M - Pure 1 - 12) Differentiation

Question 1

Equation for differentiation by first principles

Answer 1

f’(x) = lim_h→0 [f(x + h) - f(x)]/h

Question 2

The derative of every power of x and constants

Answer 2

• ax^n≥2 → anx^n-1
• bx → b
• c → 0

Question 3

Equation of the tangent to a curve with coordinates (a, f(a))

Answer 3

y - f(a) = f’(a)·(x - a)

Question 4

Information that shows if a function is increasing or decreasing on the interval (a, b)

Answer 4

• If increasing → f’(x) ≥ 0 for all values of x
• If decreasing → f’(x) ≤ 0 for all values of x

Question 5

Notations of second order derivatives

Answer 5

• f’‘(x)
• d²y/dx²

Question 6

Meaning of a stationary point

Answer 6

A point where f’(x) = 0

Question 7

Method to find the coordinates of a stationary point of a function

Answer 7

1. Solve dy/dx = 0 to find x-ordinate
2. Substitute that into f(x) to find y-ordinate

Question 8

Information that shows what type of stationary point x = a is

Answer 8

• If f’‘(a) > 0 → local minimum
• If f’‘(a) < 0 → local maximum
• If f’‘(a) = 0 → the point could be a local minimum, local maximum or point of inflection - look at points on either side

Question 9

Features of a function and the corresponding sketches of its gradient function

Answer 9

• Positive gradient → above x-axis
• Negative gradient → below x-axis
• Maximum or minimum → cuts x-axis
• Point of inflection → touches x-axis
• Vertical asymptote → vertical asymptote
• Horizontal asymptote → horizontal asymptote at x-axis