Let y = Ae x + Be − x for any constants A , B . (a) Sketch the graph of the function for ( i ) A = 1 , B = 1 ( ii ) A = 1 , B = − 1 ( iii ) A = 2 , B = 1 ( iv ) A = 2 , B = − 1 ( v ) A = − 2 , B = − 1 ( vi ) A = − 2 , B = 1 (b) Describe in words the general shape of the graph if A and B have the same sign. What effect does the sign of A have on the graph? (c) Describe in words the general shape of the graph if A and B have different signs. What effect does the sign of A have on the graph? (d) For what values of A and B does the function have a local maximum ? A local minimum ? Justify your answer using derivatives.
Let y = Ae x + Be − x for any constants A , B . (a) Sketch the graph of the function for ( i ) A = 1 , B = 1 ( ii ) A = 1 , B = − 1 ( iii ) A = 2 , B = 1 ( iv ) A = 2 , B = − 1 ( v ) A = − 2 , B = − 1 ( vi ) A = − 2 , B = 1 (b) Describe in words the general shape of the graph if A and B have the same sign. What effect does the sign of A have on the graph? (c) Describe in words the general shape of the graph if A and B have different signs. What effect does the sign of A have on the graph? (d) For what values of A and B does the function have a local maximum ? A local minimum ? Justify your answer using derivatives.
(
i
)
A
=
1
,
B
=
1
(
ii
)
A
=
1
,
B
=
−
1
(
iii
)
A
=
2
,
B
=
1
(
iv
)
A
=
2
,
B
=
−
1
(
v
)
A
=
−
2
,
B
=
−
1
(
vi
)
A
=
−
2
,
B
=
1
(b) Describe in words the general shape of the graph if A and B have the same sign. What effect does the sign of A have on the graph?
(c) Describe in words the general shape of the graph if A and B have different signs. What effect does the sign of A have on the graph?
(d) For what values of A and B does the function have a local maximum? A local minimum? Justify your answer using derivatives.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
Find the equation of the line that is perpendicular to (1/5)y + 5 x = 6 at the point (1,5). Express your answer as a function y in terms of x.
y =
E. coli is a very harmful type of bacteria.
The relation
t
N = No (2) ²0
estimates the number of E.coli, N, of
an initial sample of No bacteria
after t minutes, at 37°C (body temperature),
under optimal conditions.
According to the equation above, the doubling time of E. coli is
minutes.
Do not write the units.
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