a. Find the equation of the tangent line to $f(x)=x^{2}$ at the point $(9,81)$.
b. Use the tangent line equation to approximate $(9.01)^{2}$. Compare your result with the value from a calculator (the actual value).
c. Which value is larger, the tangent line estimate or the actual value? Why?
Answer
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.
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.
