举一反三
- 函数\( f\left( x \right) = 2{x^2} - x - 3 \)在区间\( \left[ { - 1,1.5} \right] \)上满足罗尔定理的点是 ________. ______
- 函数\( f\left( x \right) = {x^2} \)在区间\( \left[ {1,2} \right] \)上满足拉格朗日中值定理的数值\( \xi \)= ________。______
- 函数\( f\left( x \right) = {x^3} \)与\( g\left( x \right) = {x^2} + 1 \)在区间\( \left[ {1,2} \right] \)上不满足柯西中值定理。（ ）
- 函数\( f\left( x \right) = {x^3} \)在区间\( \left[ {1,2} \right] \)上满足拉格朗日中值定理的数值\( \xi \)=（ ）. A: \( { { \sqrt {21} } \over 3} \) B: \( { { \sqrt {21} } \over 7} \) C: \( {7 \over 3} \) D: \( {3 \over 7} \)
- 函数\( f\left( x \right) = 5{x^2} + x - 2 \)在区间\( \left[ {0,1} \right] \)上满足拉格朗日中值定理（ ）
内容
- 0
多项式\( f\left( x \right) = {x^3} - 3x + a \)在\( \left[ {0,1} \right] \)上有两个零点（ ）
- 1
已知函数\( f(x) \)在区间\( \left[ {a,b} \right] \)上连续，则由\( y = f(x),\;x = a,\;x = b,\;x \)轴围成的平面图形面积为（ ）。 A: \( \int_a^b {f(x)dx} \) B: \( \left| {\int_a^b {f(x)dx} } \right| \) C: \( \int_a^b {\left| {f(x)} \right|dx} \) D: \( f'(\xi )(b - a) \)
- 2
函数\( y = x \)在区间\( \left[ { - 1,1} \right] \)上满足罗尔定理（ ）
- 3
若\({y_1}\left( x \right), {y_2}\left( x \right)\)都是\(y' + P\left( x \right)y = Q\left( x \right)\)的特解，且 \({y_1}\left( x \right), {y_2}\left( x \right)\) 线性无关，则通解可表为\(y\left( x \right) = {y_1}\left( x \right) + C\left[ { { y_1}\left( x \right) - {y_2}\left( x \right)} \right]\)。
- 4
求函数$y = \root 3 \of {x + \sqrt x } $的导数$y' = $( ) A: ${{1 + 2\sqrt x } \over {\root 3 \of {{{\left( {x + \sqrt x } \right)}^2}} }}$ B: $ {{1 + 2\sqrt x } \over {6\root 3 \of {{{\left( {x + \sqrt x } \right)}^2}} }}$ C: $ {{1 + 2\sqrt x } \over {6\sqrt x \cdot \root 3 \of {{{\left( {x + \sqrt x } \right)}^2}} }}$ D: $ {{1 + 2\sqrt x } \over {\sqrt x \cdot \root 3 \of {{{\left( {x + \sqrt x } \right)}^2}} }}$