Matched Problem 3 Referring to the figure for Example 3, calculate the definite integral (A) ∫ a 0 f ( x ) d x (B) ∫ 0 c f ( x ) d x (C) ∫ 0 b f ( x ) d x EXAMPLE 3 Definite Integrals Calculate the definite integrals by referring to Figure 9. (A) ∫ a b f ( x ) d x (B) ∫ a c f ( x ) d x (C) ∫ b c f ( x ) d x
Matched Problem 3 Referring to the figure for Example 3, calculate the definite integral (A) ∫ a 0 f ( x ) d x (B) ∫ 0 c f ( x ) d x (C) ∫ 0 b f ( x ) d x EXAMPLE 3 Definite Integrals Calculate the definite integrals by referring to Figure 9. (A) ∫ a b f ( x ) d x (B) ∫ a c f ( x ) d x (C) ∫ b c f ( x ) d x
Solution Summary: The above integral indicates the sum of area of the region A, from a to 0. The regions A lies in third quadrant with its area 2.33.
Matched Problem 3 Referring to the figure for Example 3, calculate the definite integral
(A)
∫
a
0
f
(
x
)
d
x
(B)
∫
0
c
f
(
x
)
d
x
(C)
∫
0
b
f
(
x
)
d
x
EXAMPLE 3 Definite Integrals Calculate the definite integrals by referring to Figure 9.
(A)
∫
a
b
f
(
x
)
d
x
(B)
∫
a
c
f
(
x
)
d
x
(C)
∫
b
c
f
(
x
)
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.
3. [10 marks]
Let Go (Vo, Eo) and G₁
=
(V1, E1) be two graphs that
⚫ have at least 2 vertices each,
⚫are disjoint (i.e., Von V₁ = 0),
⚫ and are both Eulerian.
Consider connecting Go and G₁ by adding a set of new edges F, where each new edge
has one end in Vo and the other end in V₁.
(a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so
that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian?
(b) If so, what is the size of the smallest possible F?
Prove that your answers are correct.
Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
Chapter 5 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
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