Problem

Let f(x)=3x2
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)=6x3.
b) Using the linearization, 3(1.1)2 is approximately
(Type an integer or a decimal.)

Answer

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Answer

The difference between the approximation and the actual value is 3.633.6=0.03.

Steps

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

Step 2 :First, we need to find the derivative of f(x)=3x2. 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(x1)=6x3.

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 3.633.6=0.03.

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