- Kia ora everyone,
welcome to StudyTime's
Level 2 Calculus walkthrough video.
I'm Hattie.
- And I'm McKenzie.
- And we are walking you through
some tips to succeed in this exam.
Now first off, a lot of
students struggle to see
the purpose of calculus or why we apply it
for the kinds of questions
we do in this exam,
but it's actually really important
and because this is the first time
you've met calculus in NCEA,
it's really important that
we clear all of this up.
- Now for a lot of the questions,
they're all sort of centered
around this main idea
of finding the rate of change.
Now, rate of change can be
applied to finding things
like the speed of a moving object
or finding the height of a cylinder
based off a rate of volume increase.
There are a bunch of
other examples as well
that we can find in everyday life.
- It's really common to find
that students get stuck on a question
and they just try and differentiate it
to see what happens.
This is actually an okay strategy,
because it might get you an
achieved on some questions,
but to get those higher marks
we need to make sure that you actually
understand what you're doing
so that you can plan out a proper answer
that you have control over.
- In understanding the
purpose of calculus,
it can be handy to think
back to year 10 or 11 physics
using an example.
So if we use an example of
a runner on a track runs
100 meters in 10 seconds,
it was a really good day,
we can plot it on a distance-time graph.
Now to put it in calculus language,
we would discuss the rate
and this would be the change in distance
in relation to time, and in physics terms,
this would be our velocity.
And so our rate in this one
would be 10 meters per second.
- But what happens when
velocity isn't constant?
Well, that's when we can use calculus
to find what the velocity is
at any particular instant in time.
For this we just
differentiate the equation
to find the gradient, which
is the same as the velocity,
then we plug in the given time
to generate the velocity for
any point on the motion graph.
This idea is really the main
thing behind all of calculus,
and all this exam is
going to ask you to do
is basically, it's gonna give
you a bunch of information
with some things missing,
and it's going to ask you
to find the missing bits.
For example, it could also ask you
to find the point in time
when the velocity was four.
This is just the opposite of
what we were doing before,
so instead of plugging in a
time and finding a velocity,
we would plug in a velocity as four
and solve backwards to find the time.
- Some general tips for
the standard include
nailing the questions that ask you
for the gradient function of x.
Drawing these up can be a
little bit annoying at first
but they're well worth it.
If you can understand why a parabola
becomes a linear equation
after differentiation,
it really helps your general
understanding of calculus.
If you have another question
that you're feeling a
little bit unsure of,
try to draw it out.
This can be really helpful
in giving you some
context for the question
and let you know what
you're really searching for,
whether this be a graph
or just sketching the
movement of an object.
- On the most basic level, you
want to be really comfortable
with differentiating and
anti-differentiating.
Now obviously these would have been
about the first things
you learned in class
and that's because you really
can't progress without them.
There's no point staring at an exam paper,
not being able to do anything
because you didn't practice
those basic skills.
So make sure you do
heaps of basic problems
before you try and move
on to anything else.
Some tips to help you with this,
for example if you have x
on the bottom of a fraction,
like one over x, it
really helps to turn this
into a negative power before trying
to differentiate or anti-differentiate,
so one over x would become x
to the power of negative one.
Likewise, if you have a root
symbol like a square root,
you want to turn that
into a fractional power.
So if you had square root of x,
that would turn into
x to the power of 1/2.
And lastly, we can't stress this enough,
don't forget the plus c when
you're anti-differentiating.
- A question NCQA almost always throws in
is about local minimums
and maximums on your graph,
and this is what we
call our turning points.
Now the gradient at a
turning point is zero,
and what this means is that the rate
at this point of time is
also going to be zero.
Being familiar with the
different types of graphs
can also be really really helpful
in understanding how many turning points
there are going to be.
- We should also mention
that there's a third option
for when the gradient is negative,
and this is called a point of inflection.
This is when the graph is going up
and then it goes flat and
then it goes up again,
or you could get a negative equivalent.
So if a question asks you to show
whether something's a
maximum or a minimum,
what do we do then?
So this is when we use the
second derivative test.
So you need to go back to
your gradient function,
redifferentiate it and
plug your x value in.
If your answer comes out positive,
then your point is a minimum,
and if it comes out negative,
then it's a maximum.
Now this is a bit counter-intuitive,
so I like to remember
it as a positive face
would be a smiley face,
and the turning point
on that would be a minimum.
The other option, of course,
is if it comes out to zero then
it's a point of inflection.
- Finding the equation of a tangent
is considered one of the more
tricky parts of the standard
and you would have been
given a formula in class.
However if you find this a bit tricky,
we can use our linear
formula, y equals mx plus c.
Now m is the gradient of our line,
and the way that we find
this is differentiating
our original curve and using x to find m.
For our c, our y intercept,
what we then do is use our now known m
and two known co-ordinates,
either from our graph
or from the data table given to us,
and we use these for y
and x to solve for c.
What we then do is use
our now known variables
and rewrite our equation.
- A lot of the trickier
excellence questions
focus around finding the volume
of a three-dimensional shape.
For example, a question
might ask you to find
the height and radius of a cylinder
that would maximize its volume,
given that it had a
particular surface area.
The whole trick here is to
use that surface area equation
and rearrange it so that you have
the height in terms of the radius,
then substitute this back
into the volume equation.
The idea is to make sure
that your volume equation
only has one variable in it, the radius,
so that you can easily differentiate it.
Now differentiate it
and set it equal to zero
so you can solve to find the radius
that would give the maximum volume.
- Many of the lower level E
questions involve kinematics.
And one key tip to
remember about kinematics
is that it stands for spatium,
which in Latin actually means space.
And this is why we denote
distance with an s.
So in these kind of questions,
all they really do require you to do
is figure out whether or
not you need to integrate
or differentiate to find the equation
that you're searching for.
Now, in knowing how to do either
differentiation or integration,
you have to use SVA.
And if you have trouble
remembering the order of this,
you can compare it to PVA glue,
like the stuff in primary,
and that order is the same.
Lastly, it can be really
helpful to use a timeline
of either going left
to right or up or down,
depending on the context of the question.
What can be useful here
is to put in the point
where you begin to measure your rate
and any other relevant information
such as your motion and the
point that you're looking for.
So this has been our Level 2
strategy video for calculus.
We've walked you through some key tips
to do well on this exam, but
we haven't covered everything.
We really recommend going through
a few years of past papers
to get a feel for what to expect,
and also checking out the
StudyTime walkthrough guides,
they're available for free online
or to purchase in print
with next day delivery,
and they're designed to walk you through
everything you need to
do well in this exam.
- Thanks for watching guys and good luck.
(upbeat music)
