1. ## Theorems about Definite Integrals!!

The graph of a derivative f '(x) is shown in the figure below. This is not the graph of f(x).

Fill in the table of values for f(x) given that f(0) = 8.

i need some help on doing this, its still a little fuzzy haha, thanks

mathlete

2. Originally Posted by mathlete
The graph of a derivative f '(x) is shown in the figure below. This is not the graph of f(x).

Fill in the table of values for f(x) given that f(0) = 8.

...
I assume that the graph of f is continuous and once (but only once!) differentiable - even though that isn't stated in your problem, but otherwise you can't calculate the missing values.

Divide the graph of f' into 4 parts concerning the intervals [0,1), [1, 3), [3, 5) and [5, 6]:

to [0, 1): The graph of f is a parabola whose quadratic term is $\displaystyle -\frac12 x^2$. Therefore $\displaystyle f(1)= \frac{15}2$:....$\displaystyle f_1(x)=-\frac12 x^2+8$

to [1, 3): The graph of f is a straight line whose linear term is $\displaystyle - x$. Therefore $\displaystyle f(3)= \frac{11}2$:....$\displaystyle f_2(x) =-(x-1)+7.5$

to [3, 5): The graph of f is a parabola whose quadratic term is $\displaystyle \frac12 x^2$. Therefore $\displaystyle f(5)= \frac{11}2$:....$\displaystyle f_3(x)=\frac12 (x-4)^2+5$

to [5, 6]: The graph of f is a straight line whose linear term is $\displaystyle x$. Therefore $\displaystyle f(6)= \frac{13}2$:....$\displaystyle f_4(x) = (x-5)+5.5$