Question

Mathmatics for Calculus

Answer 1

This is a problem finding the slope of a line at some point "a"

It accomplishes this by doing an average rate of change formula over an area approaching 0.

We must know the format of this type of equation.

See first image.

The variables in this example don't match up.

Let the function input be x, and the limit being a --> 0

a = 2

f(x) = x^5

Answer 2

`\displaystyle{f(x)=x^4 \ \: \text{and} \ \: a =2}`

Explanation:

The first principal of derivative:

`\displaystyle{f'(x) = \lim_(h\to 0) (f(x+h)-f(x))/(h)}`

Since the given problem represents f'(a), meaning:

`\displaystyle{f'(a) = \lim_(h\to 0) (f(a+h)-f(a))/(h)}`

If we compare:

`\displaystyle{f'(a) = \lim_(h\to 0) (f(a+h)-f(a))/(h)}`

and

`\displaystyle{ \lim_(h\to 0) ((2+h)^4-16)/(h)}`

which can be rewriten as:

`\displaystyle{f'(a) = \lim_(h\to 0) ((2+h)^4-2^4)/(h)}`

We can conclude by comparing that:

1. a = 2
2. f(2) = 2⁴

This would mean that:

`\displaystyle{f(x)=x^4}`

Therefore:

`\displaystyle{f(x)=x^4 \ \: \text{and} \ \: a =2}`