Use the formula $L(x)=f(c)+f^{\prime}(c)(x-c)$. Find the linear approximation of the function $f(x)=x^{2}$ at the point $(1,1)$.
Select one:
a. $y=x+1$
b. $y=2 x+1$
c. $y=2 x-1$
d. $y=2 x-2$

Answer

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Answer

Therefore, the linear approximation of f(x)=x^2 at the point (1,1) is y=2x-1

Steps

Step 1 ：Find the value of f(c) by substituting c=1 into f(x)=x^2: f(c)=f(1)=1^2=1

Step 2 ：Find the derivative of f(x)=x^2: f'(x)=2x

Step 3 ：Find the value of f'(c) by substituting c=1 into f'(x)=2x: f'(c)=f'(1)=2(1)=2

Step 4 ：Substitute the values of f(c) and f'(c) into the linear approximation formula: L(x)=1+2(x-1)

Step 5 ：Simplify the expression: L(x)=2x-1

Step 6 ：Therefore, the linear approximation of f(x)=x^2 at the point (1,1) is y=2x-1