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高等数学基础归类复习
一、单项选择题
1-1下列各函数对中，（ C ）中的两个函数相等． A.
f(x)?(x)2，g(x)?x B. f(x)?x23，g(x)?x
x2?1 C.f(x)?lnx，g(x)?3lnx D. f(x)?x?1，g(x)?
x?11-⒉设函数f(x)的定义域为(??,??)，则函数f(x)?f(?x)的图形关于（C ）对称．
A. 坐标原点 B. x轴 C. y轴 D. y?x
设函数f(x)的定义域为(??,??)，则函数f(x)?f(?x)的图形关于（D ）对称．
A. y?x B. x轴 C. y轴 D. 坐标原点 e?x?ex.函数y?2的图形关于（ A ）对称．
(A) 坐标原点 (B)
x轴 (C) y轴 (D) y?x
1-⒊下列函数中为奇函数是（ B ）． A.
y?ln(1?x) B. y?xcosx C.
2ax?a?xy?2 D.
y?ln(1?x)
下列函数中为奇函数是（A ）． A.
y?x3?x B. y?ex?e?x C. y?ln(x?1) D. y?xsinx
A
下列函数中为偶函数的是（ D ）．
y?(1?x)sinx B y?x2x C y?xcosx D y?ln(1?x2)
2-1 下列极限存计算不正确的是（ D ）．
x2?1 B. limln(1?x)?0 A. lim2x??x?2x?0sinx1?0 D. limxsin?0 C. limx??x??xx2-2当x?0时，变量（ C ）是无穷小量．
1sinx1A. B. C. xsin D. ln(x?2)
xxxx1sinxx当x?0时，变量（ C ）是无穷小量．A B C e?1 D 2
xxxsinx1x.当x?0时，变量（D ）是无穷小量．A B C 2 D ln(x?1)
xx下列变量中，是无穷小量的为（ B ）
11x?2 Asin?x?0? B ln?x?1??x?0? Cex?x??? D.2?x?2?
xx?4精品 1
学 海 无 涯
f(1?2h)?f(1)3-1设f(x)在点x=1处可导，则lim?（ D ）．
h?0hA. f?(1) B. ?f?(1) C. 2f?(1) D. ?2f?(1)
f(x0?2h)?f(x0)设f(x)在x0可导，则lim?（ D ）．
h?0hA f?(x0) B 2f?(x0) C ?f?(x0) D ?2f?(x0)
设
f(x)在x0可导，则limA.
f(x0?2h)?f(x0)?（ D ）．
h?02h?2f?(x0) B. f?(x0) C. 2f?(x0) D. ?f?(x0)
?x?0设
f(x)?ex，则limf(1??x)?f(1)11?（ A ） A e B. 2e C. e D. e
?x2413-2. 下列等式不成立的是（D ）．
1dx?dx D.lnxdx?d()
x2x11dx)?arctanxdxd()??下列等式中正确的是（B ）．A.d( B.
x1?x2x2xx C.d(2ln2)?2dx D.d(tanx)?cotxdx
A.exdx?dex B ?sinxdx?d(cosx) C.
f(x)?x2?4x?1的单调增加区间是（ D ）．
A. (??,2) B. (?1,1) C. (2,??) D. (?2,??)
4-1函数函数.函数
y?x2?4x?5在区间(?6,6)内满足（A ）．
A. 先单调下降再单调上升 B. 单调下降 C. 先单调上升再单调下降 D. 单调上升
y?x2?x?6在区间（－5，5）内满足（ A ）
A 先单调下降再单调上升 B 单调下降 C先单调上升再单调下降 D 单调上升
. 函数
y?x2?2x?6在区间(2,5)内满足（D ）．
A. 先单调下降再单调上升 B. 单调下降 C. 先单调上升再单调下降 D. 单调上升
5-1若
f(x)的一个原函数是
1，则f?(x)?（D ）． A. lnxx B.
?1x2 C.
