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Câu 41 trang 216 SGK Đại số và Giải tích 11 Nâng cao

Áp dụng công thức (2), tìm giá trị gần đúng
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Áp dụng công thức (2), tìm giá trị gần đúng của các số sau (làm tròn kết quả đến hàng phần nghìn).

LG a

\({1 \over {0,9995}}\)

Phương pháp giải:

Công thức (2): \(f\left( {{x_0} + \Delta x} \right) \approx f\left( {{x_0}} \right) + f'\left( {{x_0}} \right)\Delta x\)

Lời giải chi tiết:

Xét hàm số \(f\left( x \right) = {1 \over x},\,\text{ ta có }\,f'\left( x \right) = {{ - 1} \over {{x^2}}}\) Đặt \({x_0} = 1,\Delta x =  - 0,0005\) và áp dụng công thức gần đúng \(f\left( {{x_0} + \Delta x} \right) \approx f\left( {{x_0}} \right) + f'\left( {{x_0}} \right)\Delta x\) Ta được : \({1 \over {{x_0} + \Delta x}} \approx {1 \over {{x_0}}} - {1 \over {x_0^2}}.\Delta x,\) \( \Rightarrow \frac{1}{{1 + \left( { - 0,0005} \right)}} \approx \frac{1}{1} - \frac{1}{{{1^2}}}.\left( { - 0,0005} \right)\) Hay : \({1 \over {0,9995}} \approx 1 + 0,0005 = 1,0005\)

LG b

 \(\sqrt {0,996} \)

Lời giải chi tiết:

Xét \(\eqalign{  & f\left( x \right) = \sqrt x \,\text{ ta có }\,f'\left( x \right) = {1 \over {2\sqrt x }}  \cr  & {x_0} = 1,\Delta x =  - 0,004  \cr  & f\left( {{x_0} + \Delta x} \right) \approx f\left( {{x_0}} \right) + f'\left( {{x_0}} \right)\Delta x  \cr  & \Rightarrow \sqrt {{x_0} + \Delta x}  \approx \sqrt {{x_0}}  + \frac{1}{{2\sqrt {{x_0}} }}\Delta x \cr &\Leftrightarrow \sqrt {1 + \left( { - 0,004} \right)}  \approx \sqrt 1  + \frac{1}{{2\sqrt 1 }}.\left( { - 0,004} \right)\cr &  \Leftrightarrow \sqrt {0,996}  \approx 1 - {1 \over 2}.0,004 = 0,998 \cr} \)

LG c

\(\cos 45^\circ 30'\)

Lời giải chi tiết:

Xét hàm số \(f(x) = \cos x\), ta có: \(f'\left( x \right) =  - \sin x.\) Đặt \({x_0} = {\pi  \over 4},\Delta x = {\pi  \over {360}}\) (Vì \({\pi  \over {360}} = 30'\) ) và áp dụng công thức gần đúng trên, ta được :   \(\eqalign{  & \cos \left( {{\pi  \over 4} + {\pi  \over {360}}} \right) \approx \cos {\pi  \over 4} - \sin \left( {{\pi  \over 4}} \right).{\pi  \over {360}}  \cr  & \text{Vậy }\,\cos 45^\circ 30' \approx {{\sqrt 2 } \over 2} - {{\sqrt 2 } \over 2}.{\pi  \over {360}} \approx 0,7009 \cr} \)

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