<< 
        Graphics`Colors`
temperaturedata  :=  
      List[{0,  13},  {10,  12.9},  {20,  13},  {30,  
          13.6}]

p1 = ListPlot[
          temperaturedata,  PlotStyle -> {PointSize[0.02],  Red},  
          PlotRange -> {{0,  40},  {11,  15}},  
          PlotLabel ->  "Maximum Temperature vs. Time",  
          AxesLabel -> {"Time (years)", "Temp (deg C)"},  
          TextStyle -> {FontSize -> 
                18}]

T[y_]  =  
        A*y^3  +  B*y^2  +  C*y  +  D
Solve[{
T[0] == 
            13, 
T[10] == 
            12.9, 
T[20] == 
            13, 
T[30] == 
            13.6
},  {A,  B,  C, D}]

Af = 0.00005000000000000117`
Bf = -0.0005000000000000555`
Cf = -0.009999999999999397`
Df = 13.`

T[y_] :=  
      Af*y^3  +  Bf*y^2  +  Cf*y  +  Df
p2 = 
      Plot[T[y],  {y,  0,  
          40},   PlotRange -> {{0, 40},  {12,  14}}]

Show[p1,  p2]

(T[30  +  1]  -  T[30])/1

(T[30  +   .1]  -  T[30])/ .1

(T[30  +   .01]  -  T[30])/ .01

(T[30  +   .001]  -  T[30])/ .001

(T[30  +   .0001]  -  T[30])/ .0001

Texp[y_]  =  
        A*E^(B*y)  +  
          C*E^(D*
                  y)
Solve[{
Texp[0] == 13, 
Texp[10] == 
          12.9, 
Texp[20] == 
          13, 
Texp[30] == 
          13.6
},  {A,  B,  C, D}]

Af1 = 0.08359075870575888`
Bf1 = 0.0928679591660315`
Cf1 = 12.916409241294241`
Df1 = -0.0017808927255499123`

Texp[y_] :=  
      Af1*E^(Bf1*y) + Cf1*E^(Df1*y)
pexp2 = 
      Plot[Texp[y],  {y,  0,  
          40},   PlotRange -> {{0, 40},  {12,  17}}]

Show[p1,  pexp2]

Plot[{T[y],  Texp[y]},  {y,  0,  
          80},   PlotRange -> {{0, 80},  {0,  
              50}},   PlotStyle -> {Blue,  Red}]

(* Drastically different temperatures are predicted by the two models in 2015 *)
Texp[55] - T[55]

Plot[{T[y],  Texp[y]},  {y,  0,  
        30},   PlotRange -> {{0, 30},  {12.75,  
            14}},   PlotStyle -> {Blue,  Red}]

(Texp[30  +  1]  -  Texp[30])/
        1

(Texp[30  +   .1]  -  Texp[30])/ .1

(Texp[30  +   .01]  -  Texp[30])/ .01

(Texp[30  +   .001]  -  Texp[30])/ .001

(Texp[30  +   .0001]  -  Texp[30])/ .0001

(Texp[30  +   .00001]  -  Texp[30])/ .00001

(Texp[30  +   .000001]  -  Texp[30])/ .000001