Given a function f(x)=2x^2+3, what is the average rate of change of f on the interval [2, 2+h]?

4x+2h

Step-by-step explanation:

The average rate of change of a continuous function,

f

(

x

)

, on a closed interval  

[

a

,

b

]

is given by

f

(

b

)

−

f

(

a

)

b

−

a

So the average rate of change of the function  

f

(

x

)

=

2

x

2

+

1

on  

[

x

,

x

+

h

]

is:

A

r

o

c

=

f

(

x

+

h

)

−

f

(

x

)

(

x

+

h

)

−

(

x

)

=

f

(

x

+

h

)

−

f

(

x

)

h

...

.

.

[

1

]

=

2

(

x

+

h

)

2

+

1

−

(

2

x

2

+

1

)

h

=

2

(

x

2

+

2

x

h

+

h

2

)

+

1

−

2

x

2

−

1

h

=

2

x

2

+

4

x

h

+

2

h

2

−

2

x

2

h

=

4

x

h

+

2

h

2

h

=

4

x

+

2

h

Which is the required answer.

Additional Notes:

Note that this question is steered towards deriving the derivative  

f

'

(

x

)

from first principles, as the definition of the derivative is:

f

'

(

x

)

=

lim

h

→

0

f

(

x

+

h

)

−

f

(

x

)

h

This is the function we had in [1], so as we take the limit as  

h

→

0

we get the derivative  

f

'

(

x

)

for any  

x

, This:

f

'

(

x

)

=

lim

h

→

0

4

x

+

2

h

=

4

x