21 D. 3xx
.若F(x)是 AC5-2若
f(x) 的一个原函数，则下列等式成立的是（ A ）。
?xaf(x)dx?F(x)?F(a) B
?babF(x)dx?f(b)?f(a)
f?(x)?F(x) D?f?(x)dx?F(b)?F(a)
af(x)?cosx，则?f?(x)dx?（ B ）．
A. A.
sinx?c B. cosx?c C. ?sinx?c D. ?cosx?c
下列等式成立的是（D ）．
?f?(x)dx?f(x) B. ?df(x)?f(x)
df(x)dx?f(x) C. d?f(x)dx?f(x) D.
dx?d11233233xf(x)f(x)xf(x)dx?f(x)f(x) （ B ）． A. B. C. D. ?dx33d11222xf(x)xf(x)dx xf(x)dx?f(x)dxf(x)（ D ） A B C D ?dx221f(x)dx?（ B ）． ⒌-3若?f(x)dx?F(x)?c，则?x精品
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学 海 无 涯
A.
F(x)?c B. 2F(x)?c C. F(2x)?c D.
?x?x?x ?F(e)?c, 无穷积分收敛的是 ef(e)dx??1xF(x)?c
补充：
???11dx 2x 函数
二、填空题
f(x)?10x?10?x的图形关于 y 轴 对称。
x2?9?ln(1?x)的定义域是 （3，+∞） ．
x?3⒈函数f(x)?函数
x?4?x的定义域是 （2，3） ∪ （3，4 ]
ln(x?2)1函数f(x)?ln(x?5)?的定义域是 （－5，2）
2?x?x2?1,x?0若函数f(x)??，则f(0)? 1 ．
xx?0?2,1?x?2若函数f(x)??(1?x),x?0，在x?0处连续，则k? e
?x?0?x?k,?sin2x?x?0.函数f(x)??x在x?0处连续，则k? 2
?x?0?k?x?1,x?0函数y??的间断点是 x=0 ．
?sinx,x?0x2?2x?3函数y?的间断点是 x=3 。
x?31函数y?的间断点是 x=0 x1?e3-⒈曲线f(x)?x?1在(1,2)处的切线斜率是 1/2 ．
y?曲线曲线.曲线
．
f(x)?x?2在(2,2)处的切线斜率是 1/4 ．
f(x)?ex?1在（0，2）处的切线斜率是 1 ．
f(x)?x3?1在(1,2)处的切线斜率是 3 ．
π3-2 曲线f(x)?sinx在(,1)处的切线方程是 y = 1 ．切线斜率是 0 2曲线y = sinx 在点 (0,0)处的切线方程为 y = x 切线斜率是 1
4.函数y?ln(1?x)的单调减少区间是 （－∞，0 ） ．
2f(x)?ex的单调增加区间是 （0，+∞） ．
2.函数y?(x?1)?1的单调减少区间是 （－∞，－1 ） ．
2.函数f(x)?x?1的单调增加区间是 （0，+∞） ．
函数 函数5-1d2y?e?x22的单调减少区间是 （0，+∞） ．
?x?edx?
e?xdx
2 ． .
dsinx2dx? sinx2． ?dx?(tanx)?dx? tan x +C ．
若?f(x)dx?sin3x?c，则f?(x)? －9 sin 3x ．
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5-2
15(sinx?)dx? 3 ． ??323学 海 无 涯 1x3dedx? 0 ． ln(x?1)dx? 0 ??1x2?1?1dx1下列积分计算正确的是（ B ）．
A
?1?1(ex?e?x)dx?0 B?(ex?e?x)dx?0 C?x2dx?0 D
?1?11?1?1|x|dx?0
三、计算题
（一）、计算极限（1小题，11分）
（1）利用极限的四则运算法则，主要是因式分解，消去零因子。 （2）利用连续函数性质：类型1： 利用重要极限 f(x0)有定义，则极限limf(x)?f(x0)
x?x0limsinxsinkxtankx?1 ， lim?k， lim?k 计算
x?0x?0x?0xxxsin6xsin6x6x1-1求lim． 解： limsin6x?lim?? x?0sin5xx?0sin5xx?0sin5x5xtanxtanx1tanx11 解： lim?lim??1?
