Consider the map f(x) =x^2+k .Find the values of k for
which the map f has...
Consider the map f(x) =x^2+k .Find the values of k for
which the map f has
a) two fixed points
b) only one fixed point
c) no fixed points
For what values of k there will be an attracting fixed point of the
map?
Consider the function f(x) =x/x^2+1
(a) Find all values ofxfor which f(x) is positive or negative.
Discuss the behaviour of f(x) as x gets very large (in both the
positive and negative direction).
(b) Draw a sign diagram clearly showing the values of x for
which f(x) is increasing and decreasing. Hence find the critical
points off(x) and determine their nature.
(c) Draw a sign diagram clearly showing the values of x for
which f(x) is concave up and concave...
Consider the function f(x) =x/x^2+1
(a) Find all values ofxfor which f(x) is positive or negative.
Discuss the behaviour of f(x) as x gets very large (in both the
positive and negative direction).
(b) Draw a sign diagram clearly showing the values of x for
which f(x) is increasing and decreasing. Hence find the critical
points off(x) and determine their nature.
(c) Draw a sign diagram clearly showing the values of x for
which f(x) is concave up and concave...
Find the value of k such that the graph of y = f(x) has no
vertical asymptote given by x = 2 where
f(x) = (4x 3 − 4x 2 + kx + 14)/ 4x 2 − 12x + 8 .
Then find all the intercepts, asymptotes, local extreme values,
points of inflection, monotonicity intervals, concavity intervals.
Finally sketch the graph of y = f(x).
Consider this table of values for a function:
x
f(x)
-3
15
-2
2
-1
-5
0
-3
1
4
2
8
3
-12
How many zeroes does this function appear to have?
Where are those zeroes (give intervals of x-values). Use
interval notation.
Can you be guaranteed that those are the only zeroes? Why or
why not?
If I told you that the table represented a third degree (cubic)
polynomial, is that enough to guarantee that those are the...
Problem 2
Find the locations and values for the maximum and minimum of f
(x, y) = 3x^3 − 2x^2 + y^2 over the region given by x^2 + y^2 ≤
1.
and then over the region x^2 + 2y^2 ≤ 1.
Use the outline:
INSIDE
Critical points inside the region.
BOUNDARY
For each part of the boundary you should have:
• The function g(x, y) and ∇g
• The equation ∇f = λ∇g
• The set of three equations...