Question

1. 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}.

(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).

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 \(-7.7\)