Calculus HELP ME PLZ?

Question

13. (a) Use a graph of f(x)=2-x^3 to decide whether f'(1) is positive or neg. GIve reasons

You: (b) Use a small interval to estimate f'(1).

Answer 1

f '(x) = -3x^2 which is a parabola with vertex at (0,0) and opens downward.

f '(1) = -3 which is negative

f '(x) is everywhere negative except at x = 0 where it has a maximum value of 0.

Answer 2

f(1) = 2 - 1=1 (positive)

f(1.259) = 0

1 < 3rdROOT(2)

Answer 3

im only in trig. but im 99% sure that f(1) is positive

:S sorry thts all i got lol

Answer 4

Draw the graph of f(x) = 2 - x^3.  It is pretty clear that the graph is always decreasing (except at x = 0, where it is flat), so f '(x) is negative for all x except x = 0.  Specifically, f '(1) must be negative.  If you are supposed to use a small interval to estimate f '(1), then just look at some small interval containing 1 and draw the secant line through the two points on the graph.  This line should clearly have a negative slope, and since the derivative at 1 is the limit of the slope of this secant line, f '(1) will turn out to be negative as well.  Finally, if you know derivative rules, f '(x) = -3x^2 is always less than or equal to 0.