Differentiation (P1.12 & P2.9) Flashcards
what is the gradient of a curve at a given point defined as?
the gradient of the tangent to the curve at that point
as point B moves closer to point A on the curve, what happens to the gradient of chord AB?
gets closer to the gradient of the tangent to the curve at A
label graph with x0, x0+h & the coordinates of A & B
pg 259 P1 textbook
differentiate from first principles
what is the derivative of the quadratic ax^2 + bx + c?
straight line with the gradient 2a
crosses the x-axis once at the point where the quadratic curve has 0 gradient = the turning point of the quadratic graph
what is the equation of the normal to the curve y=f(x) at the point with coordinates (a, f(a))?
y-f(a) = -1/f’(a) (x-a)
like y-y1 = m(x-x1)
describe how to use the derivative to determine whether a function is increasing or decreasing on a given interval
the function f(x) is increasing on the interval [a,b] if f’(x)≥0 when a<x<b
the function is decreasing when f’(x)≤0
what does differentiating a function twice show?
the rate of change of the gradient
= second order derivative
f’‘(x) =
d^2y / dx^2
what is a stationary point & different types?
any point where the curve has gradient zero
local maximum, local minimum, point of inflection
complete table of type of stationary point, f’(x-h), f’(x) & f’(x+h)
see P1 textbook pg 273
show the different types of stationary points graphically
see P1 pg 273
how can the second derivative be used to determine the nature of a stationary point?
if function f(x) has stationary point at x=a
f’‘(a) > 0 point is a local minimum
f’‘(a) < 0 point is a local maximum
f’‘(a) = 0 point could be local max, min or point of inflection –> must look at points either side to determine its nature
how can second derivatives be used to determine whether a curve is concave or convex on a given domain?
concave: f’‘(x) ≤ 0 looks like a cave on graph ∩
convex: f’‘(x) ≥ 0 looks like the V on graph ∪
define point of inflection
the point at which a curve changes from being concave to convex or vice versa
= the point at which f’‘(x) changes sign
when sketching gradient functions, what features of the graph y=f’(x) correspond with the following y=f(x) features?
maximum/minimum
point of inflection
+ve gradient
-ve gradient
vertical asymptote
horizontal asymptote
cuts the x-axis
touches the x-axis
above the x-axis
below the x-axis
vertical asymptote
horizontal asymptote at x-axis
differentiate y = sinx from first principles
differentiate y = cosx from first principles
what small angle approximations are used in differentiating sinx & cosx from first principles?
sinx ~ x
cosx ~ 1 - 1/2(x^2)
what do sinx & cosx differentiate to?
sinx –> cosx
cosx –> -sinx
f(x) = ln(f(x))
f’(x) =
f’(x)/f(x)
f(x) = a^kx
f’(x) =
a^kx . k . lna
chain rule
dy/dx = dy/du x du/dx
f’(g(x))g’(x)
n(f(x))^n-1f’(x)
product rule f(x)g(x)
f’(x)g(x) + f(x)g’(x)
quotient rule u/v
vu’ - uv’ / v^2
parametric differentiation equation
dy/dx = dy/dt / dx/dt
d/dx (f(y)) =
f’(y) dy/dx
d/dx (y^n) =
ny^n-1 dy/dx
chain rule
d/dx (xy) =
x dy/dx + y
product rule
d/dx (x/y) =
y-x.dy/dx / y^2
quotient rule
how do you connect rates of change involving more than 2 variables?
using chain rule to form differential equations
forming differential equations
