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Let f f be a differentiable function with f(3)=2 f(3) = 2 and f′(3)=−4 f'(3) = -4 . What is the value of the approximation of f(3.1) f(3.1) using the function's local linear approximation at x=3 x = 3 ?

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Q. Let f f be a differentiable function with f(3)=2 f(3) = 2 and f′(3)=−4 f'(3) = -4 . What is the value of the approximation of f(3.1) f(3.1) using the function's local linear approximation at x=3 x = 3 ?
  1. Identify values: Identify the given values.\newlinef(a)=f(a) f(a) = f(a) \newlinef′(a)=f′(a) f'(a) = f'(a)
  2. Write formula: \newlineStep 22: Write the formula for the local linear approximation.\newlineL(x)=f(a)+f′(a)(x−a) L(x) = f(a) + f'(a)(x - a)
  3. Substitute values: \newlineStep 33: Substitute the given values into the formula.\newlineL(x)=f(a)+f′(a)(x−a) L(x) = f(a) + f'(a)(x - a)
  4. Simplify expression: \newlineStep 44: Simplify the expression.\newlineL(x)=f(a)+f′(a)(x−a) L(x) = f(a) + f'(a)(x - a)

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