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(A) Given function: $f(x) = x^{3} - 3x + 5$. We need to find the approximate value at $x = 1.99$. Let $x = a + h$,where $a = 2$ and $h = -0.01$. The f

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$6.91$

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(A) Given function: $f(x) = x^{3} - 3x + 5$. We need to find the approximate value at $x = 1.99$. Let $x = a + h$,where $a = 2$ and $h = -0.01$. The formula for linear approximation is $f(a + h) \approx f(a) + h \cdot f'(a)$. First,calculate $f(a) = f(2) = 2^{3} - 3(2) + 5 = 8 - 6 + 5 = 7$. Next,find the derivative $f'(x) = 3x^{2} - 3$. Calculate $f'(a) = f'(2) = 3(2)^{2} - 3 = 3(4) - 3 = 12 - 3 = 9$. Now,substitute these values into the approximation formula: $f(1.99) \approx f(2) + (-0.01) \cdot f'(2) = 7 + (-0.01)(9) = 7 - 0.09 = 6.91$. Thus,the approximate value is $6.91$.

The approximate value of the function $f(x) = x^{3} - 3x + 5$ at $x = 1.99$ is

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