Suppose a function f : R → R is continuous with f(0) = 1. Show
that if there is a positive number x0 for which
f(x0) = 0, then there is a smallest positive number p
for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) =
0}.)
True and False (No need to solve).
1. Every bounded continuous function is
integrable.
2. f(x)=|x| is not integrable in [-1, 1] because the function f
is not differentiable at x=0.
3. You can always use a bisection algorithm to find a root of a
continuous function.
4. Bisection algorithm is based on the fact that If f is a
continuous function and f(x1) and f(x2) have
opposite signs, then the function f has a root in the interval
(x1,...
Consider the function and the value of a.
f(x) =
−2
x − 1
, a = 9. (a) Use mtan = lim h→0
f(a + h) − f(a)
h
to find the slope of the tangent line mtan =
f '(a).
mtan =
(b)Find the equation of the tangent line to f at x =
a.
(Let x be the independent variable and y be
the dependent variable.)
1. Determine the absolute minimum and maximum values of the
function f(x) = x^3 - 6x^2 + 9x + 1 in the following
intervals:
a) [0,5]
b) [-1,2]
2. A company produces and sells x number of calculators per
week. The functions for demand and cost are the following:
p = 500 - 0.5x and c(x) = 10,000 + 135x.
Determine:
a) Function of weekly revenue
b) Price and number of calculators that have to be sold to
maximize revenue...
1. Find the absolute minimum and maximum value of f(x) = x4 −
18x 2 + 7 (in coordinate form) on [-1,4]
2. If f(x) = x3 − 6x 2 − 15x + 3 discuss whether there are any
absolute minima or maxima on the interval (2,∞)
show work please
Find the absolute maximum and the absolute minimum of
the function f(x,y) = 6 - x² - y² over the region R = {(x,y) | -2
<= x <= 2, -1 <= y <= 1 }. Also mention the points at
which the maximum and minimum will occur.
Suppose f(x) is a very well behaved function in that it is
continuous and differential everywhere, and that f(2) = 6 and f(6)
= 34.
a. Find the slope of the line between these two points on the
graph.
b. What does the Mean Value Theorem tell you about f(x) in the
interval between x = 2, and x = 6?
Part A. If a function f has a derivative at x not. then f is
continuous at x not. (How do you get the converse?)
Part B. 1) There exist an arbitrary x for all y (x+y=0). Is
false but why?
2) For all x there exists a unique y (y=x^2) Is true but
why?
3) For all x there exist a unique y (y^2=x) Is true but why?