(message "
(pick 3 (combine (iseq 3 7) (iseq 9 11))) ;; 116
(pick 3 '(1 2 4 7 10 19 20 21 28 29 34 38 39 40)) ;; 124
(pick 3 '(2 4 5 8 16 22 23 26)) ;; 134
(pick 3 (iseq 3 12)) ;; 141
;; selected problems:
13.1: (11 3 7)
13.2: (1 7 28)
13.3: (8 23 5)
13.4: (12 10 7)
")

(message "
13.1: 3, 7, 11

#3:
(a) negative: as the prices go up, the number/amount purchased goes down.
(b) negative: as the prices go up, the number/amount purchased goes down.

#7:
(a) negative (function decreases as we move along increasing x-axis)
(b) positive (function increases as we move along increasing y-axis)

#11:
If C=f(w,T), this is C_T: the partial of C with respect to T.
")

(message "
13.2: (1 7 28)

#1:
(func f(u v) u * (u ^ 2 + v ^ 2) ^ (3 / 2))
(diff f u fu)
(calc ( (f 1.001 3) - (f 1 3)) / .001)
(fu 1 3)
;; pretty darn close!
")

(func f(u v) u * (u ^ 2 + v ^ 2) ^ (3 / 2))
(diff f u fu)
(calc ( (f 1.001 3) - (f 1 3)) / .001)
(fu 1 3)

(message "
#7:
(func f(m v r) m * v ^ 2 / r)
(diff f v)
")

(func f(m v r) m * v ^ 2 / r)
(diff f v)

(message "
#28:
(func f(x y) x * exp (sqrt (x * y)))
(diff f x)
")

(func f(x y) x * exp (sqrt (x * y)))
(diff f x)

(message "
13.3: (8 23 5)

#5: As discussed in class, a continuity argument works: the tangent plane
occurs at a z value of 9; the function just gets to zero at (0,0), but the
tangent plane will be negative there. Since the tangent plane is continuous, it
must have reached zero before the function itself.

#8: We need to
a) find f(3,1), fx(3,1), and fy(3,1):
(func f(x y) x ^ 2 * y)
(diff f x fx)
(diff f y fy)
(f 3 1)
(fx 3 1)
(fy 3 1)
;; Then z = 9 + 6*(x-3) + 9*(y-1)
")
(func f(x y) x ^ 2 * y)
(diff f x fx)
(diff f y fy)
(f 3 1)
(fx 3 1)
(fy 3 1)
(print "Then z = 9 + 6*(x-3) + 9*(y-1)")

(message "
#23:
dA = Ax*dx + Ay*dy
   = y*dx + x*dy
   = 6*.01+ 5*.01 = .11 feet^2
")

(message "
13.4: (12 10 7)

#7: for x > 2

#10:
(f(x+h/sqrt(2),y+h/sqrt(2))-f(x,y))/h
   = ((2*(x+h/sqrt(2))^2 + (y+h/sqrt(2))^2)-(2*x^2 + y^2))/h
   = (2*xh*sqrt(2) + h^2 + yh*sqrt(2) + h^2/2)/h
   = 2*x*sqrt(2) + y*sqrt(2) in the limit

(func f(x y) 2 * x ^ 2 + y ^ 2)
(diff f x fx)
(diff f y fy)
(print (list (what fx) '/ 'sqrt 2 '+ (what fy) '/ 'sqrt 2))
")

(func f(x y) 2 * x ^ 2 + y ^ 2)
(diff f x fx)
(diff f y fy)
(print (list (what fx) '/ 'sqrt 2 '+ (what fy) '/ 'sqrt 2))
(calc (fx 2 1) / (sqrt 2) + (fy 2 1) / (sqrt 2))
;; > (/ 10 (sqrt 2))
;; 7.071067811865475

(message "
#12:
(func f(x y) x ^ 2 * y)
(gradient f gradf)
(def v (vector 4 -3))
(def u (/ v (norm v)))
(dot (gradf 2 6) u)
")

(func f(x y) x ^ 2 * y)
(gradient f gradf)
(def v (vector 4 -3))
(def u (/ v (norm v)))
(dot (gradf 2 6) u)