In Problems 15–22 , use the given information to sketch the graph of f. Assume that f is continuous on its domain and that all intercepts are included in the table of values. 17. Domain: All real x , except x = −2 ; lim x → − 2 − f ( x ) = ∞ ; lim x → ∞ − 2 + f ( x ) = − ∞ ; lim x → ∞ f ( x ) = 1 x −4 0 4 6 f ( x ) 0 0 3 2
In Problems 15–22 , use the given information to sketch the graph of f. Assume that f is continuous on its domain and that all intercepts are included in the table of values. 17. Domain: All real x , except x = −2 ; lim x → − 2 − f ( x ) = ∞ ; lim x → ∞ − 2 + f ( x ) = − ∞ ; lim x → ∞ f ( x ) = 1 x −4 0 4 6 f ( x ) 0 0 3 2
Solution Summary: The author illustrates the graph of the function f(x) from the given information.
In Problems 15–22, use the given information to sketch the graph of f. Assume that f is continuous on its domain and that all intercepts are included in the table of values.
17. Domain: All real x, except x = −2;
lim
x
→
−
2
−
f
(
x
)
=
∞
;
lim
x
→
∞
−
2
+
f
(
x
)
=
−
∞
;
lim
x
→
∞
f
(
x
)
=
1
28. (a) Under what conditions do we say that two random variables X and Y are
independent?
(b) Demonstrate that if X and Y are independent, then it follows that E(XY) =
E(X)E(Y);
(e) Show by a counter example that the converse of (ii) is not necessarily true.
7. [10 marks]
Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G
of length 5. We show how to find a longer cycle in G.
(a) Let x be a vertex of G that is not on C. Show that there are three C-paths
Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect
C at their final vertices.
(b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C.
(c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.
1. Let X and Y be random variables and suppose that A = F. Prove that
Z XI(A)+YI(A) is a random variable.
Chapter 4 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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