Week 4¶

3.1 (2)(4)(5)

In [1]:

(* 2 *)
Limit[x/(x-1)-1/Log[x], x->1]
(* 4 *)
Limit[(1+1/x^2)^x, x->Infinity]
(* 5 *)
Limit[Sqrt[n-Sqrt[n]]-Sqrt[n], n->Infinity]

(* 2 *) Limit[x/(x-1)-1/Log[x], x->1] (* 4 *) Limit[(1+1/x^2)^x, x->Infinity] (* 5 *) Limit[Sqrt[n-Sqrt[n]]-Sqrt[n], n->Infinity]

Out[1]:

3.2 (5) - (8)

In [7]:

(* 5 *)
D[Exp[x]+Exp[Exp[x]],x]

(* 6 *)
D[x^(x^x),x]

(* 7 *)
D[(Sin[x])^Cos[x],x]

(* 8 *)
D[Log[Cos[ArcTan[(Exp[x]-Exp[-x])/2]]],x]

(* 5 *) D[Exp[x]+Exp[Exp[x]],x] (* 6 *) D[x^(x^x),x] (* 7 *) D[(Sin[x])^Cos[x],x] (* 8 *) D[Log[Cos[ArcTan[(Exp[x]-Exp[-x])/2]]],x]

Out[7]:

3.3 (3)(4)

In [15]:

(* 3 *)
D[1/(1-x^2),{x,60}]
(* 4 *)
D[(1+x)/Sqrt[1-x],{x,60}]

(* 3 *) D[1/(1-x^2),{x,60}] (* 4 *) D[(1+x)/Sqrt[1-x],{x,60}]

Out[15]:

3.4 (3)-(6)

In [19]:

(* 3 *)
Integrate[x^2*a^x,x]
(* 4 *)
Integrate[(2x^2-5)/(x^4-5x^2+6),x]
(* 5 *)
Integrate[Log[x+Sqrt[1+x^2]],x]
(* 6 *)
Integrate[(Exp[2x]+1)/(Exp[x]+1),x]

(* 3 *) Integrate[x^2*a^x,x] (* 4 *) Integrate[(2x^2-5)/(x^4-5x^2+6),x] (* 5 *) Integrate[Log[x+Sqrt[1+x^2]],x] (* 6 *) Integrate[(Exp[2x]+1)/(Exp[x]+1),x]

Out[19]:

3.5 (2N)

In [27]:

(* 2 *)
Integrate[Sqrt[Exp[x]-1],{x,0,Log[2]}]
(* 4 *)
Integrate[x^2/Sqrt[x^2+a^2],{x,0,a}]
(* 6 *)
Integrate[(1/2*Sqrt[4x+1]+1)/x, {x,0,1}]
(* 8 *)
Integrate[x*y^2,{x,0,1},{y,x^2,x}]
(* 10 *)
Integrate[x*y*z,{x,0,1},{y,0,x},{z,0,x+y}]
(* 12 *)
Integrate[x*Sqrt[y]*Boole[x^2<=y<=Sqrt[x]],{x,0,Infinity},{y,0,Infinity}]

(* 2 *) Integrate[Sqrt[Exp[x]-1],{x,0,Log[2]}] (* 4 *) Integrate[x^2/Sqrt[x^2+a^2],{x,0,a}] (* 6 *) Integrate[(1/2*Sqrt[4x+1]+1)/x, {x,0,1}] (* 8 *) Integrate[x*y^2,{x,0,1},{y,x^2,x}] (* 10 *) Integrate[x*y*z,{x,0,1},{y,0,x},{z,0,x+y}] (* 12 *) Integrate[x*Sqrt[y]*Boole[x^2<=y<=Sqrt[x]],{x,0,Infinity},{y,0,Infinity}]

                             1   Sqrt[1 + 4 x]
Integrate::idiv: Integral of - + ------------- does not converge on {0, 1}.
                             x        2 x

Out[27]:

3.6

In [39]:

y[x_] := a * Cosh[x/a]
Integrate[Sqrt[1+y'[x]^2],{x,-a,a}]
ClearAll["Global`*"]

y[x_] := a * Cosh[x/a] Integrate[Sqrt[1+y'[x]^2],{x,-a,a}] ClearAll["Global`*"]

Out[39]:

3.7 (3)(5)

In [42]:

(* 3 *)
Series[Log[(1+x)/(1-x)],{x,0,5}]
(* 5 *)
Series[Cos[x]*Cos[y],{x,0,5},{y,0,5}]

(* 3 *) Series[Log[(1+x)/(1-x)],{x,0,5}] (* 5 *) Series[Cos[x]*Cos[y],{x,0,5},{y,0,5}]

Out[42]:

3.8 (6)(9)

In [46]:

(* 6 *)
DSolve[{y'[x]==y[x]+x,y[0]==1},y[x],x]
(* 8 *)
DSolve[{x'[t] == 2*x[t] - y[t] + z[t],y'[t] == 2*x[t] + 2*y[t] - z[t],z'[t] == x[t] + 2 * y[t] - z[t]}, {x[t], y[t], z[t]}, t]

(* 6 *) DSolve[{y'[x]==y[x]+x,y[0]==1},y[x],x] (* 8 *) DSolve[{x'[t] == 2*x[t] - y[t] + z[t],y'[t] == 2*x[t] + 2*y[t] - z[t],z'[t] == x[t] + 2 * y[t] - z[t]}, {x[t], y[t], z[t]}, t]

Out[46]:

3.9 (1)(3)

In [50]:

(* 1 *)
DSolve[(y+z)*D[u[x,y,z],x]+(z+x)*D[u[x,y,z],y]+(x+y)*D[u[x,y,z],z]==0,u[x,y,z],{x,y,z}]
(* 3 *)
DSolve[(x*y^3-2x^4)*D[u[x,y],x]+(3y^4-x^3*y)*D[u[x,y],y]==9*u[x,y]*(x^3-y^3),u[x,y],{x,y}]

(* 1 *) DSolve[(y+z)*D[u[x,y,z],x]+(z+x)*D[u[x,y,z],y]+(x+y)*D[u[x,y,z],z]==0,u[x,y,z],{x,y,z}] (* 3 *) DSolve[(x*y^3-2x^4)*D[u[x,y],x]+(3y^4-x^3*y)*D[u[x,y],y]==9*u[x,y]*(x^3-y^3),u[x,y],{x,y}]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[50]: