Calculusi Notes
CALCULUS (I) RULES
opp. opp. hyp. hyp. θ θ adj. adj. tanθ = sinθcosθ = opp.adj. secθ = 1cosθ = hyp.adj. cscθ = 1sinθ = hyp.opp. cotθ = 1tanθ = cosθsinθ = adj.opp.
sin2θ + cos2θ = 1 sin2θ = 1 - cos2θ cos2θ = 1 - sin2θ tan2θ = sec2θ - 1 csc2θ = 1 + cot2θ sec2θ = 1 + tan2θ cot2θ = csc2θ - 1 S
S
A
A
- θ = 360 – θ
T
T
C
C
∴ sin(-θ) = - sinθ & csc(-θ) = - cosecθ cos(-θ) = + cosθ & sec(-θ) = + secθ tan(-θ) = - tanθ & cot(-θ) = - cotθ
sinπ2-x=cosx cosπ2-x=sinx tanπ2-x=cotx cscπ2-x=secx secπ2-x=cscx cotπ2-x=tanx
sin(A±B) = sinAcosB ± cosAsinB cos(A±B) = cosAcosB ∓ sinAsinB tan(A±B)= tanA ± tanB1 ∓tanAtanB
∴ sin(2A) = 2sinAcosA cos(2A) = cos2A – sin2A = 1 – 2sin2A = 2cos2A – 1 tan(2A) = 2tanA1- tan2A
sin2A = 12 - 12cos2A cos2A = 12 + 12cos2A tan2A = 1-cos(2A)1+cos(2A)
sinAsinB = 12 ( cos(A-B) – cos(A+B) ) cosAcosB = 12 ( cos(A-B) + cos(A+B) ) sinAcosB = 12 ( sin(A+B) + sin(A-B) ) cosAsinB = 12 ( sin(A+B) – sin(A-B) )
* ddxc=zero * ddxcx=c * ddxcy=cdydx * ddxf(x)±g(x)=d(fx)dx±d(gx)dx * ddxfx.g(x)=dfxdx.gx+dgxdx.f(x) * ddxf(x)±g(x)=d(fx)dx±d(gx)dx * ddxfx.gx.hx= dfxdx.gx.hx+dgxdx.fx.hx+dhxdx.fx.g(x) * ddxf(x)g(x)=dfxdx.gx-dgxdx.f(x)[gx]2 * dydx=dydu.dudv.dvdx , where y=fu, u=gv, v=h(x) * ddxxn=n.xn-1 * ddxyn=n.yn-1.dydx ,where y=f(x) * dydx=1dxdy
* ddxsinx=cosx * ddxcscx=-cscx.cotx * ddxcosx=-sinx * ddxsecx=secx.tanx * ddxtanx=sec2x * ddxcotx=-csc2x
* xn dx=xn+1n+1+c, where n≠-1 * x-1 dx=lnx+c * (ax+b)n dx= 1a(ax+b)n adx= 1a×(ax+b)n+1n+1+c * sinx dx= -cosx+c * cosx dx= sinx+c * cosec2x dx= -cotx+c * sec2x
opp. opp. hyp. hyp. θ θ adj. adj. tanθ = sinθcosθ = opp.adj. secθ = 1cosθ = hyp.adj. cscθ = 1sinθ = hyp.opp. cotθ = 1tanθ = cosθsinθ = adj.opp.
sin2θ + cos2θ = 1 sin2θ = 1 - cos2θ cos2θ = 1 - sin2θ tan2θ = sec2θ - 1 csc2θ = 1 + cot2θ sec2θ = 1 + tan2θ cot2θ = csc2θ - 1 S
S
A
A
- θ = 360 – θ
T
T
C
C
∴ sin(-θ) = - sinθ & csc(-θ) = - cosecθ cos(-θ) = + cosθ & sec(-θ) = + secθ tan(-θ) = - tanθ & cot(-θ) = - cotθ
sinπ2-x=cosx cosπ2-x=sinx tanπ2-x=cotx cscπ2-x=secx secπ2-x=cscx cotπ2-x=tanx
sin(A±B) = sinAcosB ± cosAsinB cos(A±B) = cosAcosB ∓ sinAsinB tan(A±B)= tanA ± tanB1 ∓tanAtanB
∴ sin(2A) = 2sinAcosA cos(2A) = cos2A – sin2A = 1 – 2sin2A = 2cos2A – 1 tan(2A) = 2tanA1- tan2A
sin2A = 12 - 12cos2A cos2A = 12 + 12cos2A tan2A = 1-cos(2A)1+cos(2A)
sinAsinB = 12 ( cos(A-B) – cos(A+B) ) cosAcosB = 12 ( cos(A-B) + cos(A+B) ) sinAcosB = 12 ( sin(A+B) + sin(A-B) ) cosAsinB = 12 ( sin(A+B) – sin(A-B) )
* ddxc=zero * ddxcx=c * ddxcy=cdydx * ddxf(x)±g(x)=d(fx)dx±d(gx)dx * ddxfx.g(x)=dfxdx.gx+dgxdx.f(x) * ddxf(x)±g(x)=d(fx)dx±d(gx)dx * ddxfx.gx.hx= dfxdx.gx.hx+dgxdx.fx.hx+dhxdx.fx.g(x) * ddxf(x)g(x)=dfxdx.gx-dgxdx.f(x)[gx]2 * dydx=dydu.dudv.dvdx , where y=fu, u=gv, v=h(x) * ddxxn=n.xn-1 * ddxyn=n.yn-1.dydx ,where y=f(x) * dydx=1dxdy
* ddxsinx=cosx * ddxcscx=-cscx.cotx * ddxcosx=-sinx * ddxsecx=secx.tanx * ddxtanx=sec2x * ddxcotx=-csc2x
* xn dx=xn+1n+1+c, where n≠-1 * x-1 dx=lnx+c * (ax+b)n dx= 1a(ax+b)n adx= 1a×(ax+b)n+1n+1+c * sinx dx= -cosx+c * cosx dx= sinx+c * cosec2x dx= -cotx+c * sec2x