`f(x) = (1/3)xsqrt(x^2 + 5), (2,2)` (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) = (1/3)xsqrt(x^2 + 5), (2,2)` (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) = (1/3)*x*sqrt(x^2 + 5)` . The slope of a line tangent to this curve at x = a is f'(a).

The derivative `f'(x) = (1/3)*(x*sqrt(x^2 + 5))'`

= `(1/3)*(x*(sqrt(x^2+5))' + sqrt(x^2+5))`

= `(1/3)*(x*x/sqrt(x^2+5) + sqrt(x^2+5))`

= `(1/3)*(x^2/sqrt(x^2+5) + sqrt(x^2+5))`

At x = 2, `(1/3)*(x^2/sqrt(x^2+5) + sqrt(x^2+5))`

= `(1/3)*(4/sqrt(4+5) + sqrt(4+5))`

= `(1/3)*(4/3 + 3)`

= `13/9`

The equation of the tangent at (2,2) is `(y - 2)/(x - 2) = 13/9`

9y - 18 = 13x - 26

y = `(13x - 8)/9`

The graph of the curve and the tangent is: