Chapter 12 Differentiation Flashcards
what is the gradient of a curve at a given point defined as?
the gradient of a tangent to the curve at that point
what is the derivative of the curve y = f(x) written as?
f’(x) or dy/dx
how do you differentiate from first principles?
use the formula f’(x) = lim h->0 (f(x + h) - f(x))/h
if f(x) = x^n then what is f’(x)?
f’(x) = nx^n-1
if f(x) = ax^n then what is f’(x)?
f’(x) = anx^n-1
what is the derivative of the quadratic curve y = ax^2 + bx + c?
2ax + b
what is the equation to the tangent to the curve y = f(x) at the point (a, f(a))?
y - f(a) = f’(a) * (x - a)
what is the equation to the normal to the curve y = f(x) at the point (a, f(a))?
y - f(a) = -1/f’(a) * (x - a)
how can you use the derivative to determine if a function is increasing/decreasing?
- increasing if f’(x) is > (or =) 0
- decreasing f’(x) is < (or =) 0
what are the symbols for second-order derivative?
f’‘(x) and d^2y/dx^2
what is a stationary point?
any point on the curve where f’(x) = 0
how can you tell if a point is a local maximum, local minimum or a point of inflection with the derivative?
for a local maximum:
f’(x - h) is positive
f’(x + h) is negative
for a local minimum:
f’(x - h) is negative
f’(x + h) is positive
for a point of inflection:
f’(x - h) and f’(x + h) must be both positive or both negative
how can you tell if a point is a local maximum, local minimum or a point of inflection with the second order derivative?
if the stationary point is when x = a:
- if f’‘(a) > 0, the point is a local minimum
- if f’‘(a) < 0, the point is a local maximum
- if f’‘(a) = 0, the point could be either
if a point on y = f(x) is a maximum or minimum, what will it be on y = f’(x)?
it will cut the x-axis
if a point on y = f(x) is a point of inflection, what will it be on y = f’(x)?
it will touch the x-axis
if a point on y = f(x) has a positive gradient, what will it be on y = f’(x)?
it will be above the x-axis
if a point on y = f(x) has a negative gradient, what will it be on y = f’(x)?
it will be below the x-axis
what will a vertical asymptote on y = f(x) by on y = f’(x)?
a vertical asymptote
what will a horizontal asymptote on y = f(x) by on y = f’(x)?
a horizontal asymptote at the x-axis
