y := 3*x^3-2*x^2+5; lth := Int(sqrt(1+ (diff(y,x))^2),x=0..1); sol_a := evalf(lth); k = diff(diff(f(x),x),x)/ (1+diff(f(x),x)^2)^(3/2); y1 := diff(y,x); y2 := diff(y1,x); k := y2/(1+y1^2)^(3/2); sol_b := evalf(subs(x=1/2,k)); r := t -> t*i+(3*t^3-2*t^2+5)*j: 'r(t)' = r(t); sol_d := subs(t=1/2, [1, 9*t^2-4*t]/sqrt(1+(9*t^2+4*t)^2)); sol_d := evalf(evalm(sol_d)); k := 'k': v := t -> ln(1+t)*i - (t^2+1)*j - t*cos(t)*k: 'v(t)' = v(t); R := R0 + Int(v(s),s=0..t); R0 := 0: i:= 'i': j:='j':k:='k': R := subs(t=1,R); R := evalf(int(ln(1+s),s=0..1))*i-evalf(int(s^2+1,s=0..1))*j - evalf(int(s*cos(s),s=0..1))*k; x := 'x':y:='y': f := (x,y) -> (x^2+y^2-2*x-2*y)/(x^2+y^2-2*x+2*y+2): 'f(x,y)' = f(x,y); # f(1,-1); '(x-1)^2+(y+1)^2' = expand((x-1)^2+(y+1)^2); # x:='x':y:='y':f:=(x,y)->x*exp(-y)+3*y*cos(x): 'f(x,y)'= f(x,y); fx := unapply(diff(f(x,y),x),x,y)(-1,2); fy := subs({x=0,y=1/2},diff(f(x,y),y)); fx := evalf(fx); fy := simplify(fy); z := sqrt(3*x+y^2); A*(x-x0)+B*(y-y0)-('z'-z0) = 0; A := subs({x=1,y=-1},diff(z,x)); B := subs({x=1,y=-1},diff(z,y)); C:=-1; N := [simplify(A),simplify(B),-1]; # f := (x,y) -> sqrt(9*x^2+y^2): 'f(x,y)' = f(x,y); 'f(1.95,8.1)' = 'f(2,8)'+ 'fx'*dx + 'fy'*dy + bit_more; fx :=simplify(subs({x=2,y=8},diff(f(x,y),x))); fy :=simplify(subs({x=2,y=8},diff(f(x,y),y))); dx := 1.95-2; dy := 8.1-8; apprx := f(2,8)+fx*dx+fy*dy; exact := f(1.95,8.1); f :='f': z := f(x,y); x := r^2+s^2; y := 2*r*s; zs := Diff(z,s): Zrs := Diff(zs,r); Zrs := simplify(diff(diff(z,s),r));