6. let f be a function with f(2) = -8 such that for all points (x,y) on the graph of f, the slope is given…

6. let f be a function with f(2) = -8 such that for all points (x,y) on the graph of f, the slope is given by \\frac{3x^{2}}{y}.\n(a) write an equation of the line tangent to the graph of f at the point where x = 2 and use it to approximate f(1.8).

6. let f be a function with f(2) = -8 such that for all points (x,y) on the graph of f, the slope is given by \\frac{3x^{2}}{y}.\n(a) write an equation of the line tangent to the graph of f at the point where x = 2 and use it to approximate f(1.8).

Answer

Explanation:

Step1: Find the slope of the tangent line

Given the slope formula (\frac{dy}{dx}=\frac{3x^{2}}{y}). When (x = 2) and (y=f(2)=-8), substitute into the slope formula: [m=\frac{3\times2^{2}}{-8}=\frac{12}{-8}=-\frac{3}{2}]

Step2: Use the point - slope form of a line

The point - slope form of a line is (y - y_{1}=m(x - x_{1})). Here (x_{1}=2), (y_{1}=-8) and (m =-\frac{3}{2}). [y+8=-\frac{3}{2}(x - 2)] [y+8=-\frac{3}{2}x+3] [y=-\frac{3}{2}x - 5]

Step3: Approximate (f(1.8))

Let (x = 1.8) in the tangent line equation (y=-\frac{3}{2}x - 5) [y=-\frac{3}{2}\times1.8-5] [y=-2.7-5] [y=-7.7]

Answer:

The equation of the tangent line is (y =-\frac{3}{2}x - 5) and the approximation of (f(1.8)) is (-7.7)