Chapter 12 Differention - Year 1

Question 1

What do you do when you are asked to find the tangent or the normal?

Answer 1

Substitute x into the equation to calculate the y value so that you know what co-ordinate you are working with.
Take the first derivative of the equation.
Then substitute x into the gradient function f’(x) to find the gradient of the tangent
You can find the gradient of the normal by finding the negative reciprocal

Then use the equation Y-y=m(X-x) to find the equation of the normal/tangent

Question 2

What is an increasing function?

Answer 2

A function which is always greater than 0

Question 3

What is a decreasing function?

Answer 3

A function which has a gradient that is always less than 0

Question 4

Give an example of set notation?

Answer 4

(2,4) (this means between the values of 2 and 4
The curved brackets signal that
The square brackets signal that

Guvugvuvugvuhvuhvuv

Question 5

How so you show that a function is increasing fir all values of x?

Answer 5

You take the first derivative of f(x)
You then complete the square

You then say that the expression containing x and which is squared is >(or equal to) 0 for all real values of x
Therefore 3(x+2)^2 +9 > 0 for all real values of x
Therefore f(x) is an increasing function for all real values of x

Question 6

How do you find the interval of which a function is decreasing?

Answer 6

You take the first derivative of f(x)
You have been told that the function is decreasing so you know that f’(x)<0
You can then solve the equation as it is <0

How do u know weather to out it into set notation or not

Question 7

What is a stationary point?

Answer 7

Where the gradient of the tangent is = 0

Question 8

What are the 3 types of stationary points?

Answer 8

Local maximum
Local minimum
Points of inflexion

Question 9

How do you try to find the turning points an equation with a x power greater than 2. Eg a cubic?

Answer 9

You differentiate
As you know that it is a stationary point (stated in the question) you can set f’(x) = 0
Then you solve it
Then you substitute your x values back into the original equations to get a corresponding y value

Question 10

What does finding the least value of a equation mean?

Answer 10

Finding the turning point

Question 11

What is a point of inflexion?

Answer 11

Where the curve changes from convex to concave

Question 12

Briefly explain the two methods of finding the point of inflexion

Answer 12

Method 1 - you work out the gradients before and after the point of the stationary point

Method 2 - you use the second derivative

Question 13

Use method 1 to work out weather the stationary point is a local maximum or a local minimum or a point of inflexion?

Answer 13

You work out the stationary point
You know that the gradient of a stationary point (x) is 0
You then work out what the gradient of x-0.1 and x+0.1
If the gradient is negative then the gradient is decreasing
If the gradient is positive then the gradient is increasing
Draw it out so you can visualise the graph
State what type of stationary point it is

Question 14

Use method 2 to work out the classification of the stationary point?

Answer 14

You need to find the second derivative
You can then substitute you value for x into the second derivative so solve the equation

If the answer is greater than 0 then it is a minimum
If the answer is less than 0 then it is a maximum
If the answer = 0 then you need to use method 1 as it could be either

Question 15

How do you use differentiation to model?

How would you justify that your answer is the maximum?

Answer 15

It is usually used in optimisation problems
You need 2 variables perimeter and volume
You will have a constraint “the surface area is 20cm”
And you will have something that you want to maximise the area for eg volume

You then use the constraint to eliminate one of the variables
Now you have an equation that is in terms of just 1 variable

You can now differentiate
Set the answer to 0 and solve

Put your variable answer (x) back into the equation that it is asking you to maximise eg volume

Use method 2 (taking the second derivative) or method 1 (crude sketch)

Part c will be on a different flashcard

Question 16

What is the definition of the dervivative?

Answer 16

Question 17

Substitute 3x^2 into the definition of the derivative?

Answer 17

Question 18

How do you do a show that question the definition of a derivative and it gives you f(x) and a point on that line?

Answer 18

Write out the definition of the derivative. Since you know x you can substitute it in
Then rearrange

Question 19

When sketching the gradient function what is the firs thing you should map out?

Answer 19

Any stationary point draw a dotted line downwards as that will be a root of your derivative

Question 20

What is the general method of sketching the gradient function?

Answer 20

Draw in the roots first (stationary points of f(x)
On f(x) draw in + or - signs where the graphs gradient is increasing / decreasing

Where you have + the graph will be above the x axis. Where they are - the gradient function will be below the x axis

Question 21

When sketching the gradient function what do you know if you see a point of inflexion on f(x)?

Answer 21

It will be a repeated root on f’(x)

Question 22

When sketching the gradient function what happens to a vertical asymptote?

Answer 22

You will have a vertical asymptote

Question 23

When sketching the gradient function what happens to a horizontal asymptote?

Answer 23

Horizontal asymptote at the x axis