Consider the parabola y = 6x − x2.
(a) Find the slope of the tangent line to the parabola at the point (1, 5)
the slope of the tangent is equal to the 1st derivative
slope = 6 - 2x
sub in 1 in the x and the 5 in the y
no, ghost ... y is not a factor in the slope equation
okok now i get it
To find the slope of the tangent line to the parabola at a particular point, we need to find the derivative of the function at that point.
Step 1: Start with the equation of the parabola, y = 6x - x^2.
Step 2: Differentiate the equation with respect to x to find the derivative.
dy/dx = d/dx (6x - x^2)
To differentiate, we apply the power rule: the derivative of a constant multiplied by x to the power of n is equal to the constant multiplied by n times x to the power of n-1.
So, using the power rule:
dy/dx = 6 - 2x
Step 3: Now that we have the derivative, we can find the slope of the tangent line at the point (1, 5) by substituting x = 1 into the derivative.
dy/dx = 6 - 2(1)
= 6 - 2
= 4
Therefore, the slope of the tangent line to the parabola at the point (1, 5) is 4.