Calculus UNIT204 Finding the Derivative!

2809 days ago by MATH4R2013

#1) find the derivative of f(x)=sin(x), what if the power doesn't work? f(x)=sin(x) var('h') show((f(x+h)).simplify_trig()) show((f(x+h)-f(x)).simplify_trig()) show(((f(x+h)-f(x))/h).simplify_trig()) show(lim((f(x+h)-f(x))/h,h=0)) p1=plot(sin(x),0,2*pi,color='red') p2=plot(cos(x),0,2*pi,color='green') (p1+p2).show(aspect_ratio=1) 
       




#1a) find the hole at h=0 var('h') g(h)=(cos(h)-1)/h print "h\t\t\tg(h)" for i in range(6): print .1^i,"\t",g(.1^i) plot(g(h),-2*pi,2*pi) 
       
h			g(h)
1.00000000000000 	-0.459697694131860
0.100000000000000 	-0.0499583472197418
0.0100000000000000 	-0.00499995833347366
0.00100000000000000 	-0.000499999958325503
0.000100000000000000 	-0.0000499999996961264
0.0000100000000000000 	-5.00000041370185e-6
h			g(h)
1.00000000000000 	-0.459697694131860
0.100000000000000 	-0.0499583472197418
0.0100000000000000 	-0.00499995833347366
0.00100000000000000 	-0.000499999958325503
0.000100000000000000 	-0.0000499999996961264
0.0000100000000000000 	-5.00000041370185e-6
#1b) find the hole at h=0 var('h') g(h)=(sin(h))/h print "h\t\t\tg(h)" for i in range(6): print .1^i,"\t",g(.1^i) plot(g(h),-2*pi,2*pi) 
       
h			g(h)
1.00000000000000 	0.841470984807897
0.100000000000000 	0.998334166468282
0.0100000000000000 	0.999983333416666
0.00100000000000000 	0.999999833333342
0.000100000000000000 	0.999999998333333
0.0000100000000000000 	0.999999999983333
h			g(h)
1.00000000000000 	0.841470984807897
0.100000000000000 	0.998334166468282
0.0100000000000000 	0.999983333416666
0.00100000000000000 	0.999999833333342
0.000100000000000000 	0.999999998333333
0.0000100000000000000 	0.999999999983333
#2) find the derivative of f(x)=1/x, does the power rule work? var('h') f(x)=1/x show(f(x+h)) show((f(x+h)-f(x)).simplify_rational()) show(((f(x+h)-f(x))/h).simplify_rational()) show(lim((f(x+h)-f(x))/h,h=0)) 
       



#3) find the derivative of f(x)=1/x, does the power rule work? var('h') f(x)=sqrt(x) show(f(x+h)) show(f(x+h)-f(x)) show((f(x+h)-f(x))/h) show(lim((f(x+h)-f(x))/h,h=0))