参变分离再认识(课件)

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title: 一题一课(参变分离再认识)
speaker: 魏刚

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一题一课\((7)\)<br >

\(\hspace{0.3cm}\)------参变分离再认识

数学研究的四个方面： 数值、变化、结构、图像！

\(\hspace{6cm}\) 2016\(\hspace{0.05cm}\cdot\hspace{0.1cm}\)四川\(\hspace{0.05cm}\cdot\hspace{0.1cm}\)成都\(\hspace{0.05cm}\cdot\hspace{0.1cm}\)魏刚老师

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问题回顾

方程\(x^2-ax+1=0\)在区间\((\dfrac{1}{2},3)\)上有根\(,\hspace{0.13cm}\)

则实数\(a\)的取值范围为\(\underline{\qquad\blacktriangle\qquad}.\)

[note]演算[/note]

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问题回顾

[note]演算[/note]

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问题1

已知函数 \(\textit{f}(\textit{x})=\textit{kx}-1\) \(,\) \(\textit{g}(\textit{x})=\ln\textit{x}\) \(,\)

若 \(\textit{f}(\textit{x})\geqslant\textit{g}(\textit{x})\) 在 \((0,+\infty)\)上恒成立 \(,\)

求 \(\textit{k}\) 的取值范围.

[note]演算[/note]

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动态图形绘制

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问题2

已知函数 \(\textit{f}(\textit{x})=\textit{k}(\textit{x}-1)\) \(,\) \(\textit{g}(\textit{x})=\ln\textit{x}\) \(,\)

若 \(\textit{f}(\textit{x})\geqslant\textit{g}(\textit{x})\)在 \([1,+\infty)\)上恒成立 \(,\)

求 \(\textit{k}\) 的取值范围.

[note]演算[/note]

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动态图形绘制

[slide data-transition = 'cards'] ####幂函数的图像 [slide data-transition = 'cards'] ####幂函数在\\((1,+\infty)\\)上的图像

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课堂练习1

已知函数\(\textit{f}(\textit{x})=\sin\textit{x} (\textit{x}\geqslant 0),\) \(\textit{g}(\textit{x})=\textit{a}\textit{x} (\textit{x}\geqslant 0),\)

若不等式\(\textit{f}(\textit{x})\leqslant\textit{g}(\textit{x})\)恒成立\(,\) 求实数\(\textit{a}\) 的取值范围\(.\)

[note]演算[/note]

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课堂练习2

已知函数\(\textit{f}(\textit{x})=\text{e}^\textit{x}+\textit{ax}\) \( (\)其中\(\text{e}=2.71828\cdots)\)

若对于任意的\(\textit{x}\geqslant 0, \textit{f}(\textit{x})\geqslant\text{e}^{-\textit{x}}\) 恒成立\(,\)

求实数\(\textit{a}\) 的取值范围\(.\)

[note]演算[/note]

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计时屏幕

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课堂练习3

已知函数\(\textit{f}(\textit{x})=\textit{ax}^2-\textit{a}-\ln\textit{x}, \) 其中\(\textit{a}\in\textbf{R}.\)

试确定\(\textit{a}\) 的所有可能值\(, \) 使得\(\textit{f}(\textit{x})>\dfrac{1}{\textit{x}}-\text{e}^{1-\textit{x}}\)

在区间\((1,+\infty)\)内恒成立\(.(\)其中\(\text{e}=2.71828\cdots)\)

[note]演算[/note]

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课堂练习4

当\(\textit{x}\geqslant 0\) 时\(,\) \(\text{e}^{\textit{x}}-1-\textit{x}\geqslant \textit{a}\textit{x}^2\) 恒成立\(,\)

求实数\(\textit{a}\) 的取值范围\(.(\)其中\(\text{e}=2.71828\cdots)\)

[note]演算[/note]

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课堂练习5

已知函数\(f(x)=1-\text{e}^{-x}, \forall x\in[0,+\infty), \)

\(f(x)\leqslant \dfrac{x}{ax+1},\) 求\(a\)的取值范围\(.\)

[note]演算[/note]

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思考题1

已知函数\(\textit{f}(\textit{x})\)和函数\(\textit{g}(\textit{x})\)都在\(\textbf{R}\)上连续

且可导\(, f(x_0)=g(x_0).\)

p： \(\forall x\in [x_0,+\infty), f(x)\leqslant g(x)\)恒成立\(;\)

q： \(f'(x_0)\leqslant g'(x_0).\)

则q是p的\(\underline{\qquad\qquad\qquad}\)条件\(.\)

\((\)其中\(f'(x)\)为\(f(x)\)的导函数\(, g'(x)\)为\(g(x)\)的导函数\()\)

[note]演算[/note]

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思考题2

已知函数\(\textit{f}(\textit{x})\)在\(\textbf{R}\)上连续且可导\(, f(0)=f(1)=0.\)

p： 函数\(f(x)\)在区间\(（0,1）\)上有零点\(;\)

q： \(f'(0)f'(1)>0.\)

则q是p的\(\underline{\qquad\qquad\qquad}\)条件\(.\)

\((\)其中\(f'(x)\)为\(f(x)\)的导函数\()\)

[note]演算[/note]