[PDF] Calculus Cheat Sheet Visit http://tutorial.math.lamar.

0 x!af(x) =L f(x) L x a a x=a x!a+f(x) =L x > a x!af(x) =L x < a x!1f(x) =L f(x) L x x!1f(x) =L x x!af(x) =1 f(x) x a a x=a x!af(x) =1 f(x) x!af(x) =L)x!a+f(x) =x!af(x) =Lx!a+f(x) =x!af(x) =L)x!af(x) =L x!a+f(x)6= x!af(x))x!af(x) x!af(x) x!ag(x) c x!a[cf(x)] =cx!af(x) x!a[f(x)g(x)] =x!af(x)x!ag(x) x!a[f(x)g(x)] =x!af(x)x!ag(x) x!a f(x) g(x) x!af(x) x!ag(x) x!ag(x)6= 0 x!a[f(x)]n= h x!af(x) in x!a h npf(x) i =nqx!af(x) 1 x!1x=1 x! 1x= 0 x!1(x) =1 x!0+(x) =1 r >0 x!1 b xr= 0 r >0xr x x!1 b xr= 0 n x!1xn=1 n x!1xn=1 x! 1xn=1 n x!1axn++bx+c=(a)1 n x!1axn++bx+c=(a)1 n x!1axn++cx+d=(a)1 (a) = 1a >0 (a) =1a <0 f(x) a x!af(x) =f(a) f(x) b x!ag(x) =b x!af(g(x)) =f x!ag(x) =f(b) x!2 x2+ 4x12 x22x=x!2 (x2)(x+ 6) x(x2) =x!2 x+ 6 x=8 2= 4 x!9 3px x281=x!9 3px x281 3 +px 3 +px =x!9 9x (x281)(3 +px)=x!9 1 (x+ 9)(3 +px) =1 (18)(6)=1 108
h!0 1 h 1 x+h1 x =h!0 1 h x(x+h) x(x+h) =h!0 1 h h x(x+h) =h!0 1 x(x+h)=1 x2 x!a f(x) g(x)=0 0 x!a f(x) g(x)=1 1 x!a f(x) g(x)=x!a f0(x) g0(x)a 11 p(x)q(x) x!1 p(x) q(x) xq(x) p(x)q(x) x!1 3x24

5x2x2=x!1

x234 x2 x25 x2 =x!1 34
x2 5 x2=3 2 x!2g(x)g(x) = x2+ 5x <2

13xx 2

x!2g(x) = x!2x2+ 5 = 9 x!2+g(x) =x!2+13x= 7 x!2g(x) x!2g(x) x x x npx x npx x0 x x (x)x >0 (x) (x) x (x) (x) x6=;3 2; 2; 2;3 2; (x) (x) x6=;2;;0;;2; f(x) [a;b] M f(a)f(b) c a < c < bf(c) =M y=f(x) f0(x) =h!0 f(x+h)f(x) h y=f(x) f0(x) =y0=df dx=dy dx=d dx(f(x)) =Df(x) y=f(x) x=a f0(a) =y0jx=a=df dx x=a =dy dx x=a =Df(a) y=f(x) m=f0(a) y=f(x)x=a x=a y=f(a) +f0(a)(xa) f0(a) f(x)x=a f(t) t f0(a) t=a f(x)g(x) cn d dx c = 0 cf(x)

0=cf0(x)

d dx xn =nxn1 f(x)g(x)

0=f0(x)g0(x)

f(x)g(x)

0=f0(x)g(x) +f(x)g0(x)

f(x) g(x) 0 =f0(x)g(x)f(x)g0(x) g(x) 2 d dx f g(x) =f0 g(x) g0(x) d dx x = 1 d dx (x) =(x) d dx (x) =(x) d dx (x) =2(x) d dx (x) =(x)(x) d dx (x) =(x)(x) d dx (x) =2(x) d dx 1(x) =1p1x2 d dx 1(x) =1p1x2 d dx 1(x) =1 1 +x2 d dx ax =ax(a) d dx x =x d dx (x) =1 x; x >0 d dx jxj =1 x; x6= 0 d dx a(x) =1 x(a); x >0 d dx h f(x) in =n h f(x) in1f0(x) d dx f(x) =f0(x)f(x) d dx h f(x) i =f0(x) f(x) d dx h f(x) i =f0(x) h f(x) i d dx h f(x) i =f0(x) h f(x) i d dx h f(x) i =f0(x)2h f(x) i d dx h f(x) i =f0(x) h f(x) i h f(x) i d dx 1h f(x) i =f0(x) 1 + h f(x) i2 2nd f00(x) =f(2)(x) =d2f dx2 f00(x) = f0(x) 0 f0(x) nth f(n)(x) =dnf dxn f(n)(x) = f(n1)(x) 0 (n1)stf(n1)(x) y02x9y+x3y2=(y) + 11x y=y(x) xy y y y0 y0

2x9y(29y0) + 3x2y2+ 2x3y y0=(y)y0+ 11

22x9y9y02x9y+ 3x2y2+ 2x3y y0=(y)y0+ 11

2x3y92x9y(y)y0= 1122x9y3x2y2

)y0=1122x9y3x2y2

2x3y92x9y(y)

x=c f(x) f0(c) = 0 f0(c) f0(x)>0 x I f(x) I f0(x)<0 x I f(x) I f0(x) = 0 x I f(x) I f00(x)>0 x I f(x) I f00(x)<0 x I f(x) I x=c f(x) x=c x=c f(x) f(c)f(x) x x=c f(x) f(c)f(x) x f(x) x=c x=c f(x) f(x) [a;b] cd ac;db f(c) [a;b] f(d) [a;b] f(x) [a;b] f(x)[a;b] f(x) f(a)f(b) x=c f(x) f(c)f(x) xc x=c f(x) f(c)f(x) xc 1st x=c f(x)x=c f(x)f0(x)>0 x=cf0(x)<0 x=c f(x)f0(x)<0 x=cf0(x)>0 x=c f(x)f0(x x=c 2nd x=c f(x) f0(c) = 0 x=c f(x)f00(c)<0 f(x)f00(c)>0 f00(c) = 0 f(x) 1st 2nd f(x) [a;b] (a;b) a < c < b f0(c) =f(b)f(a) ba xn nth f(x) = 0(n+ 1)st xn+1=xnf(xn) f0(xn) f0(xn) t t 1 4 x0 x x2+y2= 152)2xx0+ 2y y0= 0 x= 10121 4 = 7 y=p15272=p176 y0 7 1 4 +p176y0= 0)y0=7 4p176 = 0:5

0= 0:01 x0

() =x

50)()()0=x0

50
= 0:5 0 (0:5)(0:5)(0:01) =x0 50
x0= 0:3112

A=xy x+2y= 500

quotesdbs_dbs12.pdfusesText_18