Differentiation | NCEA Level 3 Calculus Strategy Video | StudyTime NZ

Differentiation | NCEA Level 3 Calculus Strategy Video | StudyTime NZ


– Got it, guys, welcome to our level three
calculus differentiation strategy video. I’m Mackenzie. – And I’m Gabrielle. – And we’re going to
be giving you some tips in getting through the standard. – Cool. So, calculus has so many different kinds of equations, but you can basically say, break them down to three different types. So, the first one is pure equations and so that’s just
differentiating them straight away and that can involve
exponential trigonometry or solving some of the tiniest equations or even finding equations
for a straight line. The second is graphing. And so, that’s noticing points where there might be continuity or where a function might concave
up or down, turning points. And, also where a function is increasing or decreasing. And the third one is application. And so that’s when you’re finding the maximum or minimum of a variable or finding the rate of change
for a variable as well. – So for the standard, you
only need to have three of the questions up to excellence to actually get full marks, which is pretty neato. and we’re going to be,
throughout this video, giving you tips on how to get to there. A key thing to recognize
about the standard is, the examiners like to word things in ways that you often have to re-form it before you can really work with it. So for example, you cannot differentiate this root of a. However, we can turn it into a to the power of a half and a differentiate that so it becomes one over two times the root of a. – So, an important term
to keep yourself familiar with is normal line. And, that’s basically just another way of saying perpendicular line. And this means set at a point
two gradients multiplied by each other are equal to negative one and in equations, you can, it can be worded as a is normal line to the curve or a and b are perpendicular to each other and that’s just talking about the relationship
between the two lines. – Another key skill to bear in mind, especially if you’re
aiming for excellence, is using simultaneous
equations in being able to find the equation for a straight line. A really good place to start with this is by using the general formula for a straight line,
so y equals ax plus c, and filling that in with
information along the way. You don’t want to mix
and match your methods, too much just for the simple
fact that it can get confusing. So find a method that works
for you and really stick to it. – So it’s likely that
you’re going to get a graph which has different types of lines on it and you’ll be asked for different points where the graph is increasing
or decreasing, et cetera And so one thing that is really helpful in that case is if you take the first derivative of that graph, points where the value is positive, as where the graph is
increasing and points where the value is
negative, is decreasing. And then same if you took the second derivative of that graph. So you’ll be able to find where a graph is concave up or concave down. – So we’ve finally come
to the part of the video where we’re answering the
excellence type questions. There are sort of two
main branches of this. The first of which being,
finding the minimum or maximum. So in doing this we can have
an example a question like, you have a piece of paper that’s 50 centimeters by 50 centimeters and they want you to find the maximum regular cube using this paper. So, the first step for
this is to make an equation with your wanted variable on one side and everything else on the other. Secondly, we want to be
making sure everything is in terms of the wanted variable. What this means is that
we only have one like term and so we’re only dealing
with one like unknown variable and everything is in relation to that. Thirdly, we want to differentiate this and make it equal to zero and then there you go. There’s is the excellence
part of your question done. Last optional step for
these kind of questions is popping x back into the equation to a, make sure it’s right and b, look at the second derivatives. Now if it is a negative,
what you’re getting is your maximum value
and if it’s positive, you’re gonna be getting
your minimum value. Use these and compare them to
zero and clearly state this in your answer right at the end. – So the second type
of excellence question that you’ll come across is a rate of change question. So, for example our one
we’re working with today is the volume of the cone changes at a rate of three
centimeters cubed per minute. So what’s the rate of change
for the height of the cone? Now you’re going to break this down into four separate steps. The first thing that you’re
gonna do is write down the differentiated
equation in this question that we already know, so this would be, dv over dt equals three, and so you’re going to also
add in the necessary units and change them if you need to as well, but we don’t need to do that. We can just keep it as
three centimeters cubed. The second thing that you’re gonna do, is write down the equation for the rate of change that you want to find and you can use the equation that we just created there before and change it using the triangle so that’ll equal dh over dt equals dv. over dt times, dt over dh. Now the third thing
that you’re gonna do is, find dt of a dh, using the table, so you can do this by using trigonometry and similar triangles if
that’s useful for you. The fourth thing that you’re going to do is substitute that number in and solve for the final answer and you’re good to go. – So the level three calculus
differentiation standard requires that you’re familiar with a bunch of different kinds of
equations and you need to have the foundational skills to be able to answer them and recognize
them under an exam condition. – Excellence questions are
always going to be the hardest to do as they’re designed to be the most difficult to know where to start, but if you’re using the strategies that we just talked about today, then you should be all good. Also, remembering though,
that it’s really important to study for at least
three of the past exams for that external in order
to get a really good grade. – So we’ve covered the fundamentals of the standard in this video. There’s heaps more in
the walkthrough guides that we have online and you can order a hard copy with next day shipping. And, good luck.

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