12. Differentiation

Question 1

Gradient function

Answer 1

Derivative - a function to find the gradient at any point

Question 2

Gradient function symbol for the curve f(x)

Answer 2

f’(x)

Question 3

Gradient function symbol for the curve y = …

Answer 3

dx

Question 4

Differentiation by first principles

Answer 4

    h

Simplify and then remove all values with a h as it is infinitesimally small

Question 5

What to write for each step of a differentiation by first principles?

Answer 5

f’(x) = lim

h–> 0

Question 6

If y = ax^n dy/dx =

Answer 6

an x^n-1

Question 7

How to differentiate with multiple parts?

Answer 7

Differentiate each individually

Question 8

(1/3x^2)

Answer 8

1/3(x^-2)

Question 9

Normal to a curve at point A

Answer 9

Perpendicular to the tangent of the curve at point A

Question 10

Increasing functions

Answer 10

m is always >= 0

Question 11

[a,b]

Answer 11

Means that x is between a and b

Question 12

How to show that a function is always increasing or decreasing

Answer 12

Complete the square and use that anything squared is positive to prove

Question 13

How to find an interval where a function is increasing/decreasing

Answer 13

Find the derivative and solve, sketch to show whether m is >= or <= 0

Question 14

Second/ second order derivative

Answer 14

Differentiate twice
The rate of change of the gradient function

d^2y
——– or f’‘(x)
dx^2

Question 15

Stationary Points

Answer 15

Where the gradient f’(x) = 0

Question 16

Local minimum/maximum points

Answer 16

Not the actual min/max but in the region where the turning points are they are

Question 17

Point of inflection

Answer 17

The point at which the curve of the line changes direction

Question 18

How to find the type of stationary point

Answer 18

Use the second derivative
>0: minimum point
<0: maximum point
=0: substitute a value just above and just below x into f’(x), if both of those have the same sign it’s a point of inflection

Question 19

Max/min value of y = f(x) on the graph y = f’(x)

Answer 19

Root at that x

Question 20

Point of inflection of y = f(x) on the graph y = f’(x)

Answer 20

Turning point at that x

Question 21

Positive gradient of y = f(x) on the graph y = f’(x)

Answer 21

Above x-axis

Question 22

Negative gradient of y = f(x) on the graph y = f’(x)

Answer 22

Below x-axis

Question 23

Vertical asymptote of y = f(x) on the graph y = f’(x)

Answer 23

Vertical asymptote at the same point

Question 24

Horizontal asymptote of y = f(x) on the graph y = f’(x)

Answer 24

Horizontal asymptote on the x-axis

Question 25

What to make sure you do when modelling?

Answer 25

Write dy/dx in terms of the right variables

Question 26

How to solve modelling questions

Answer 26

• make 2 variables from your information
• make 2 equations from these, one equalling a constraint (number) and one equalling the value you are minimising/maximising
• use the constraint to eliminate one variable by rearranging in terms of the other
• substitute that into the other equation
• differentiate for the min/max and check which