Question

Find the slope of the graph of the function at the indicated point. Use the derivative feature of a graphing utility to confirm your results. $$ f(t)=3-\frac{3}{5 t} $$ $$ \left(\frac{3}{5}, 3\right) $$

Short answer

The slope of the function \( f(t) = 3 - 3/5t \) at the point \((3/5, 3)\) is 20.

Step-by-step solution

Find the Derivative of Function

The derivative of the function \(f(t) = 3 - 3/(5t)\) can be found using the rule of derivatives which states that the derivative of \(1/x\) is \(-1/x^2\). \nSo if we take the derivative of our function we obtain: \(f'(t) = 0 - (-3/(5t^2)) \) = \(3/(5t^2)\).

Substitute the Value of 't' in Derivative Function

The coordinate given in the question is \((3/5, 3)\). We are interested in the x-coordinate (t-value). So, we substitute \(t = 3/5\) into the derivative function: \n \(f'(3/5) = 3/(5*(3/5)^2) \)

Simplify the Expression to Get the Answer

On simplifying, we find the result is 20. So, the slope of the function at the given point is 20.