with(plots); -1 

Section 3.5 #7 

`:=`(f, proc (x) options operator, arrow; `+`(`*`(`/`(1, 5), `*`(`^`(x, 5))), `-`(`*`(`/`(8, 3), `*`(`^`(x, 3)))), `*`(16, `*`(x))) end proc) 

proc (x) options operator, arrow; `+`(`*`(`/`(1, 5), `*`(`^`(x, 5))), `-`(`*`(`/`(8, 3), `*`(`^`(x, 3)))), `*`(16, `*`(x))) end proc (1)
 

`:=`(fp, D(f)) 

proc (x) options operator, arrow; `+`(`*`(`^`(x, 4)), `-`(`*`(8, `*`(`^`(x, 2)))), 16) end proc (2)
 

`:=`(fpp, D(fp)) 

proc (x) options operator, arrow; `+`(`*`(4, `*`(`^`(x, 3))), `-`(`*`(16, `*`(x)))) end proc (3)
 

solve(fp(x) = 0, x) 

2, 2, -2, -2 (4)
 

solve(fpp(x) = 0, x) 

0, 2, -2 (5)
 

plot(f(x), x = -4 .. 4) 

Plot_2d
 

3.5 #26 

`:=`(g, proc (x) options operator, arrow; `*`(x, `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)))) end proc) 

proc (x) options operator, arrow; `*`(x, `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)))) end proc (6)
 

`:=`(gp, D(g)) 

proc (x) options operator, arrow; `+`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)), `-`(`/`(`*`(`^`(x, 2)), `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)))))) end proc (7)
 

normal(`+`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)), `-`(`/`(`*`(`^`(x, 2)), `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2))))))) 

`+`(`-`(`/`(`*`(2, `*`(`+`(`*`(`^`(x, 2)), `-`(1)))), `*`(`^`(`+`(`-`(`*`(`^`(x, 2))), 2), `/`(1, 2)))))) (8)
 

`:=`(gpp, D(gp)) 

proc (x) options operator, arrow; `+`(`-`(`/`(`*`(3, `*`(x)), `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2))))), `-`(`/`(`*`(`^`(x, 3)), `*`(`^`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)), 3))))) end proc (9)
 

normal(`+`(`-`(`/`(`*`(3, `*`(x)), `*`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2))))), `-`(`/`(`*`(`^`(x, 3)), `*`(`^`(sqrt(`+`(`-`(`*`(`^`(x, 2))), 2)), 3)))))) 

`+`(`/`(`*`(2, `*`(x, `*`(`+`(`*`(`^`(x, 2)), `-`(3))))), `*`(`^`(`+`(`-`(`*`(`^`(x, 2))), 2), `/`(3, 2))))) (10)
 

solve(gp(x) = 0) 

-1, 1 (11)
 

solve(gpp(x) = 0) 

0, `*`(`^`(3, `/`(1, 2))), `+`(`-`(`*`(`^`(3, `/`(1, 2))))) (12)
 

plot(g(x), x = -3 .. 3) 

Plot_2d
 

 

S3.5 #23 

`:=`(h, proc (x) options operator, arrow; sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))) end proc) 

proc (x) options operator, arrow; sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))) end proc (13)
 

`:=`(hp, D(h)) 

proc (x) options operator, arrow; `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(x)), 1))), `*`(sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2)))))) end proc (14)
 

`:=`(hpp, D(hp)) 

proc (x) options operator, arrow; `+`(`/`(1, `*`(sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))))), `-`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(`*`(2, `*`(x)), 1), 2))), `*`(`^`(sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))), 3)))... (15)
 

normal(`+`(`/`(1, `*`(sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))))), `-`(`/`(`*`(`/`(1, 4), `*`(`^`(`+`(`*`(2, `*`(x)), 1), 2))), `*`(`^`(sqrt(`+`(`*`(`^`(x, 2)), x, `-`(2))), 3)))))) 

`+`(`-`(`/`(`*`(`/`(9, 4)), `*`(`^`(`+`(`*`(`^`(x, 2)), x, `-`(2)), `/`(3, 2)))))) (16)
 

solve(hp(x) = 0) 

