6.12. EXERCISES 161

6. As an important application of Theorem 6.4.3 consider the following. Experimentsare done at n times, t1, t2, · · · , tn and at each time there results a collection of nu-merical outcomes. Denote by {(ti,xi)}p

i=1 the set of all such pairs and try to findnumbers a and b such that the line x = at + b approximates these ordered pairs aswell as possible in the sense that out of all choices of a and b, ∑

pi=1 (ati +b− xi)

2

is as small as possible. In other words, you want to minimize the function of twovariables f (a,b) ≡ ∑

pi=1 (ati +b− xi)

2. Find a formula for a and b in terms of thegiven ordered pairs. You will be finding the formula for the least squares regressionline.

7. 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.

8. 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.

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

10. 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.

11. 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).

12. 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.