1—Basic Stuff 231.44 Start from the definition of a derivative, manipulate some terms: (a) derive the rule for differen-
tiating the functionh, whereh(x) =f(x)g(x)is the product of two other functions.
(b) Integrate the resulting equation with respect toxand derive the formula for integration by parts.
1.45 Show that in polar coordinates the equationr= 2acosφis a circle. Now compute its area in
this coordinate system.
1.46 The cycloid* has the parametric equationsx=aθ−asinθ, andy=a−acosθ. Compute the
area,
∫
ydxbetween one arc of this curve and thex-axis. Ans: 3 πa^2
1.47 An alternate approach to the problem1.13: Change variables in the integral definition oferfto
t=αu. Now differentiate with respect toαand of course the derivative must be zero and there’s your
answer. Do the same thing for problem1.14and the Gamma function.
1.48 Recall section1.5and compute this second derivative to show
d^2
dt^2
∫t0dt′(t−t′)F(t′) =F(t)
1.49 From the definition of a derivative show that
If x=f(θ) and t=g(θ) then
dx
dt
=
df/dθ
dg/dθ
Make up a couple of functions that let you test this explicitly.