#1a) what's the slope of the tangent line at t=1sec? [h]=ft of ball off the ground
h(t)=1024-16*t^2
show(h(t))
show((h(1)-h(8))/(1-8))
show((h(1)-h(7))/(1-7))
show((h(1)-h(6))/(1-6))
show((h(1)-h(5))/(1-5))
show((h(1)-h(4))/(1-4))
show((h(1)-h(3))/(1-3))
show((h(1)-h(2))/(1-3))
plot(h(t),0,8)
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#1b) apply definition of f'(x) to f(x)=1024-16*x^2
f(x)=1024-16*x^2
var('h')
show(f(x+h))
show(expand(f(x+h)-f(x)))
show(((expand(f(x+h)-f(x)))/h).simplify_rational())
show(lim((f(x+h)-f(x))/h,h=0))
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#2) apply definition of f'(x) to f(x)=x^2
f(x)=x^2
var('h')
show(expand(f(x+h)))
show(expand(f(x+h)-f(x)))
show(((expand(f(x+h)-f(x)))/h).simplify_rational())
show(lim((f(x+h)-f(x))/h,h=0))
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#3) apply definition of f'(x) to f(x)=x^3
f(x)=x^3
var('h')
show(expand(f(x+h)))
show(expand(f(x+h)-f(x)))
show(((expand(f(x+h)-f(x)))/h).simplify_rational())
show(lim((f(x+h)-f(x))/h,h=0))
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#4) apply definition of f'(x) to f(x)=x^4
f(x)=x^4
var('h')
show(expand(f(x+h)))
show(expand(f(x+h)-f(x)))
show(((expand(f(x+h)-f(x)))/h).simplify_rational())
show(lim((f(x+h)-f(x))/h,h=0))
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#5) apply definition of f'(x) to f(x)=x^5
f(x)=x^5
var('h')
show(expand(f(x+h)))
show(expand(f(x+h)-f(x)))
show(((expand(f(x+h)-f(x)))/h).simplify_rational())
show(lim((f(x+h)-f(x))/h,h=0))
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