Differentiation 2

Question 1

When sketching the gradient function, what happenes to the following:

Stationary points

Maximum or minimum

Positive gradient

Negative gradient

Stationary point of inflection

Vertical asymptote

Horizontal asymptote

Answer 1

Stationary points- Become roots

Maximum or minimum- Where graph cuts the x axis

Positive gradient- Goes above x axis

Negative gradient- Goes below x axis

Stationary point of inflection- Touches the x axis

Vertical asymptote- Stays in same place

Horizontal asymptote- Becomes y = 0

Question 2

When is the function 𝑓(𝑥) increasing between 𝑥-coordinates 𝑎 and 𝑏?

And

When is the function 𝑓(𝑥) decreasing between 𝑥-coordinates 𝑎 and 𝑏?

Answer 2

The function 𝑓(𝑥) is increasing between 𝑥-coordinates 𝑎 and 𝑏 when 𝒇′(𝒙) ≥ 𝟎 for all real values of 𝑥 such that 𝑎 ≤ 𝑥 ≤ 𝑏.

The function 𝑓(𝑥) is decreasing between 𝑥-coordinates 𝑎 and 𝑏 when 𝒇′(𝒙) ≤ 𝟎 for all real values of 𝑥 such that 𝑎 ≤ 𝑥 ≤ 𝑏.

Question 3

How do you differentiate from first principles?

Answer 3

M1 – State 𝑓(𝑥) = ⋯
AND expand and simplify 𝑓(𝑥 + ℎ)

B1 – Sub 𝑓(𝑥) and 𝑓(𝑥 + ℎ) into formula
AND attempt to simplify

A1 – Show factorising
AND cancelling separately

A1 – “as ℎ → 0, … → ⋯”
AND “∴ 𝑓′(𝑥) = ⋯”
AND “𝑓′(𝑥) = limℎ→0( …