Determine equation of the tangent line to f (x) = 4x - 8 √x at x = 16 .
Solution : We already know that the equation of a tangent line is specified by,
y = f ( a ) + f ′ ( a ) ( x - a )
Hence, we will have the derivative of the function (don't forget to get rid of the radical).
f ( x ) = 4x - 8x ½ ⇒ f ′ ( x ) = 4 - 4x -1/2 = 4 - 4/x 1/2
Again, notice as well that we remove the negative exponent in the derivative solely for the sake of the evaluation. All we have to do then is evaluate the function & the derivative at the point in question, x = 16 .
f (16) = 64 - 8 ( 4) = 32 f ′ ( x ) = 4 - 4 /4= 3
Then the tangent line is,
y = 32 + 3( x -16) = 3x -16