2011年电大《经济数学基础》形成性考核册答案
大题必须写步骤,否则0分处理
作业(一)
(一)填空题 1.limx?0x?sinx?___________________.答案:0 x?x2?1,x?02.设f(x)??,在x?0处连续,则k?________.答案:1
?k,x?0?3.曲线y?x+1在(1,2)的切线方程是 .答案:x-2y+1=0
__.答案:2x 4.设函数f(x?1)?x2?2x?5,则f?(x)?__________5.设f(x)?xsinx,则f??()?__________.答案:?π2π 2(二)单项选择题
1.当x→+∞时,下列变量为无穷小量的是( )答案:D
A.
ln(1?x)B.D.x2C.exx?1x2x?1 sinxx2. 下列极限计算正确的是( )答案:B A.limx?0?1 B.lim?x?0xx?1
C.limxsinx?01sinx?1 D.lim?1
x??xx3. 设y?lg2x,则dy?( ).答案:B A.
11ln101dx B.dx C.dx D.dx 2xxln10xx4. 若函数f (x)在点x0处可导,则( )是错误的.答案:B
A.函数f (x)在点x0处有定义 B.limf(x)?A,但A?f(x0)
x?x0 C.函数f (x)在点x0处连续 D.函数f (x)在点x0处可微
1)=x,则f’(x)=( ). 答案:B x1111A.2 B.—2 C. D.—
xxxx5.若f ((三)解答题
1
1.计算极限
x2?3x?21(x?2)(x?1)x?2(1)lim = = ??limlim2x?1x?1x?12x?1(x?1)(x?1)(x?1)x2?5x?61(x?2)(x?3)x?3(2)lim2=lim = lim =
x?2x?6x?8x?2(x?2)(x?4)x?2(x?4)2(1?x?1)(1?x?1)1?x?1lim(3)lim= x?0x?0xx(1?x?1) =limx?0?x?11=lim??
2x(1?x?1)x?0(1?x?1)2?5x2?2 43x23?2x2?3x?5x?lim(4)lim2x??3x?2x?4x??23??x(5)lim5xsin3x33sin3x?lim= x?03xsin5x55x?0sin5xx2?4(x?2)(x?2)(6)lim?lim?4
x?2sin(x?2)x?2sin(x?2)
1?xsin?b,x?0?x?2.设函数f(x)??a,x?0,
?sinxx?0?x?问:(1)当a,b为何值时,f(x)在x?0处有极限存在? (2)当a,b为何值时,f(x)在x?0处连续.
1lim?f(x)?lim?(xsin?b)?bx?0x?0x答案:(1)
sinxlim?f(x)?lim??1x?0x?0x当b?1,a任意时,f(x)在x?0处有极限存在; (2)f(0)= a =limf(0)?b?1
x?0当a?b?1时,f(x)在x?0处连续。
2
3.计算下列函数的导数或微分:
(1) y=x2+2x+log2x-22 求y' 解:y'??x2?'??2x?'??log2x?'??22?'
?2x?2xln2?1 xln2
(2) y=(ax+b)/(cx+d), 求y' 解:y'???ax?b?'?cx?d???ax?b??cx?d?' 2?cx?d?a?cx?d???ax?b?c?cx?d?2?ad?bc?cx?d?2
(3)y?1,求y' 3x?51?2'1313??1???解:y'???3x?5?????3x?5?2??3x?5?'???3x?5?2。
22??
(4)y?x?xex,求y'
1??1xxx2?'??xe?'?x?x'e?x?e?'
2??12解:y'??x???1?1?11?x2??ex?xex??x2?ex?1?x? 22
(5)y=eax sinbx,求dy。
解 y'??eax?'sinbx?eax?sinbx?'?eax?ax?'sinbx?eaxcosbx?bx?'
?aeaxsinbx?beaxcosbx,
3
dy?y'dx?(aeaxsinbx?beaxcosbx)dx
(6)
y?e?xx,求dy
1x'32'1x'11????13?x?2y'?e?x?e?x??????解:
x2??????1x?1x??'11313?x2??exx?2?x2, 22?e?2?(?exdy?y'dx1x31?x2)dx 2
(7)y?cosx?e?x2,求dy。
?x2?x2)'??sinx?(x)'?e???x?'
2解:y'?(cosx)'?(e12??sinx?(x)'?2xe?x21?1?x22??sinx?x?2xe,
21?1?x22dy?y'dx?(?sinx?x?2xe)dx
2
(8)y=sinnx+sin nx,求y'。
解:y'=(sinnx)’+(sin nx)’ =n sinn-1x (sinx)’+cos nx (nx)’
=n cosx sinn-1x +ncos nx
(9)y?lnx?1?x2,求dy。
?? 4
解:y'?1x?1?x2?x?1?x2'??1x?1?x2?1?(1?x)'?
21??12?2?1?x?' ?1?1?x?????x?1?x2?21??12x??1?? 222x?1?x?1?x?1?1x???1???222x?1?x?1?x?x?1?x1?11?x?21?x2?x1?x2
dx1?x2dy?y'dx
(10)y?2。
cot1x1?3x2?2x。 ?,求y'
x?x?12解:因y?2所以y'?2cotcot1x?x21?32?2?2cot1x?x?12?x?2,
161x153cot??1?1?31111???2xln2?cot?'?x2?1??csc?2ln2???'?x2?x6
x?2x6??x?2?56?csc21?2x1cotx1ln2?x?2?x23?2?x 6
1、 下列各方程中y是x的隐函数,试求y'或dy
(1)
x2+y2-xy+3x=1 ,求 dy
解:方程两边对x求导,
2x?2yy'?y?xy'?3?0, (2y?x)y'?y?3?2x,
y'?y?3?2xy?3?2x'dx。 , dy?ydx?2y?x2y?x(2)
sin(x+y)+exy=4x , 求y'
5