Let $f(x)=3 x^{2}$
a) Find the linearization $L(x)$ of $f$ at $a=1$.
b) Use the linearization to approximate $3(1.1)^{2}$.
c) Find $3(1.1)^{2}$ using a calculator.
d) What is the difference between the approximation and the actual value of $3(1.1)^{2}$.
a) The linear approximation is $L(x)=6 x-3$.
b) Using the linearization, $3(1.1)^{2}$ is approximately
(Type an integer or a decimal.)

Step 1 ：The linearization of a function \(f(x)\) at a point \(a\) is given by \(L(x) = f(a) + f'(a)(x-a)\).

Step 2 ：First, we need to find the derivative of \(f(x) = 3x^2\). Using the power rule, we get \(f'(x) = 6x\).

Step 3 ：Substituting \(a = 1\) into the derivative, we get \(f'(1) = 6\).

Step 4 ：Substituting \(a = 1\) into the original function, we get \(f(1) = 3\).

Step 5 ：Substituting these values into the linearization formula, we get \(L(x) = 3 + 6(x - 1) = 6x - 3\).

Step 6 ：To approximate \(3(1.1)^2\), we substitute \(x = 1.1\) into the linearization, getting \(L(1.1) = 6(1.1) - 3 = 3.6\).

Step 7 ：Using a calculator, we find that \(3(1.1)^2 = 3.63\).

Step 8 ：The difference between the approximation and the actual value is \(3.63 - 3.6 = 0.03\).