Презентация, доклад по математике Производная, 10 класс

Содержание

1:: Find the derivative of polynomials.2:: Find equations of tangents and normal to curves.Those who have done either IGCSE Mathematics, IGCSE Further Mathematics or Additional Mathematics would have encountered this content. Otherwise it will be completely

Слайд 1Made by ALEX
P1 Chapter 6
«Differentiation»

Made by ALEXP1 Chapter 6«Differentiation»

Слайд 21:: Find the derivative of polynomials.
2:: Find equations of tangents and

normal to curves.

Those who have done either IGCSE Mathematics, IGCSE Further Mathematics or Additional Mathematics would have encountered this content. Otherwise it will be completely new!

3:: Identify increasing and decreasing functions.

5:: Find stationary points and determine their nature.

6:: Sketch a gradient function.

7:: Model real-life problems.

1:: Find the derivative of polynomials.2:: Find equations of tangents and normal to curves.Those who have done

Слайд 3For a straight line, the gradient is constant:
At GCSE, you found

the gradient of a curve at a particular point by drawing a tangent.

?

?

?

?

?

?

?

?

For a straight line, the gradient is constant:At GCSE, you found the gradient of a curve at

Слайд 4The question is then: Is there a method to work out

the gradient function without having to draw lots of tangents and hoping that we can spot the rule?

?

?

?

As the second point gets closer and closer, the gradient becomes a better approximation of the true gradient:

?

?

The question is then: Is there a method to work out the gradient function without having to

Слайд 5?
?
?
?
?
?

??????

Слайд 6“Lagrange’s notation”
“Leibniz’s notation”

“Lagrange’s notation”“Leibniz’s notation”

Слайд 7!
Use the “differentiation by first principles” formula.
a
b
?
?

!Use the “differentiation by first principles” formula.ab??

Слайд 8?
Helping Hand:
“Row 4” of Pascal’s Triangle is:
1 4 6 4 1
You’re

welcome.
?Helping Hand:“Row 4” of Pascal’s Triangle is:1 4 6 4 1You’re welcome.

Слайд 9ME-WOW!
?

ME-WOW!?

Слайд 10Examples:
Power is 5, so multiply by 5 then reduce power by

5.

?

?

?

?

?

?

?

?

Examples:Power is 5, so multiply by 5 then reduce power by 5.????????

Слайд 11?
?
?
?
?
?
?
1
2
3
4
5

???????12345

Слайд 131
2
3
4
5
?
?
?
?
?

12345?????

Слайд 161. Turn roots into powers:
2. Split up fractions.
3. Expand out brackets.
4.

Beware of numbers in denominators!

?

?

?

?

?

?

?

?

?

?

1. Turn roots into powers:2. Split up fractions.3. Expand out brackets.4. Beware of numbers in denominators!??????????

Слайд 17Differentiate the following.
?
?
?

Differentiate the following.???

Слайд 18This just means “differentiate twice”. We’ll be looking at the ‘second

derivative’ later in this chapter.

?

This just means “differentiate twice”. We’ll be looking at the ‘second derivative’ later in this chapter.?

Слайд 19?
?
?
?
tangent

????tangent

Слайд 20?
The normal to a curve is the line perpendicular to the

tangent.

tangent

normal

Fro Exam Tip: A very common error is for students to accidentally forget whether the question is asking for the tangent or for the normal.

?The normal to a curve is the line perpendicular to the tangent.tangentnormalFro Exam Tip: A very common

Слайд 21? y
? gradient
? Final equation

? y? gradient? Final equation

Слайд 22Pearson Pure Mathematics Year 1/AS
Pages 269-270
Extension
1
2

Pearson Pure Mathematics Year 1/ASPages 269-270Extension12

Слайд 24What do you think it means for a function to be

an ‘increasing function’?

A function can also be increasing and decreasing in certain intervals.

?

?

?

What do you think it means for a function to be an ‘increasing function’?A function can also

Слайд 25Fro Tip: To show a quadratic is always positive, complete the

square, then indicate the squared term is always at least 0.

?

?

Fro Tip: To show a quadratic is always positive, complete the square, then indicate the squared term

Слайд 27When you differentiate once, the expression you get is known as

the first derivative.
Unsurprisingly, when we differentiate a second time, the resulting expression is known as the second derivative. And so on…

Leibniz’s

Newton’s

Lagrange’s

Original Function

First Derivative

Second Derivative

When you differentiate once, the expression you get is known as the first derivative.Unsurprisingly, when we differentiate

Слайд 28ME-WOW!

ME-WOW!

Слайд 31Local maximum
Local minimum
Fro Note: It’s called a ‘local’ maximum because it’s

the function’s largest output within the vicinity. Functions may also have a ‘global’ maximum, i.e. the maximum output across the entire function. This particular function doesn’t have a global maximum because the output keeps increasing up to infinity. It similarly has no global minimum, as with all cubics.

?

Local maximumLocal minimumFro Note: It’s called a ‘local’ maximum because it’s the function’s largest output within the

Слайд 32? Method 1: Differentiation
? Method 2: Completing the Square
Fro Note: Method 2 is

only applicable for quadratic functions. For others, differentiation must be used.

?