x?03xx?03x3x?0x33tan3xtan3xtan3x.3?1?3?3 1-3 求lim 解：lim=limx?0x?0x?0xx3xsin(x?a)x?a?1， lim?1 化简计算。 类型2： 因式分解并利用重要极限 limx?a(x?a)x?asin(x?a)1-2 求 lim(x?1)x2?1x2?1.(x?1)?1?(?1?1)??2 2-1求lim． 解： lim=limx??1sin(x?1)x??1sin(x?1)x??1sin(x?1)sin?x?1?sin(x?1)sin(x?1)111lim?lim.?1??2-2lim 解： 22x?1x?1x?1(x?1)(x?1)1?12x?1x?1x2?4x?3x2?4x?3(x?3)(x?1)?lim?lim(x?1)?2 2-3lim 解： limx?3sin(x?3)x?3x?3x?3sin(x?3)sin(x?3)类型3：因式分解并消去零因子，再计算极限
x2?6x?8x2?6x?8(x?4)(x?2)x?22?lim? 3-1 lim2 解： lim2=limx?4x?5x?4x?4x?5x?4x?4(x?4)(x?1)x?4x?13x2?x?6?x?3??x?2??limx?2?5 x2?x?63-2 lim2 lim2?limx??3x?x?12x??3x?x?12x??3?x?3??x?4?x??3x?47x2?3x?2x2?3x?2(x?2)(x?1)x?113-3 lim 解 lim?lim?lim? 22x?2x?2x?2x?2x?4(x?2)(x?2)x?24x?412x2sinxsin1?x?12其他： lim?lim?2 ?lim?0， limx?0x?0x?0x?01sinxsinxx?1?1x22222x?6xx?6x?5x2x22lim2?lim2?1， lim2?lim2?
x??3x?4x?5x??x?4x?5x??xx??3x3tan8xtan8xx.?8?2 （0807考题）计算lim． 解： lim=limx?0sin4xx?0sin4xx?0sin4x4x精品
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tan8x学 海 无 涯
sinxsinx1sinx1（0801考题. ）计算lim． 解 lim?lim?
x?02xx?02xx?02x2(x?1).(x?3)x2?2x?3?1?(?1?3)??4 （0707考题.）lim=limx??1sin(x?1)x??1sin(x?1)（二） 求函数的导数和微分（1小题，11分）
（1）利用导数的四则运算法则
(u?v)??u??v? (uv)??u?v?uv?
（2）利用导数基本公式和复合函数求导公式
1aa?1 (x)??ax xuuxx (e)??e (e)??e.u? (sinx)??cosx222(ex)??ex.(x2)??2xex(cosx)???sinxsinxsinxsinx?? (e)?e.(sinx)?ecosx 2?(tanx)?secx(ecosx)??ecosx.(cosx)???ecosxsinx2(cotx)???cscx(sinu)??cosu.u?(cosu)???sinu.u?(lnx)??(sinx2)??cosx2.(x2)??2xcosx2(sinex)??cosex.(ex)??excosex1-1 (cosx2)???sinx2(x2)???2xsinx2(cose)???sinex.(ex)???exsinex 类型1：加减法与乘法混合运算的求导，先加减求导，后乘法求导；括号求导最后计算。 y?(xx?3)ex 13?133??x???3???33?xxxx 解：y?＝?x2?3?e??x2?3??e??x2e??x2?3?e??x2?x2?3?e
2?2???????21-2 y?cotx?xlnx
22222 解：y??(cotx)??(xlnx)???cscx?(x)?lnx?x(lnx)???cscx?2xlnx?x
x1-3 设y?etanx?lnx，求y?．
11xxxxx2解： y??(etanx)??(lnx)??(e)?tanx?e(tanx)???etanx?esecx?
xx类型2：加减法与复合函数混合运算的求导，先加减求导，后复合求导 2-1
y?sinx2?lnx，求y? 解：y??(sinx2)??(lnx)??2xcosx2?1 x2-2 y?cosex?sinx2，求
x2xx22xx2解：y??(cose)??(sinx)???sine.(e)??cosx.(x)???esine?2xcosx
2-3
y?ln5x?e?5x，求
2， 解：
y??(ln5x)??.(e?5x)??2254lnx?5e?5x x2类型3： 乘积与复合函数混合运算的求导，先乘积求导，后复合求导 y?excosx，求y? 。 解：y??(ex)?cosx?ex(cosx)??2xexcosx?exsinx
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(2020年整理)电大高等数学基础考试答案完整版(整理).doc
