Find f(3) and f'(3), assuming that the tangent line to y=f(x) at a=3 has the equation y=5x+2
i;m so sad
There are any number of curves that are tangent to the line y=5x+2 at x=3.
The line y=5x+2 has slope 5, so f'(3)=5
and f(3) touches the line at x=3, so f(3) = 17.
i hope you feel better by now
i hope you dont
Well, if the tangent line to y = f(x) at a = 3 has the equation y = 5x + 2, then we can determine f'(3) just by looking at the slope of the tangent line. Since the slope of the tangent line is 5, we know that f'(3) = 5.
As for f(3), unfortunately, I can't determine the actual value without any further information. But hey, at least we know the derivative! Keep calm and calculate on!
To find \( f(3) \), we can use the equation of the tangent line. Recall that the equation of a tangent line at a point \((a, f(a))\) on a curve \(y=f(x)\) is in the form \(y = f'(a)(x-a) + f(a)\).
In this case, the equation of the tangent line is given as \(y = 5x + 2\). We can match this equation with the general form of a tangent line and determine the values of \(a\) and \(f(a)\) by comparing the coefficients.
Comparing the coefficients, we have:
\(f'(a) = 5\) and \(f(a) = 2\).
Since we are asked to find \(f(3)\), we substitute \(a = 3\) into the equation we derived earlier:
\(f'(3) = 5\) and \(f(3) = 2\).
Therefore, \(f(3) = 2\) and \(f'(3) = 5\).