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Câu 24 trang 205 SGK Đại số và Giải tích 11 Nâng cao

Viết phương trình tiếp tuyến của đồ thị hàm số
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Viết phương trình tiếp tuyến của đồ thị hàm số

LG a

\(y = {{x - 1} \over {x + 1}}\), biết hoành độ tiếp điểm là x0 = 0

Phương pháp giải:

Phương trình tiếp tuyến tại điểm \(M(x_0;y_0)\) là: \(y-y_0=f'(x_0)(x-x_0)\)

Lời giải chi tiết:

\(\eqalign{  & f\left( x \right) = {{x - 1} \over {x + 1}}  \cr  & {x_0} = 0 \Rightarrow {y_0} = f\left( 0 \right) =  - 1  \cr  & f'\left( x \right) \cr & = \frac{{\left( {x - 1} \right)'\left( {x + 1} \right) - \left( {x - 1} \right)\left( {x + 1} \right)'}}{{{{\left( {x + 1} \right)}^2}}} \cr &= \frac{{x + 1 - x + 1}}{{{{\left( {x + 1} \right)}^2}}}\cr & = {2 \over {{{\left( {x + 1} \right)}^2}}} \cr &\Rightarrow f'\left( 0 \right) = 2 \cr} \) Phương trình tiếp tuyến cần tìm là : \(y - \left( { - 1} \right) = 2\left( {x - 0} \right) \Leftrightarrow y = 2x - 1\)

LG b

 \(y = \sqrt {x + 2} ,\) biết tung độ tiếp điểm là y0 = 2.

Lời giải chi tiết:

\(\eqalign{  & f\left( x \right) = \sqrt {x + 2} \cr &f\left( {{x_0}} \right) = 2 \Leftrightarrow \sqrt {{x_0} + 2}  = 2 \cr &\Leftrightarrow {x_0} = 2  \cr  & f'\left( x \right) = {1 \over {2\sqrt {x + 2} }} \Rightarrow f'\left( 2 \right) = {1 \over 4} \cr} \) Phương trình tiếp tuyến cần tìm là : \(y - 2 = {1 \over 4}\left( {x - 2} \right) \Leftrightarrow y = {{x + 6} \over 4}\)

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