6.
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Tafu is working with subatomic particles in the Physics lab. A positron is traveling in a straight line down the particle accelerator. Tafu
models its position with the function
p(t)=
t
2
t
– 1
where
t>1
. In the limit calculations below, if you need to use
∞
or
∞
, enter INFINITY or INFINITY.
(a) Calculate the limit
lim_(t>infty) p(t)
=
INFINITY
.
(b) Calculate the limit
lim_(t>1^+) p(t)
=
INFINITY
.
(c) Calculate the limit
lim_(t>2) p(t)
=
$$4
.
(d) Let
a>1
be some fixed number. Let
h>0
be some small number. Write out a formula for the slope of the secant line to the graph
of
x=p(t)
between the point at
t=a
and the point at
t=a+h
. You don't need to simplify this formula yet.
(WARNING: When you write out an expression involving
a
and
h
, anytime you want to multiply
a
times
h
, you need to leave a single
space between the two; i.e. "
a h
" means "
a
times
h
". In contrast, if you write "
ah
" the computer=webassign thinks that you are
talking about a new variable called "
ah
". Alternatively, use the arithmetic buttons in the popup box whenever you want to add,
subtract, multiply or divide.):

$$(
a
+
h
)2(
a
+
h
−1)−
a
2
a
−1
h
(e) Keep
a
fixed and think of your formula in (d) as a function in the variable
h
. Call it
f(h)
. Now simplify your formula for
f(h)
so
that it is a fraction with polynomials in the numerator and denominator, and the denominator does not vanish at
h=0
; i.e. your final
answer should be a rational function in
h
which is defined at
h=0
.
(WARNING: When you write out an expression involving
a
and
h
, anytime you want to multiply
a
times
h
, you need to leave a single
space between the two; i.e. "
a h
" means "
a
times
h
". In contrast, if you write "
ah
" the computer=webassign thinks that you are
talking about a new variable called "
ah
". Alternatively, use the arithmetic buttons in the popup box whenever you want to add,