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1

a2

f12[x_]:=2x^5-6x^3-16x-7

f12'[x]

f12''[x]

b2

Solve[f12'[x]==0,{x}]

Solve[f12'[x]==0,{x}]//N

c2

f2[x_]:=(4x+3)(-5x^4+3x-2); f2[x]//Expand

f2'[x]

f2'[x]//Expand

d2

f3[x_]:=(a+x)/x^2-(4x^3+4)^9; f3[x]//Expand

f3'[x]

f3'[x]//Expand

e2

f4[x_]:=Log[1/x]+x^Sin[x]-Cos[Sin[x]]; f4[x]//Expand

f4'[x]

f4'[x]//Expand

2

a2

f5[n_]:=(n^2+5n-4)/(n^2+5n+4); f5[n]//Expand

Limit[f5[n],{n->Infinity}]

1-f5[n]//Together//Simplify

Plot[1-f5[n],{n,1,20}];

Solve[1-f5[n]==0.001,{n}]

b2

p3[x_,a_]:=x^2+a x; p4[x_,a_]:=x^2+a;

Solve[p3[x,a]==p4[x,a],{x}]

ArcTan[D[p3[x,a],x]]

ArcTan[D[p4[x,a],x]]

Plot[ArcTan[a+2]-ArcTan[2],{a,-5,5}];

FindRoot[ArcTan[a+2]-ArcTan[2]==-Pi/2,{a,-1}]

FindRoot[ArcTan[a+2]-ArcTan[2]==-Pi/2,{a,-2}]

c2

h[x_,a_]:=a/x

D[h[x0,a],x0]

g[x_]:=h[x0,a]+D[h[x0,a],x0](x-x0);
g[0]

solv=Solve[g[x]==0,{x}]//Flatten

x1=x/.solv

A=1/2 g[0] x1

d2

f[x_]:=3 Cos[x]

A[x_]:=1/2 (x-(-x)) f[x]; A[x]

A'[x]

Evaluate[A'[y]==0/.y->x]

FindRoot[Evaluate[A'[y]==0/.y->x],{x,1}]

3

a2

f[x_]:=1/x+2 x^2 -2.3811 +E^-(x+283.11)^2

Plot[f[x],{x,0.5,3}];

FindRoot[f[x],{x,0.6}]

f[0.62996]

b2

f'[x]

Solve[f'[x]==0,{x}]

Solve[f'[x]==0,{x}] // N

4

a2

f[x_]:=Sin[Cos[x]]

Table[Nest[f,0.5,k],{k,1,20}]

Plot[{f[x],x},{x,0,3}]

b2

Show[Graphics[{PointSize[0.03],Point[{1,0}],Point[{2,0}],Point[{3,0}],Point[{4,1}]}]];

h0[x_]:=2x^3-3x^2+1;
h1[x_]:=x^3-2x^2+x;
h2[x_]:=x^3-x^2;
h3[x_]:=1-h0[x];
pH[x_,y1_,m1_,m2_,y2_]:=y1 h0[x]+m1 h1[x]+m2 h2[x]+y2 h3[x];

y1=0;y2=0;m1=0;m2=1/2;

xL[1]=2;
xL[2]=3;
t[x_,k_]:=(x-xL[k])/(xL[k+1]-xL[k]); t[x,1]

delta[k_]:= xL[k+1]-xL[k];
pH[x_,y1_,m1_,m2_,y2_,k_]:=y1 h0[t[x,k]]+ delta[k] m1 h1[t[x,k]]+ delta[k] m2 h2[t[x,k]]+y2 h3[t[x,k]];

Plot[{h0[t[x,1]],h1[t[x,1]],h2[t[x,1]],h3[t[x,1]],pH[x,y1,m1,m2,y2,1]},{x,2,3}];

pH[x,y1,m1,m2,y2,1]

D[pH[x,y1,m1,m2,y2,1],x]/.x->3

pH[x,y1,m1,m2,y2,1]//Expand

Plot[{pH[x,y1,m1,m2,y2,1]},{x,2,3},Epilog->{PointSize[0.03],Point[{2,0}],Point[{3,0}]}];