chapter 12 differentiation

Question 1

How would you differentiate from 1st principle if it was ^2

Answer 1

(x,x^2) (x+h), (x+h)^2 then work out the gradient.

Question 2

What are two important aspects of differentiating from first principles.

Answer 2

Lim
h—0
Also as h—–0 2x+h——- 2x.

Question 3

What gives you the coordinates when differentiating?

Answer 3

Sub into the original equation y=f(x)

Question 4

How would you get the gradient after differentiating?

Answer 4

sub into dy/dx

Question 5

What does it mean if a differentiating question asks for you to find the gradients of A and B at y= something

Answer 5

First factorise the quadratic and x can have two values at that y coordinate.
Then work out the gradient function and sub in to find gradient at that point.

Question 6

How to work out the equation of a tangent to the curve at a point?

Answer 6

Differentiate the original curve.
Then sub in the point of x.
Then put that into the original y-y1=m(x-x1.

Question 7

what is the relationship between tangents and normals?

Answer 7

They are perpendicular so the products of the gradient =-1

Question 8

How to find the coordinates where the tangent to the curve meets the normal of the curve?

Answer 8

You work out the equation of the tangent then the equation of the normal and then solve them simultaneously.

Question 9

What does an increasing or decreasing function noted as?

Answer 9

increasing f’(x) is greater than or equaled to zero.

Decreasing f’(x) is less than or equaled to zero.

Question 10

What does a strictly increasing or decreasing function noted as?

Answer 10

Increasing strictly f’(x) is greater than zero.

Decreasing strictly f’(x) is less than zero.

Question 11

What happens when the answer asks for an interval?

Answer 11

Put it in square brackets.

Question 12

How would you find the interval when you do not have a graph?

Answer 12

Find the gradient function and then equate that to the inequality you are trying to satisfy.

Question 13

What is a stationary point?

Answer 13

A stationary point has a gradient of 0.

Question 14

What are the different types of stationary points?

Answer 14

Local maximum.
Point of inflexion (origin)
Local minimum.

Question 15

How to check that points are minimum or maximum?

Answer 15

Get two coordinates .1 either side of the x and see how the gradient changes.
When gradient goes negative then positive it is min
When gradient is positive then negative.

Question 16

What is the second derivative (rate of change of gradient) at a maximum?

Answer 16

d2y/dx2<0 at a maximum is is decreasing

d2y/dx2>0 at a minimum is increasing.

Question 17

A positive gradient f(x) means what on f’(x)

Answer 17

Above x axis

Question 18

A maximum/ negative gradient means what on f’(x)

Answer 18

Cuts x axis

Question 19

what does a negative gradient means what on f’(x)

Answer 19

below x axis

Question 20

What does a minimum on f(x) means what on f’(x)

Answer 20

cuts x axis

Question 21

What does positive gradient mean on f’(X)

Answer 21

Above x axis.

Question 22

How would you approach a diff modelling question?

Answer 22

Draw a diagram in respect to the problem
Define the variables
Express the restriction algebraically
Write the statement to be maximised or minimised