the slope of the tangent line to the curve y=(3/x) at the point (6,0.5000) is?
the equation of this tangent line can be written in the form y=mx+b where m is? and where b is?
y = 3/x
y' = -3/x^2
y'(6) = -3/36 = -1/12
y-1/2 = -1/12 (x-6)
y = -1/12 x + 1
To find the slope of the tangent line to the curve, we need to take the derivative of the curve and evaluate it at the given point.
Step 1: Take the derivative of the curve y = 3/x.
The derivative of y with respect to x can be found using the power rule for derivatives: d/dx (x^n) = nx^(n-1).
Taking the derivative of y = 3/x, we get:
dy/dx = d/dx (3/x) = (-3/x^2)
Step 2: Evaluate the derivative at the point (6, 0.5000).
To evaluate the derivative at a point, substitute the x-coordinate of the point into the derivative.
Substituting x = 6 into (-3/x^2), we get:
dy/dx = (-3/6^2) = (-3/36) = (-1/12)
So, the slope of the tangent line to the curve y = 3/x at the point (6, 0.5000) is -1/12.
Now, let's find the equation of the tangent line in the form y = mx + b.
Step 3: Use the point-slope form of the equation of a line to find the equation of the tangent line.
The point-slope form of a line is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope of the line.
Substituting the values of the slope (-1/12) and the given point (6, 0.5000) into the point-slope form, we get:
y - 0.5000 = (-1/12)(x - 6)
Simplifying, we have:
y - 0.5000 = (-1/12)x + (1/2)
Multiplying both sides by 12 to get rid of the fraction, we get:
12y - 6 = -x + 6
Re-arranging the terms, we get:
x + 12y = 12
In the equation y = mx + b, m represents the slope. So, in this case, m = -1/12.
Similarly, b represents the y-intercept of the line, which can be found by solving the equation for y when x = 0. In our case, the equation of the line is:
x + 12y = 12
Substituting x = 0, we get:
0 + 12y = 12
Simplifying, we solve for y:
12y = 12
y = 1
Thus, b = 1.
Therefore, the equation of the tangent line to the curve y = 3/x at the point (6, 0.5000) is y = (-1/12)x + 1, where m = -1/12 and b = 1.