(message "
(pick 3 '(2 3 13 14 16 20 26))
(pick 3 '(3 7 8 10))
;; selected problems:
13.8: (2 3 16)
13.9:  (3 8 10)
")

(message "
13.8: (2 3 16)

#2

(func f1(x) 1 - (x - 2 * 0) ^ 2)
(func f2(x) 1 - (x - 2 * 1) ^ 2)
(func f3(x) 1 - (x - 2 * 2) ^ 2)
(plot f1 f2 f3)

(func f(tt x) 1 - (x - 2 * tt) ^ 2)
(range -5 2)
(animate f (grid 0 2 10) -1 5 :loops 3)
")

(func f1(x) 1 - (x - 2 * 0) ^ 2)
(func f2(x) 1 - (x - 2 * 1) ^ 2)
(func f3(x) 1 - (x - 2 * 2) ^ 2)
(plot f1 f2 f3)

(func f(tt x) 1 - (x - 2 * tt) ^ 2)
(range -5 2)
(animate f (grid 0 2 10) -1 5 :loops 3)

(message "
#3 

That c = d * (a^2 + b^2).

)

(message "
#16

Use symmetry and show that d^2v/dx^2 will cancel with the others.
)

(message "
13.9:  (3 8 10)

#3

(func f(x y) ln (1 + x ^ 2 - y))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(setq a '(0 0))
(list (f 0 0 ) (fx 0 0 ) (fy 0 0 )  (/ (fxx 0 0 ) 2) (fyx 0 0 ) (/ (fyy 0 0 ) 2))

;; (0.0 0 -1 1 0 -0.5)
")

(func f(x y) ln (1 + x ^ 2 - y))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(list (f 0 0) (fx 0 0) (fy 0 0)  (/ (fxx 0 0) 2) (fyx 0 0) (/ (fyy 0 0) 2))

(message "
#8
(func f(x y) sqrt (x ^ 2 + y ^ 2))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(list (f 1 1) (fx 1 1) (fy 1 1)  (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))

(sum (f 1 1)
     (* .1 (+ (fx 1 1) (fy 1 1) 
	      (* .1 (+ (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))))))
(f 1.1 1.1)

;; 1.5556349186104046
;; 1.5556349186104046
")

(func f(x y) sqrt (x ^ 2 + y ^ 2))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(list (f 1 1) (fx 1 1) (fy 1 1)  (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))

(sum (f 1 1)
     (* .1 (+ (fx 1 1) (fy 1 1) 
	      (* .1 (+ (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))))))
(f 1.1 1.1)

(message "
#10

(func f(x y) atan (x / y))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(list (f 1 1) (fx 1 1) (fy 1 1)  (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))

(sum (f 1 1)
     (* .1 (+ (fx 1 1) (fy 1 1) 
	      (* .1 (+ (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))))))
(f 1.1 1.1)
;; 0.7853981633974483
;; 
")

(func f(x y) arctan (x / y))
(diff f x fx)
(diff f y fy)
(diff fx x fxx)
(diff fy y fyy)
(diff fy x fyx)
(list (f 1 1) (fx 1 1) (fy 1 1)  (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))

(sum (f 1 1)
     (* .1 (+ (fx 1 1) (fy 1 1) 
	      (* .1 (+ (/ (fxx 1 1) 2) (fyx 1 1) (/ (fyy 1 1) 2))))))
(f 1.1 1.1)