Calculus Flashcards
How is the nth derivative denoted?
f(n)(x)
How is a composite function denoted?
y = g(f(x)) = (g o f)(x)
What is the derivative of ax?
axlna
What is the derivative of the inverse of a function?
The reciprocal of the derivative of the original function
If g(y) = g(f(x)) = x, g’(y) = 1/f’(x) = 1/f’(g(y))
What is the general definition of elasticity?
If f is differentiable at x and f(x) ≠ 0, the elasticity of f with respect to x is given by f’(x)x/f(x)
This is also the logarithmic derivative as dln(f(x))/dlnx = (dln(f(x))/dx)/(dlnx/dx) = (f’(x)/f(x))/(1/x) = f’(x)x/f(x)
What is the definition of an integrable function?
f:[a, b] –> R is Riemann integrable if limn–>∞ Ln = limn–>∞ Un = I = ∫ ab f(x)dx where Ln is the lower bound with n rectangles and Un is the upper bound with n rectangles
Continuous functions are integrable over every bounded interval
How do you deal with areas below the x axis?
Integrating finds the net signed area (above x axis - below x axis) so if you want the net unsigned area you have to add the sections above the x axis and subtract the sections below the x axis or sum the modulus of all areas
What is the Integral Mean Value Theorem?
For a continuous function f defined over [a, b] there exists c in [a, b] such that (b - a)f(c) = ∫ ab f(x)dx
i.e. f(c) is the average value of f in the interval [a. b], this follows from the Intermediate Value Theorem
What is the formula for integration by parts?
∫ uv’ dx = uv - ∫ u’v dx
Found by integrating and rearranging the product rule
What is integration by substitution?
∫ t0t1 f’(u(t))u’(t)dt = f(u(t))|t0t1 = ∫ u(t0)u(t1) f’(u)du = f(u)|u(t0)u(t1)
How are partial derivatives and second partial derivatives denoted?
∂f(x)/∂xi = fi(x)
fij = (fi)j = ∂/∂xj * df/dxi = ∂f2/dxj∂xi
What is Young’s Theorem?
If f:Rn has continuous second partial derivatives at x = a then fij(a) = fji(a)
