月25日精講34課件一元函數(shù)微分學(xué)

?等數(shù)學(xué)（?）?元函數(shù)微分學(xué)主講老?、導(dǎo)數(shù)與微分f'(x)limylimf(x0x)f x0 f'(x)

y

f(x0x)f(x0 x0 f'(x)

y

f(x0x)f(x0 x0 ?、導(dǎo)數(shù)與微分（通常為分段函數(shù)的分段點(diǎn)

f(x)3x1,x11x,x則limf(x)f(0) A.- B. C. D.

f(x)x(1x)x,x

0 x左導(dǎo)數(shù) ?、導(dǎo)數(shù)與微分（通常為分段函數(shù)的分段點(diǎn)3x1,x

limf(x)f

f(x)1x,xf

1x(1則（則（

f(0) x

xA.- B. C. D.

f(x)x(1x)x,x

0 x

1f(x)f(0)limx(1x)x 1ln(1x) lim(1x)xlim e ?、導(dǎo)數(shù)與微分x2sin2sin

x(3

f(x)

x用導(dǎo)數(shù)定義計(jì)算f ?、導(dǎo)數(shù)與微分x2sin2sin

x(3

f(x)

x用導(dǎo)數(shù)定義計(jì)算f f'(0)limf(x)f(0)

x2sin2sin lim(xsin2sin2x)0limsin

2limsin2x

?、導(dǎo)數(shù)與微分limf(x02x)f(x0)

?、導(dǎo)數(shù)與微分limf(x02x)f(x0)

limf(x02x)f(x0)2limf(x02x)f(x0)2f'(x)

limf(x02x)f(x0)limf'(x02x)(2)02f'(x) ?、導(dǎo)數(shù)與微分導(dǎo)數(shù)的?何意義：yy0f(x0)(xyy0

f(x0

(xx0可導(dǎo)?定⑵連續(xù)不?定不連續(xù)?定⑷不可導(dǎo)不?定?、導(dǎo)數(shù)與微分 法線?→整理?(1)曲線y=xlnx在點(diǎn)（1，0）處的 法線?程 ?、導(dǎo)數(shù)與微分 法線?→整理?(1)曲線y=xlnx在點(diǎn)（1，0）處的 yx1 法線?程 yx y'lnxxlnxx?、導(dǎo)數(shù)與微分(1)下列函數(shù)中，在點(diǎn)x=0但不可導(dǎo)的是 y yyln

y

x1?、導(dǎo)數(shù)與微分(1)下列函數(shù)中，在點(diǎn)x=0但不可導(dǎo)的是 y yyln

y

x1?、導(dǎo)數(shù)與微分[u(x)v(x)]'u'(x)v'[u(x)v(x)]'u'(x)v(x)u(x)v'[u(x)]'u'(x)v(x)u(x)v' v2[f1(x)]'

f'?、導(dǎo)數(shù)與微分yf udydydu y'[g(x)]f'(u)g?、導(dǎo)數(shù)與微分常見(jiàn)的導(dǎo)數(shù)公式對(duì)數(shù)函數(shù)g三角函數(shù)：y=sinx等反三角函數(shù)：y=arcsinx?、導(dǎo)數(shù)與微分x2x2(1)已知y

x21

ln

，求o?、導(dǎo)數(shù)與微分x2x2(1)已知y

x21

ln

，求'y' x21ln 'x2x2 1x2x21x2

ln x2 1x2 x2

x2 x2 x2 x2

x2

xln ln x2 ln x2?、導(dǎo)數(shù)與微分x(2)設(shè)函數(shù)ysin212x，求x?、導(dǎo)數(shù)與微分x(2)設(shè)函數(shù)ysin212x，求x11111)12)xxxx2x (2x)'2xln 1 sin2 ) sin2 (2x)' 2xln

x

?、導(dǎo)數(shù)與微分(1)設(shè)函數(shù)yy(x)由?程arcsinxlnye2xy3

x?、導(dǎo)數(shù)與微分(1)設(shè)函數(shù)yy(x)由?程arcsinxlnye2xy3

x(yx0)31 1 lnyarcsinxdy2e2x3y2dy1 23

0

x0232?、導(dǎo)數(shù)與微分(2)求由?程xylnyy

x?、導(dǎo)數(shù)與微分(2)求由?程xylnyy

xye2*0yxy'lnyxy'y'00

2 ?、導(dǎo)數(shù)與微分dydty

x?、導(dǎo)數(shù)與微分參數(shù)?

