L'alçada d'una pilota després de ser llançada a l'aire des de la part superior d'un edifici es pot modelar per
h
(
t
)
=
−
2
t
2
+
8
t
+
1
{\displaystyle h(t)=-2t^{2}+8t+1}
h
{\displaystyle h}
Completa la taula amb els valors numèrics que corresponguin:
t
0
0.5
1
h
(
t
)
{\displaystyle h(t)}
Per emplenar la casella davall el 0, hem de calcular
h
(
0
)
{\displaystyle h(0)}
h
(
0
)
=
−
2
⋅
(
0
)
2
+
8
⋅
(
0
)
+
1
=
−
2
⋅
0
+
8
⋅
0
+
1
=
0
+
0
+
1
=
1
{\displaystyle h(0)=-2\cdot (0)^{2}+8\cdot (0)+1=-2\cdot 0+8\cdot 0+1=0+0+1=1}
Per tant, a la primera casella hi va un 1.
De la mateixa manera, calculam
h
(
0.5
)
{\displaystyle h(0.5)}
h
(
1
)
{\displaystyle h(1)}
h
(
0.5
)
=
−
2
⋅
(
0.5
)
2
+
8
⋅
(
0.5
)
+
1
=
−
2
⋅
0.25
+
8
⋅
0.5
+
1
=
−
0.5
+
4
+
1
=
4.5
{\displaystyle h(0.5)=-2\cdot (0.5)^{2}+8\cdot (0.5)+1=-2\cdot 0.25+8\cdot 0.5+1=-0.5+4+1=4.5}
h
(
1
)
=
−
2
⋅
(
1
)
2
+
8
⋅
(
1
)
+
1
=
−
2
⋅
1
+
8
⋅
1
+
1
=
−
2
+
8
+
1
=
7
{\displaystyle h(1)=-2\cdot (1)^{2}+8\cdot (1)+1=-2\cdot 1+8\cdot 1+1=-2+8+1=7}
Aleshores la taula queda emplenada de la forma següent:
t
0
0.5
1
h
(
t
)
{\displaystyle h(t)}
1
4.5
7
Quina és la taxa de variació mitjana de la funció
f
(
x
)
=
10
{\displaystyle f(x)=10}
x
=
1
{\displaystyle x=1}
x
=
8
{\displaystyle x=8}
La TVM entre 1 i 8 es defineix de la manera següent:
T
V
M
[
1
,
8
]
=
f
(
8
)
−
f
(
1
)
8
−
1
{\displaystyle TVM[1,8]={\dfrac {f(8)-f(1)}{8-1}}}
Aleshores hem de calcular
f
(
8
)
{\displaystyle f(8)}
f
(
1
)
{\displaystyle f(1)}
Com que
f
(
x
)
=
10
{\displaystyle f(x)=10}
x
{\displaystyle x}
f
(
8
)
=
10
{\displaystyle f(8)=10}
f
(
1
)
=
10
{\displaystyle f(1)=10}
Retornam a la definició de TVM
T
V
M
[
1
,
8
]
=
f
(
8
)
−
f
(
1
)
8
−
1
=
10
−
10
8
−
1
=
0
7
=
0
{\displaystyle TVM[1,8]={\dfrac {f(8)-f(1)}{8-1}}={\dfrac {10-10}{8-1}}={\dfrac {0}{7}}=0}
Per tant, la TVM és 0.
Quina és la taxa de variació mitjana de la funció
f
(
x
)
=
3
x
2
+
7
x
−
2
{\displaystyle f(x)=3x^{2}+7x-2}
x
=
2
{\displaystyle x=2}
x
=
5
{\displaystyle x=5}
La TVM es calcula com:
T
V
M
[
2
,
5
]
=
f
(
5
)
−
f
(
2
)
5
−
2
{\displaystyle TVM[2,5]={\dfrac {f(5)-f(2)}{5-2}}}
Aleshores hem de calcular
f
(
5
)
{\displaystyle f(5)}
f
(
2
)
{\displaystyle f(2)}
Calculam el primer resultat:
f
(
5
)
=
3
⋅
(
5
)
2
+
7
⋅
(
5
)
−
2
=
3
⋅
25
+
7
⋅
5
−
2
=
75
+
35
−
2
=
108
{\displaystyle f(5)=3\cdot (5)^{2}+7\cdot (5)-2=3\cdot 25+7\cdot 5-2=75+35-2=108}
f
(
2
)
=
3
⋅
(
2
)
2
+
7
⋅
(
2
)
−
2
=
3
⋅
4
+
7
⋅
2
−
2
=
12
+
14
−
2
=
24
{\displaystyle f(2)=3\cdot (2)^{2}+7\cdot (2)-2=3\cdot 4+7\cdot 2-2=12+14-2=24}
Per tant,
T
V
M
[
2
,
5
]
=
108
−
24
5
−
2
=
84
3
=
28
{\displaystyle TVM[2,5]={\dfrac {108-24}{5-2}}={\dfrac {84}{3}}=28}
La TVM és 28.
![{\displaystyle TVM[1,8]={\dfrac {f(8)-f(1)}{8-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b74ae587baa2b0fd39ea232b007f62c7c67936)
![{\displaystyle TVM[1,8]={\dfrac {f(8)-f(1)}{8-1}}={\dfrac {10-10}{8-1}}={\dfrac {0}{7}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36c8b05b49d41122fc686e593b283f533039a5d7)
![{\displaystyle TVM[2,5]={\dfrac {f(5)-f(2)}{5-2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8667384366a67f3a2eeaad2d393e7a5e2fbc2cd2)
![{\displaystyle TVM[2,5]={\dfrac {108-24}{5-2}}={\dfrac {84}{3}}=28}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50519e85052faa303f463ce44b0fe5595bc83451)