7.13. EXERCISES 203

7. As an important application of Problem 6 consider the following. Experiments aredone at n times, t1, t2, · · · , tn and at each time there results a collection of numericaloutcomes. Denote by {(ti,xi)}p

i=1 the set of all such pairs and try to find numbers aand b such that the line x = at + b approximates these ordered pairs as well as pos-sible in the sense that out of all choices of a and b, ∑

pi=1 (ati +b− xi)

2 is as smallas possible. In other words, you want to minimize the function of two variablesf (a,b) ≡ ∑

pi=1 (ati +b− xi)

2. Find a formula for a and b in terms of the given or-dered pairs. You will be finding the formula for the least squares regression line.

8. Let f be a function which has continuous derivatives. Show that u(t,x) = f (x− ct)solves the wave equation utt−c2∆u = 0. What about u(x, t) = f (x+ ct)? Here ∆u =uxx.

9. Show that if ∆u = λu where u is a function of only x, then eλ tu solves the heatequation ut −∆u = 0. Here ∆u = uxx.

10. Show that if f (x) = o(x), then f ′ (0) = 0.

11. Let f (x,y) be defined on R2 as follows. f(x,x2

)= 1 if x ̸= 0. Define f (0,0) = 0,

and f (x,y) = 0 if y ̸= x2. Show that f is not continuous at (0,0) but that

limh→0

f (ha,hb)− f (0,0)h

= 0

for (a,b) an arbitrary vector. Thus the Gateaux derivative exists at (0,0) in everydirection but f is not even continuous there.

12. Let

f (x,y)≡

{xy4

x2+y8 if (x,y) ̸= (0,0)0 if (x,y) = (0,0)

Show that this function is not continuous at (0,0) but that the Gateaux derivativelimh→0

f (ha,hb)− f (0,0)h exists and equals 0 for every vector (a,b).

13. Let U be an open subset of Rn and suppose that f : [a,b]×U → R satisfies

(x,y)→ ∂ f∂yi

(x,y) ,(x,y)→ f (x,y)

are all continuous. Show that∫ b

a f (x,y)dx,∫ b

a∂ f∂yi

(x,y)dx all make sense and that infact

∂

∂yi

(∫ b

af (x,y)dx

)=∫ b

a

∂ f∂yi

(x,y)dx

Also explain why y →∫ b

a∂ f∂yi

(x,y)dx is continuous. Hint: You will need to usethe theorems from one variable calculus about the existence of the integral for acontinuous function. You may also want to use theorems about uniform continuityof continuous functions defined on compact sets.