1)yxsinx(xlnysinxlny'cosxlnxsinx (x1)(x(x3)(xdy(x1)(x(x3)(x

x

lny1ln(x1)(x (x3)(x(x4xx1y'1

2x x x x(x4xx?、導(dǎo)數(shù)與微分分別將x，y對(duì)t求導(dǎo)→y對(duì)t的導(dǎo)數(shù)除以xx(1)曲線ytan

o?、導(dǎo)數(shù)與微分分別將x，y對(duì)t求導(dǎo)→y對(duì)t的導(dǎo)數(shù)除以xx(1)曲線ytan

y (x dx cos2o?、導(dǎo)數(shù)與微分

xln3t2t3t(2)設(shè)函數(shù)y=f(x)由參數(shù)3t?、導(dǎo)數(shù)與微分

xln3t2t3t3tdx 333333dy 3dyt

t ?、導(dǎo)數(shù)與微分求導(dǎo)–練習(xí)xy2eysin(3x2y)xtsinty2ty(sin?、導(dǎo)數(shù)與微分求導(dǎo)–練習(xí)1)xy2eysin(3x2y)y2x2yyeyycos(3x2y)(32y)xtsint dycostty2t

sintt3)y(siny(sinx)x(lnsinxxcot?、導(dǎo)數(shù)與微分?階導(dǎo)數(shù)（?、三階導(dǎo)數(shù)dyf'(x)dxd(uv)dud(uv)duvuvdu )?、導(dǎo)數(shù)與微分(1)已知函數(shù)f(x)的導(dǎo)數(shù)f(xxln(1x2求fo?、導(dǎo)數(shù)與微分(1)已知函數(shù)f(x)的導(dǎo)數(shù)f(xxln(1x2求ff(x)ln(1x2)

1f

4x(1x2) 1 (1x2f(1)

1 (1o?、中值定理與導(dǎo)數(shù)的應(yīng)用f(b)f(a)0fbf(b)f(a)fbf(b)f(a)fF(b)F F?、中值定理與導(dǎo)數(shù)的應(yīng)用 定理端點(diǎn)值相等 2 yx34 yx3

By5Dyo?、中值定理與導(dǎo)數(shù)的應(yīng)用 定理端點(diǎn)值相等 是 2 yx34 yx3

By5Dyo?、中值定理與導(dǎo)數(shù)的應(yīng)用 定理端點(diǎn)值相等(2)函數(shù) o?、中值定理與導(dǎo)數(shù)的應(yīng)用 定理端點(diǎn)值相等(2)函數(shù) f # f(1)f(0)f()(1 322

1

?、中值定理與導(dǎo)數(shù)的應(yīng)用limf(x)limf'(x)xx0 xx0g鄰域內(nèi)可導(dǎo)，且鄰域內(nèi)分母的導(dǎo)數(shù)不為0/0∞/∞ 轉(zhuǎn)換后使 法則

10*∞lim

ex

∞-∞limx1

x1x1

lim( 1

(4)

limxx(5)

lim(1x2)x?、中值定理與導(dǎo)數(shù)的應(yīng)用1ex

2e

lim 1 x

x2

1lnxx

x1

lim

x1(x1)lnx

x1lnxx1

x111

x22lim

ln(2arctanx

xln(2arctanxlim lim

ln(arctanx)x 2arctanx1

ln(arctanx)lim

2)

x1 lim

arctanx1 2x01 2limexln e 2x1x2lim 2

2ln(1x)

