a curve i s given the equation y=3x^2+3+1/x^2,where x>0. at point A,B, and C on the curve,x=1,2 and 3. find the gradients of A,B and C.
To find the gradients of points A, B, and C on the curve, we need to find the derivative of the equation y = 3x^2 + 3 + 1/x^2 with respect to x. The derivative will give us the slope or gradient of the curve at any given point.
Let's start by finding the derivative:
1. Take the derivative of the term 3x^2. Using the power rule, the derivative of x^n is nx^(n-1). So the derivative of 3x^2 is 6x.
2. Take the derivative of the term 3. The derivative of a constant term is 0.
3. Take the derivative of the term 1/x^2. Using the power rule again, we can write 1/x^2 as x^(-2). The derivative of x^(-n) is -nx^(-n-1). So the derivative of 1/x^2 is -2/x^3.
Now, let's put all the derivatives together:
dy/dx = 6x + 0 - 2/x^3
Simplifying the expression, we get:
dy/dx = 6x - 2/x^3
Now we can find the gradients at points A, B, and C.
At x = 1 (point A):
dy/dx = 6(1) - 2/(1^3)
dy/dx = 6 - 2/1
dy/dx = 6 - 2
dy/dx = 4
At x = 2 (point B):
dy/dx = 6(2) - 2/(2^3)
dy/dx = 12 - 2/8
dy/dx = 12 - 1/4
dy/dx = 47/4
At x = 3 (point C):
dy/dx = 6(3) - 2/(3^3)
dy/dx = 18 - 2/27
dy/dx = 18 - 2/27
dy/dx = 532/27
Therefore, the gradients at points A, B, and C are:
At point A: 4
At point B: 47/4
At point C: 532/27