-`/`(1, 2) (17)
 

solve(denom(hp(x)) = 0) 

1, -2 (18)
 

solve(numer(hp(x)) = 0) 

-`/`(1, 2) (19)
 

plot(h(x), x = -4 .. 2) 

Plot_2d
 

3.5 #27 

`:=`(f, proc (x) options operator, arrow; `/`(`*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))), `*`(x)) end proc) 

proc (x) options operator, arrow; `/`(`*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))), `*`(x)) end proc (20)
 

`:=`(fp, D(f)) 

proc (x) options operator, arrow; `+`(`-`(`/`(1, `*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))))), `-`(`/`(`*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))), `*`(`^`(x, 2))))) end proc (21)
 

normal(`+`(`-`(`/`(1, `*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))))), `-`(`/`(`*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))), `*`(`^`(x, 2)))))) 

`+`(`-`(`/`(1, `*`(`^`(`+`(1, `-`(`*`(`^`(x, 2)))), `/`(1, 2)), `*`(`^`(x, 2)))))) (22)
 

`:=`(fpp, D(fp)) 

proc (x) options operator, arrow; `+`(`-`(`/`(`*`(x), `*`(`^`(sqrt(`+`(1, `-`(`*`(`^`(x, 2))))), 3)))), `/`(1, `*`(x, `*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))))), `/`(`*`(2, `*`(sqrt(`+`(1, `-`(`*`(`^`(x... (23)
 

normal(`+`(`-`(`/`(`*`(x), `*`(`^`(sqrt(`+`(1, `-`(`*`(`^`(x, 2))))), 3)))), `/`(1, `*`(x, `*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2)))))))), `/`(`*`(2, `*`(sqrt(`+`(1, `-`(`*`(`^`(x, 2))))))), `*`(`^`(x, 3)))... 

`+`(`-`(`/`(`*`(`+`(`*`(3, `*`(`^`(x, 2))), `-`(2))), `*`(`^`(`+`(1, `-`(`*`(`^`(x, 2)))), `/`(3, 2)), `*`(`^`(x, 3)))))) (24)
 

solve(fpp(x) = 0) 

`+`(`-`(`*`(`/`(1, 3), `*`(`^`(6, `/`(1, 2)))))), `+`(`*`(`/`(1, 3), `*`(`^`(6, `/`(1, 2))))) (25)
 

plot(f(x), x = -1.5 .. 1.5, y = -5 .. 5) 

Plot_2d
 

 

S 3.5 # 34 

`:=`(f, proc (x) options operator, arrow; `+`(x, cos(x)) end proc) 

proc (x) options operator, arrow; `+`(x, cos(x)) end proc (26)
 

`:=`(fp, D(f)) 

proc (x) options operator, arrow; `+`(1, `-`(sin(x))) end proc (27)
 

`:=`(fpp, D(fp)) 

proc (x) options operator, arrow; `+`(`-`(cos(x))) end proc (28)
 

solve(fp(x) = 0) 

`+`(`*`(`/`(1, 2), `*`(Pi))) (29)
 

solve(fpp(x) = 0) 

`+`(`*`(`/`(1, 2), `*`(Pi))) (30)
 

plot(f(x), x = `+`(`-`(`*`(3, `*`(Pi)))) .. `+`(`*`(3, `*`(Pi)))) 

Plot_2d
 

`:=`(f, proc (x) options operator, arrow; `+`(`*`(.5, `*`(x)), sin(x)) end proc) 

proc (x) options operator, arrow; `+`(`*`(.5, `*`(x)), sin(x)) end proc (31)
 

`:=`(fp, D(f)) 

proc (x) options operator, arrow; `+`(.5, cos(x)) end proc (32)
 

`:=`(fpp, D(fp)) 

proc (x) options operator, arrow; `+`(`-`(sin(x))) end proc (33)
 

solve(fp(x) = 0) 

2.094395102 (34)
 

solve(fpp(x) = 0) 

0 (35)
 

plot(f(x), x = `+`(`-`(`*`(3, `*`(Pi)))) .. `+`(`*`(3, `*`(Pi)))) 

Plot_2d