Video Transcript
Which of the following is the slope
field of the differential equation d𝑦 by d𝑥 is equal to the sinh of 𝑥? Option (A), (B), (C), (D), or
(E).
We’re given five sketches of a
slope field. And we need to determine which of
these sketches is a sketch of the slope field of the differential equation d𝑦 by
d𝑥 is equal to the sinh of 𝑥. To answer this question, we first
need to recall what the slope field graph of a differential equation is.
We recall, to plot the slope field
of a differential equation, we just need to plot the slope of the differential
equation at each point. In this case, we’re told the slope
d𝑦 by d𝑥 is equal to the sinh of 𝑥. So we can substitute values of 𝑥
into this equation to give us values of slope. We can then see which of our
sketches matches this information.
And before we start answering this
question, there’s one thing we’re going to need to recall. The sinh of 𝑥 is equal to 𝑒 to
the power of 𝑥 minus 𝑒 to the power of negative 𝑥 all divided by two. We now want to find the value of
our slope at several different values of 𝑥. One way of doing this is using a
table.
So we’re now ready to start finding
out information about our slope field graph. We’re going to substitute values of
𝑥 into our differential equation d𝑦 by d𝑥 is equal to the sinh of 𝑥. Let’s start with 𝑥 is equal to
zero. First, we substitute 𝑥 is equal to
zero into our differential equation. That’s the sinh of zero, which we
know is 𝑒 to the power of zero minus 𝑒 to the power of negative zero divided by
two. And of course 𝑒 to the zeroth
power is one and negative 𝑒 to the negative zeroth power is negative one. So this simplifies to give us
zero.
So when 𝑥 is equal to zero, our
slope d𝑦 by d𝑥 is equal to zero. And we know that a slope being
equal to zero means that we will have a horizontal line. So what this tells us is, on our
slope field graph, whenever 𝑥 is equal to zero, we must have a horizontal slope
line. So let’s see which of these graphs
has this property.
Let’s start with graph (A). We can see on the line 𝑥 is equal
to zero, all of our slope lines are horizontal. So option (A) could be the sketch
of our slope field of this differential equation. However, if we look at option (B),
we can see, on the line 𝑥 is equal to zero, our slope lines are positive. So option (B) can’t possibly be a
sketch of this slope field graph because, on the line 𝑥 is equal to zero, we would
need horizontal slope lines.
If we look at option (C), we can
see, on the line 𝑥 is equal to zero, all of our slope lines are horizontal. We can also see the same is true
for option (D) and option (E). So any of these could be the slope
field graph of our differential equation. This means we’re going to need to
try more values of 𝑥.
Let’s now try 𝑥 is equal to
one. We substitute 𝑥 is equal to one
into our differential equation. This gives us d𝑦 by d𝑥 will be
equal to the sinh of one, which we know is 𝑒 to the power of one minus 𝑒 to the
power of negative one all divided by two. And if we evaluate this expression
to one decimal place, we get 1.2. So on the line 𝑥 is equal to one,
we expect our slope lines to have slope approximately 1.2. This is very similar to the line 𝑦
is equal to 𝑥.
So let’s now check which of our
options has this property. Let’s start with sketch (A). We can see, on the line 𝑥 is equal
to one, all of our slope lines have approximate slope of one. This means that sketch (A) could
possibly be a sketch of our slope field of this differential equation. If we look at sketch (C) and sketch
(D), we can see that both of these have positive slope lines on the line 𝑥 is equal
to one. And these seem to have approximate
slope of one. So we won’t eliminate either of
these options.
However, if we look at option (E),
on the line 𝑥 is equal to one, we can see that all of our slopes have negative
slope. And since these slopes are
negative, this means that option (E) can’t possibly be the sketch of our slope
field. And now we could do the same again
by picking another value of 𝑥 and seeing which of our sketches has this
property. However, it’s worth discussing how
do we pick our value of 𝑥.
At the moment, we have three
possible options left: option (A), option (C), and option (D). We want to pick a value of 𝑥 where
all three of these sketches have a different property for the slope lines on this
value. For example, let’s consider 𝑥 is
equal to negative five. On sketch (A), we can see that our
slope lines are positive and the slope’s approximately equal to one. On sketch (C), when 𝑥 is equal to
negative five, our slopes are negative and approximately negative one.
However, on sketch (D), when 𝑥 is
equal to negative five, we can see that our slope lines are essentially vertical
lines. And we know, for our slope lines to
appear vertical, the absolute value of the slope must be very large. So all three of our sketches have
different properties when 𝑥 is equal to negative five. So by choosing 𝑥 is equal to
negative five next, we can guarantee we can determine which of our sketches is
correct.
So, we’ll substitute 𝑥 is equal to
negative five into our differential equation, giving us d𝑦 by d𝑥 is the sinh of
negative five, which we know is equal to 𝑒 to the power of negative five minus 𝑒
to the power of negative one times negative five all divided by two. And if we calculate this, to one
decimal place, we get negative 74.2. And the absolute value of this is
very, very large. So on this scale, this will appear
like a vertical line.
And we can then use this to
determine which of our sketches is correct. We can see that option (A) is not
correct since when 𝑥 is equal to negative five, our slope lines are positive. In sketch (C), we can see when 𝑥
is equal to negative five, our slope lines are negative. But we can see they’re
approximately negative one. So (C) can’t be the correct
sketch. And finally, in sketch (D), when 𝑥
is equal to negative five, we can see that our slope lines are vertical. This means, of all the options,
only sketch (D) can be the correct sketch of the slope field of this differential
equation.
In this question, we were given
five possible sketches of the slope field of the differential equation d𝑦 by d𝑥 is
equal to the sinh of 𝑥. We substituted values of 𝑥 into
this differential equation to determine information about the sketch of the slope
field graph. We were then able to determine that
only the sketch (D) could possibly be the sketch of the slope field graph of this
differential equation.
