Methods in Calculus

Question 1

Leibniz’s theorem for the nth derivative of y = uv

Answer 1

n
dy/dx = Σ (nCk d^ku/dx^k d^(n-k)v/dy^(n-k))
k=0
Like a binomial expansion with increasing n of derivative for u and decreasing for v and coefficients from Pascal’s triangle

Question 2

L’Hospital’s rule

Answer 2

lim x->a (f(x)/g(x)) = lim x->a (f’(x)/g’(x))

Use for as many differentials is required to give a determinate form

Question 3

When is L’Hospital’s rule valid

Answer 3

0/0 or +-∞/+-∞

Question 4

L’Hospital’s Rule 0 x ∞

Answer 4

0/(1/∞)

Question 5

L’Hospital’s Rule 1^∞

Answer 5

e^ln(1^∞)

Question 6

Weierstrass Substitution

Answer 6

For integration of difficult trig functions, use substitution with t = tan(x/2) and use t-formulae
dx = k/(1+t^2) dt for t = tan(x/k)