#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)
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#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))
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#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))
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