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高中数学专题1.1.3导数的几何意义测试新人教A版选修2_2

导数的几何意义 (时间:25 分,满分 55 分) 班级 1.(5 分)下列说法正确的是( ) 姓名 得分 A.若 f′(x0)不存在,则曲线 y=f(x)在点(x0,f(x0))处就没有切线 B.若曲线 y=f(x)在点(x0,f(x0))处有切线,则 f′(x0)必存在 C.若 f′(x0)不存在,则曲线 y=f(x)在点(x0,f(x0))处的切线斜率不存在 D.若曲线 y=f(x)在点(x0,f(x0))处没有切线,则 f′( x0)有可能存在 【答案】C 2.(5 分)已知 y=f(x)的图象如图所示,则 f′(xA)与 f′(xB)的大小关系是( ) A.f′(xA)> f′(xB) C.f′(xA)=f′(xB) 【答案】B B.f′(xA)<f′(xB) D.不能确定 【解析】由导数的几何意义, f′(xA) , f′(xB) 分别是切线在 点 A 、 B 处 切线的斜率,由图象可 知 f′(xA)<f′(xB). π 2 3.(5 分)在曲线 y=x 上切线倾斜角为 的点是( 4 A.(0,0) 1 1 C. ( , ) 4 16 【答案】D 【解析】∵y′=Δ lim x→0 ∴令 2x=tan B.(2,4) 1 1 D.( , ) 2 4 ) x+Δ x 2-x2 =Δ lim (2x+Δ x)=2x, x→0 Δx π 1 1 2 1 =1,得 x= ∴y=( ) = . 4 2 2 4 2 4.(5 分)设曲线 y=ax 在点(1,a)处的切线与直线 2x-y-6=0 平行, 则 a 等于( 1 1 A.1 B. C.- D.-1 2 2 【答案】A ) 【解析】∵y′=Δ lim x→0 a 1+Δ x Δx 2 -a×1 2 =Δ lim (2a+aΔ x)=2a, x→0 ∴可令 2a=2,∴a=1. 5.(5 分)设 f(x)为可导函数,且满足lim x→0 切线的斜率是( A.1 B.-1 C. 【答案】B 【解析】∵lim x→0 ) 1 2 D.-2 f 1 -f 1-x =-1,则曲线 y=f(x)在点(1,f(1))处的 x f 1 -f 1-x f 1-x -f 1 =-1,∴lim =-1,∴f′(1)=-1. x →0 x -x ). 6.(5 分)如图,函数 y=f(x)的图象在点 P 处的切线方程是 y=-x+8,则 f(5)+f′(5)等于( A. 2 【答案】A B.3 C.4 D.5 7.(5 分)设 y=f(x)为可导函数,且满足条件 lix→0 m f 1 -f 1-x =-2,则曲线 y=f(x)在点(1, 2x f(1))处的切线的斜率是________. 【答案】-4 【解析】由 lix→0 m f 1 -f 1-x 1 =-2, ∴ f′(1)=-2,f′(1)=-4. 2x 2 1 8. (5 分) 已知函数 y=f(x)的图象在点 M(1, f(1))处的切线方程是 y= x+2, 则 f(1)+f′(1) =________. 2 【答案】3 1 1 5 【 解析】由在 M 点处的切线方程是 y= x+2,得 f(1)= ×1+2= , 2 2 2 1 1 1+Δ x +2- -2 2 2 =liΔ m x→0 Δx 1 Δx 2 1 5 1 = .∴f(1)+f′(1)= + =3. Δx 2 2 2 f′(1)=liΔ m x→0 ? π? 2 9.(5 分)设 P 为曲线 C:y=x +2x+3 上的点,且曲线 C 在点 P 处的切线倾斜角的范围为?0, ?,则点 4? ? P 横坐标的取值范围为________. 1? ? 【答案】?-1,- ? 2? ? 【解析】∵f′(x) =Δ lim x→0 =Δ lim x→0 x+Δ x 2 +2 x+Δ x +3- Δx 2 x2+2x+3 2x+2 ·Δ x+ Δ x Δx =Δ lim (Δ x +2x+2)=2x+2. x→0 ∴可设 P 点横坐标 为 x0,则曲线 C 在 P 点处的切线斜率为 2x 0+2. 1 由已知得 0≤2x0+2≤1,∴-1≤x0≤- , 2 1? ? ∴点 P 横坐标的取值范围为?-1,- ?. 2? ? 10.(10 分)求过点 P(-1,2)且与曲线 y=3x -4x+2 在点 M(1,1)处的切线平行的直线. 2

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