Homework, Section 2.2

pp. 122--, #10, 22, 42, 43, 55
Andy Long

#10: I’m going to actually model the derivative, and then graph what’s known as an anti-derivative (which will look like the given function). This is far more than you should do: you should just be able to graph g, based on the graph of f.

g[x_]:=(x+2)^2(x-1)
f[x_]=Integrate[g[p],{p,-2,x}]
Plot[{f[x]+4,g[x]},{x,-4,2},AspectRatio->1.5]

#22: for this one, you’re to use the limit definition to compute f’. Here I just plot f and f’, to show the relationship.

f[x_]=1.5x^2-x+3.7
f'[x]
Plot[{f[x],f'[x]},{x,-1,2},AspectRatio->1.5]

#42: I’ll model the behavior of the function (graph c), and deduce b as the derivative, and a as the second derivative (we can use symmetry to determine which is which!).

f[x_]:=(x^4-1)x
Plot[{f[x],f'[x],f''[x],f'''[x]},{x,-2,2},PlotRange->{-5,5},AspectRatio->1]

#43: I’ll model the behavior of the function (graph c), and deduce b as the derivative, and a as the second derivative (we can use symmetry to determine which is which!).

f[x_]:=ArcTan[1.5(x-5)]+Pi/2
Plot[{f[x],f'[x],f''[x]},{x,0,10},AspectRatio->1]

#55, p. 126:
Use the definition to prove this in general: assume f(-x)=f(x) for even, f(-x)=-f(x) for odd. What happens in the limit as h goes to 0 with
(f(x+h)-f(x))/h?