`f(x) = (9 - x^2)^(2/3), (1,4)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the...

`f(x) = (9 - x^2)^(2/3), (1,4)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of the graphing utility to confirm your results.

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1 Answer

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `f(x) = (9 - x^2)^(2/3)` . At the point (1, 4) the equation of the tangent to this curve is:

`(y - 4)/(x - 1) = f'(1)`

`f(x) = (9 - x^2)^(2/3)`

`f'(x) = (2/3)*(9 - x^2)^(-1/3)*(-2x)`

`f'(1) = -2/3`

The equation of the tangent is `(y - 4)/(x - 1) = -2/3`

`y = (-2/3)*(x - 1) + 4`

The graph of the curve and the tangent is: