(* Graded Homework Section 2.4: #6, 23, 36, 42, 48, p. 146 *)

#6

(* #6 *)
g[t_]:=4Sec[t]+Tan[t];
g'[t]

#23

(* #23 *)
f[x_]:=Cos[x]-Sin[x]
f'[x]
m=f'[Pi]
Simplify[m(x-Pi)+(-1)]
Plot[{f[x],m(x-Pi)+(-1)},{x,Pi/2,3Pi/2}]

#36

(* #36 *)
Clear[x]
f[x_]:=2 Cos[x]+ 3Sin[x]
Plot[{f[x]},{x,0,2Pi}]
(* a. *)
f'[x]
f''[x]
(* b. *)
Plot[{f'[x],f''[x]},{x,0,2Pi}]
(* c. *)
time1=1.0*x/.First[Solve[f[x]==0,x]]
(* d. *)
time2=1.0*x/.First[Solve[f'[x]==0,x]]
Abs[f[time2]]
(* e. *)
time1+n Pi

#42

(* #42 *)
f[x_]:=(Cos[x]-1)/Sin[x]
Plot[f[x],{x,-Pi/8,Pi/8}]
g[x_]:=(Cos[x]-1)/x * x/Sin[x]
Plot[g[x],{x,-Pi/8,Pi/8}]
Limit[f[x],x -> 0]
Limit[g[x],x -> 0]

#48

(* #48 *)
Limit[Sin[x-1]/(x^2+x-2),x->1]