π/2) = -1
So, the estimate of the slope at x = π/2 is -1.
As x gets closer to π/2, the slope approaches -1. This means that the tangent line to the curve at this point becomes steeper as x approaches π/2 from both the left and right sides. The slope indicates the rate at which the value of y is changing with respect to x, and in the case of y = cosx, it decreases at a rate of 1 unit for every 1 unit increase in x as x approaches π/2.
To estimate the slope of the curve y = cos(x) at x = π/2, we can use the concept of derivatives. The derivative of a function represents the rate at which the function is changing at a particular point.
To find the derivative of y = cos(x), we can use the standard derivative formula for cosine:
dy/dx = -sin(x)
Now, let's find the slope at x = π/2 by plugging the value into the derivative equation:
dy/dx = -sin(π/2) = -1
Therefore, the slope of the curve y = cos(x) at x = π/2 is -1.
Now, let's consider what happens to the slope as x gets closer to π/2. As x approaches π/2, the value of sin(x) approaches 1. So, the slope will approach -1 as x gets closer to π/2.
In summary, the slope of the curve y = cos(x) at x = π/2 is -1, and as x gets closer to π/2, the slope approaches -1.