Directional Derivatives, Implicit Functions & Taylor Series Flashcards
What is the definition of a directional derivative?
Duf(a, b) = ∇f(a, b) · u
where u is a unit vector.
How does one find the unit vector in the direction of vector v?
u =v/∥v∥
Consider the definition of a directional derivative
Duf(a, b)= ∥∇f(a, b)∥ cos θu,
where u is some unit vector.
At which point will the derivative be at its largest and smallest (steepest and least steep), and when is u orthogonal to ∇f(a, b)?
The derivative is at its largest when cos θu = 1, its smallest when cos θu = -1, and orthogonal to ∇f(a, b) when
cos θu = 0.
Give the definition of a tangent plane of a function or scalar field.
∇f(a, b, c) · (r − p) = 0
at point (a, b, c), where
r is the position vector of some point lying on the tangent plane,
and
p is the position vector of f(a, b, c)
if F(x, y) = 0 defines y implicitly as a function of x, what is dy/dx?
dy/dx= −[Fx(x, y)/Fy(x, y)].
What is the difference between a Taylor series and a Maclaurin series?
A Maclaurin series is a special case of a Taylor series where
a = 0.
Define the 3rd value of a Taylor series for function f(x).
1/2! x [f’‘(a)(x - a)^2].
Provide the general equation for a Taylor series.
The sum between r = 0 and infinity of 1/r!, times f(a) differentiated r times, times (x - a) to the power of r.
What is the truncating error/remainder of the Taylor polynomial of degree n?
Rn = 1/(n + 1)! x [f^(n+1)(c) x (x - a)^(n + 1)]
where
f^(n+1)(c) = f of c differentiated n+1 times.
c is some value between a and x. We are usually interested in the maximum value of this.
How are functions of more than one variable affected by the Taylor series?
instead of simply f(a) differentiated n times, we must find the binomial expansion of degree n, times the partial derivatives of f.
