Questions: Find the equation of the line tangent to the given curve at (x=a). Use a graphing utility to graph the curve and the tangent line on the same set of axes. (y=frac5 xx-3 ; a=4) The equation for the tangent line is (y=) (Type an expression using x as the variable.)

Solution Steps

To find the equation of the tangent line, we first need to find the derivative of the function \( y = \frac{5x}{x-3} \).

Using the quotient rule, which states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \), we have:

• \( u = 5x \) and \( u' = 5 \)
• \( v = x - 3 \) and \( v' = 1 \)

Thus, the derivative \( y' \) is: \[ y' = \frac{(5)(x-3) - (5x)(1)}{(x-3)^2} = \frac{5x - 15 - 5x}{(x-3)^2} = \frac{-15}{(x-3)^2} \]

Now, we evaluate the derivative at \( x = 4 \): \[ y'(4) = \frac{-15}{(4-3)^2} = \frac{-15}{1} = -15 \]

Substitute \( x = 4 \) into the original function to find the y-coordinate: \[ y = \frac{5(4)}{4-3} = \frac{20}{1} = 20 \]

The equation of the tangent line is given by the point-slope form: \[ y - y_1 = m(x - x_1) \] where \( m = -15 \), \( x_1 = 4 \), and \( y_1 = 20 \).

Substituting these values, we get: \[ y - 20 = -15(x - 4) \]

Simplifying, the equation of the tangent line is: \[ y = -15x + 60 + 20 = -15x + 80 \]

Final Answer