Step 1 :The function given is \(f(x) = x^2\). The derivative of this function is \(f'(x) = 2x\).
Step 2 :At the point \((9,81)\), the slope of the tangent line is \(f'(9) = 2*9 = 18\).
Step 3 :The equation of the tangent line is given by \(y = f'(a)(x - a) + f(a)\), where \(a\) is the x-coordinate of the point of tangency.
Step 4 :Substituting \(a = 9\), \(f'(a) = 18\), and \(f(a) = 81\) into the equation, we get \(y = 18(x - 9) + 81\).
Step 5 :Simplifying the equation, we get \(y = 18x - 162 + 81\), which simplifies to \(y = 18x - 81\).
Step 6 :To approximate \((9.01)^2\), we substitute \(x = 9.01\) into the equation of the tangent line, getting \(y = 18*9.01 - 81\).
Step 7 :Calculating the above expression, we get \(y = 162.18 - 81 = 81.18\).
Step 8 :The actual value of \((9.01)^2\) calculated using a calculator is approximately 81.1801.
Step 9 :Comparing the two values, the tangent line estimate is slightly less than the actual value.
Step 10 :The reason for this is that the function \(f(x) = x^2\) is concave up, meaning that the tangent line at any point will lie below the curve to the right of the point of tangency.