Consider the function ƒ(x) = eˣ
(a) differentiate the taylor series about 0 of ƒ(x)
(b) identify the function represented by the differentiated series.
(c) give the interval of convergence of the power series for the derivative.

a) The Taylor series expansion of about is

Differentiating the expansion with respect to 'x' we get

b)

As we can identify from the expansion calculated above we have

c)

The interval of convergence of the given power series is

to plot points   for x just start at the origin (middle) and go to the left however many times it say(if the number is a positive) if the number is a negative then go right however many times it says if it is 0 go nowhere

now keep you pencil there and go up however many times it says on y(if it is positive) if it is negative then go down however many times it says. if it says 0 go nowhere.

you have just plotted a point! now when you finish plotting all of them connect them with a ruler

plot each point as (-3, 7), (0,4), (3,1), (8,-4)   in the parenthesis on the left of the comma is x and the right is y.

the slope of the line is 60

one point on the line is (40,10)

step-by-step explanation:

in the equation outside the parenthesis the equation always tells you the slope. in this case it is 60.

in the parenthesis is also says 40 which is x because it is near the x and 10 is y because it is near y.