`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.

Expert Answers
justaguide eNotes educator| Certified Educator

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: