Tangentengleichung und Normalengleichung aufstellen. f(x)= e ^x +x an Stelle x = 1

Question

für die Fkt. f(x)= e ^x  +x und x = 1

Answer 1

 f(x)= e ^x  +x und x = 1

f '(x) = e^x + 1
f '(1) = e^1 + 1 = m = e+1
f(1) = e + 1
Punkt P(1 | e+1)
Tangente
Ansatz
t: y = (e+1)* x + q
P einsetzen
e+1= e+1 + q ==> q = 0
t: y = (e+1)*x 

Comments

Graph dazu:

~plot~e^{x} + x; (e+1)*x~plot~

Nun noch die Normale.

m = -1/(1+e)

n: y = -1/(1+e) * x + q

P einsetzen

e+1 = -1/(1+e) + q

1+e + 1/(1+e) =  q

n: y = -1/(1+e) * x + 1 + e + 1/(1+e) 

vgl: ~plot~e^{x} + x; (e+1)*x; -1/(1+e) * x + 1 + e + 1/(1+e) ~plot~

Answer 2

. f(x)= e ^x  +x und x = 1      f ' (x ) = e ^x  +1    m= f ' (1) =    e+1   P(1; e ^x  +1 )

Tangente   y = m * x + n

                 e   +1 = (e  +1 ) * 1 + n   also n=0  

Tangentengl:     y = (e   +1 ) * x

Normale hat Steigung  -1 / (e   +1 )  also

Normale    y = m * x + n
                

               (    e   +1 ) + 1/ (e   +1 )  = n 

                 ((    e   +1 )^2  + 1)/ (e   +1 )  = n 

                     e² + 2e +2  ) / (e  +1 ) = n

                    

Normalengl:     y =     -x  / (e   +1 ) +      (  e² + 2e +2  ) / (e   +1 )