? Method 1: Differentiation? Method 2: Completing the SquareFro Note: Method 2 is only applicable for

Слайд 33There’s a third type of stationary point (that we’ve encountered previously):
A

point of inflection is where the curve changes from convex concave (or vice versa).

convex

concave

(the same terms used in optics!)

i.e. the line curves in one direction before the point of inflection, then curves in the other direction after.

Technically we could label these either way round depending on where we view the curve from. What’s important is that the concavity changes.

There’s a third type of stationary point (that we’ve encountered previously):A point of inflection is where the

Слайд 34?
?
?
?
?
?
?
?
?
Method 1: Look at gradient just before and just after point.

?????????Method 1: Look at gradient just before and just after point.

Слайд 35Method 1: Look at gradient just before and just after point.
?

Turning Point

? Determine point type

?

Method 1: Look at gradient just before and just after point.? Turning Point? Determine point type?

Слайд 36+ve
gradient
0
gradient
-ve
gradient

+vegradient0gradient-vegradient

Слайд 38Edexcel C2 May 2013 Q9
?
?

Edexcel C2 May 2013 Q9??

Слайд 39? Vertical Asymptotes
? Turning Points
? Graph

? Vertical Asymptotes? Turning Points? Graph

Слайд 40Pearson Pure Mathematics Year 1/AS
Page 276
Extension
1
2
3
4

Pearson Pure Mathematics Year 1/ASPage 276Extension1234

Слайд 41Therefore answer is 3.
1
2

Therefore answer is 3.12

Слайд 44The gradient at the turning point is 0.

The gradient at the turning point is 0.

Слайд 45>
>
>
>
>
>
>
Gradient is negative, but increasing.
Fro Tip: Mentally describe the gradient of

each section of the curve, starting your mental sentence with “The gradient is…” and using terms like “positive”, “but increasing”, “but not changing”, etc.

Gradient is 0.

Gradient is positive: initially increases but then decreases.

Gradient is 0, but positive before and after.

Gradient is positive: initially increases but then decreases.

Gradient is 0.

Gradient is negative. Initially decreases but then tends towards 0.

>>>>>>>Gradient is negative, but increasing.Fro Tip: Mentally describe the gradient of each section of the curve, starting

Слайд 46Solution >
Gradient is negative, but increasing.
Gradient is 0.
Gradient is positive; initially

increases but then decreases.

Gradient is positive, but tending towards 0.

Solution >Gradient is negative, but increasing.Gradient is 0.Gradient is positive; initially increases but then decreases.Gradient is positive,

Слайд 47Pearson Pure Mathematics Year 1/AS
Page 278
Extension
1
?

Pearson Pure Mathematics Year 1/ASPage 278Extension1?

Слайд 48These are examples of optimisation problems: we’re trying to maximise/minimise some

quantity by choosing an appropriate value of a variable that we can control.

We have a sheet of A4 paper, which we want to fold into a cuboid. What height should we choose for the cuboid in order to maximise the volume?

These are examples of optimisation problems: we’re trying to maximise/minimise some quantity by choosing an appropriate value

Слайд 49Real Life Maths!
A sewage container fills at a rate of 20

cm3 per second.

How could we use appropriate notation to represent this?
Real Life Maths!A sewage container fills at a rate of 20 cm3 per second.How could we use

Слайд 50Optimisation problems in an exam usually follow the following pattern:
There are

2 variables involved (you may have to introduce one yourself), typically lengths.
There are expressions for two different physical quantities:
One is a constraint, e.g. “the surface area is 20cm2”.
The other we wish to maximise/minimise, e.g. “we wish to maximise the volume”.
We use the constraint to eliminate one of the variables in the latter equation, so that it is then just in terms of one variable, and we can then use differentiation to find the turning point.

We need to introduce a second variable ourselves so that we can find expressions for the surface area and volume.

These are the two equations mentioned in the guidance: one for surface area and one volume.

?

?

Optimisation problems in an exam usually follow the following pattern:There are 2 variables involved (you may have

Слайд 51? a
? b
? c

? a? b? c

Слайд 52Extension
1
?

Extension1?
Скачать презентацию

Похожие презентации

Презентация к уроку алгебры в 10 классе Схема Горнера Презентация к уроку алгебры в 10 классе Схема Горнера 548
Презентация преобразование рациональных выражений Презентация преобразование рациональных выражений 177
Презентация по математике на тему Мистер Совершенство, или Все, что мы знаем о круге Презентация по математике на тему Мистер Совершенство, или Все, что мы знаем о круге 202
Презентация по математике на тему Тригонометрические уравнения и неравенства. Презентация по математике на тему Тригонометрические уравнения и неравенства. 140
Презентация по математике на тему Умножение многочлена на одночлен (7 класс) Презентация по математике на тему Умножение многочлена на одночлен (7 класс) 183
Презентация по математике на тему Төртбұрыштың аудандарын табуға есептер шығару (8 класс) Презентация по математике на тему Төртбұрыштың аудандарын табуға есептер шығару (8 класс) 235

Что такое shareslide.ru?

Это сайт презентаций, где можно хранить и обмениваться своими презентациями, докладами, проектами, шаблонами в формате PowerPoint с другими пользователями. Мы помогаем школьникам, студентам, учителям, преподавателям хранить и обмениваться учебными материалами.

Для правообладателей

Яндекс.Метрика

Обратная связь

Email: Нажмите что бы посмотреть