1 lim lime x

ex2x?、中值定理與導(dǎo)數(shù)的應(yīng)用limxsin x0tanxsin2lim(1 x0 ex1lim(1xlnx)cscx?、中值定理與導(dǎo)數(shù)的應(yīng)用

1cos

1lim 1x0tanxsin2

lim(1 x0

ex1ex1

ex1

ex

x

x(ex

x0 lim(1xln limxlnx

ln

limxlim(x)

x0 1xlnxcscx

limlim(1xlnx)xlnx

x0 ?、中值定理與導(dǎo)數(shù)的應(yīng)用函數(shù)的單調(diào)性：導(dǎo)數(shù)>0→導(dǎo)數(shù)<0→導(dǎo)數(shù)=0→鄰域內(nèi)的值>該點(diǎn)值→鄰域內(nèi)的值<該點(diǎn)值→極??、中值定理與導(dǎo)數(shù)的應(yīng)用的?定極值的判斷：

f(x) x x?、中值定理與導(dǎo)數(shù)的應(yīng)用→判定極值點(diǎn)與極值（分析單調(diào)性f(xxsinxcos

o?、中值定理與導(dǎo)數(shù)的應(yīng)用→判定極值點(diǎn)與極值（分析單調(diào)性f(xxsinxcos

f'(x)sinxxcosxsinf''(x)cosxcosxxsinxcoscosxxsino?、中值定理與導(dǎo)數(shù)的應(yīng)用則x=2是函數(shù)F(x)x22f(x)的 極小值 B.最小值C.極?值 D.最?值o?、中值定理與導(dǎo)數(shù)的應(yīng)用則x=2是函數(shù)F(x)x22f(x)的 B.最小值C.極?值 D.最?值F'(x)2x2f(x)x22fF''(x)2f(x)2x2f'(x)2x2f'(x)x22fo?、中值定理與導(dǎo)數(shù)的應(yīng)用arctan確定函數(shù)f(x)x1e 的單調(diào)區(qū)間與極o?、中值定理與導(dǎo)數(shù)的應(yīng)用arctan(3)確定函數(shù)f(x)x1e 的單調(diào)區(qū)間與極arctan arctan f'(x)e

1arctane

x(11o極大值f(12極小值f(0e?、中值定理與導(dǎo)數(shù)的應(yīng)用(4)求函數(shù)f(x)3x

(x

在區(qū)間[-1,2]上的最?o?、中值定理與導(dǎo)數(shù)的應(yīng)用(4)求函數(shù)f(x)3x

(x

在區(qū)間[-1,2]上的最?2)Uf(x)1

(x令f'(x0解得of(1)0,f(0)2,f(2)o4最大值f(02最小值f(-1?、中值定理與導(dǎo)數(shù)的應(yīng)用

f(x1+x2)f(x1)f(x22?階導(dǎo)數(shù)判斷（凹

f''(x)拐點(diǎn)：兩側(cè)的??階導(dǎo)數(shù)?、中值定理與導(dǎo)數(shù)的應(yīng)用(1)

yx x(k

o?、中值定理與導(dǎo)數(shù)的應(yīng)用(1)

yx x(k

y'

22

2k3x 21k

令y0,,0),凸區(qū)間是(0),拐點(diǎn)是?、中值定理與導(dǎo)數(shù)的應(yīng)用x2(2)求曲線yln x2o?、中值定理與導(dǎo)數(shù)的應(yīng)用x2求曲線yln x2y'

x24x2x24x2x23x2o令y0,,0),凸區(qū)間是(0?、中值定理與導(dǎo)數(shù)的應(yīng)用limf(x)xlimf(x)?、中值定理與導(dǎo)數(shù)的應(yīng)用?平漸近線：兩側(cè)求?窮?曲線yln(1x

的?平漸近線?o?、中值定理與導(dǎo)數(shù)的應(yīng)用?平漸近線：兩側(cè)求?窮?曲線