Question

Use the given graph to estimate the value of each derivative. (Round all answers to one decimal place.)

f'(0)
f'(1)
f'(2)
f'(3)
f'(4)
f'(5)

Answer 1

f'(x) is the derivative of f(x). the y value of derivative is the same value as the slope of f(x) at whatever point your looking at so at x=1 the tangent line of f(x) will be horizontal (or zero) so f'(x)=0

Answer 2

We can define the derivative of a function in the point x as the slope of the tangent line to the graph at that point.

Thus to estimate the value of each derivate we just need to look at the graph and estimate the slope.

Another way can be using the aproximate slope, for example, if we want to look at f'(0), then we need to analyze a small interval that contains 0, for example we can use the interval [a, b]

Then we can estimate:

$f'(0) = \frac{f'(1) - f'(-1)}{1 - (-1)} = \frac{1 - 5}{2} = -2$

Now we can do the same for all the others.

for f'(1) we don't need to analyze an interval, we can see that we have a minimum, thus, the value of the derivative must be zero:

f'(1) = 0.

For f'(2) we can analyze the interval [1, 3]

We can get:

$f'(2) = \frac{f(3) - f(1)}{3 - 1} = \frac{4 - 1}{2} = 3/2$

For f'(3) we can analyze the interval [2, 4]

$f'(3) = \frac{f(4) - f(2)}{4 - 2} = \frac{5 - 2}{4 - 2} = 3/2$

For f'(4) we can see that we have a maximum, thus the slope is zero,

f'(4) = 0.

For f'(5) we can analyze the interval [4, 6]

$f'(5) = \frac{f(6) - f(4)}{6 - 4} = \frac{3 - 5}{2} = -1$

If you want to learn more, you can read:

https://brainly.com/question/9964510