find f(3) and f'(3), assuming that the tangent line to y=f(x) at a=3 has the equation y=5x+2.
To find \(f(3)\) and \(f'(3)\), we need to use the information provided about the tangent line to find the equation of the original function \(y = f(x)\).
The equation of a tangent line to a function \(y = f(x)\) at a specific point \((a, f(a))\) can be expressed in the form \(y = mx + b\), where \(m\) is the slope of the tangent line and \(b\) is the y-intercept.
In this case, the equation of the tangent line is \(y = 5x + 2\), which implies that the slope of the tangent line is \(m = 5\).
To find \(f(3)\), we substitute \(x = 3\) into the equation of the tangent line:
\[y = 5x + 2\]
\[f(3) = 5(3) + 2\]
\[f(3) = 15 + 2\]
\[f(3) = 17\]
Therefore, \(f(3) = 17\).
To find \(f'(3)\), we can observe that the slope of the tangent line is equal to the derivative of the original function evaluated at the specific point \((3, f(3))\). Therefore, \(f'(3) = 5\).
In summary:
\(f(3) = 17\) and \(f'(3) = 5\) based on the given equation of the tangent line.