Revision of Calculus and Elementary Curve Sketching Flashcards
Define derivative
dy/dx is the slope of the (local) tangent to y(x) at a point
What are the conditions for a function to be differentiable at x=x(0)?
Differentiable at x=x(0) if:
lim as δx->0 for (f(x(0) + δx) - f(x(0)))/δx
must exist and not depend on the sign of δx
How can a function be continuous at x=x(0)?
Continuous at x=x(0) if:
lim as x->x(0) f(x) exists and equals f(x(0))
Define e^x by the power series
e^x = 1 + x + 1/2! x^2 + 1/3! x^3 + …
What are the two log rules?
ln(xy) = ln(x) + ln(y) ln(x^y) = yln(x)
What do the hyperbolic functions differentiate to?
cosh(x) -> sinh(x)
sinh(x) -> cosh(x)
tanh(x) -> 1/cosh^2 (x)
What are the four rules of differentiation?
Product rule: d(uv)/dx = u’v + uv’
Quotient rule d(u/v) = (vu’ - uv’)/(v^2)1
Chain rule: y(x) = f(g(x)), dy/dx = df/dg dg/dx = f’(g(x))g’(x)
Reciprocal rule: dx/dy = 1/(dy/dx)
How would implicit differentiation work for y^3?
d(y^3)/dx = d(y^3)/dy dy/dx = 3y^2 dy/dx
The general form of Leibnitz’s formula
d^n (fg)/dx^n = Sum (between n and m=0) of nCm f^(n-m) g^m
=f^(n)g^(0) + nf^(n-1) g^(1) + … + f^(0) g^(n)
Pascal’s rule
nCm + nC(m+1) = (n+1)C(m+1)
Define a maximum point
dy/dx = 0 and d^2 y/dx^2 < 0
Define a minimum point
dy/dx = 0 and d^2 y/dx^2 > 0
What is an inflection point?
The point at which d^2 y/dx^2 changes sign
Things to think when drawing a graph
Symmetry x and y-intercepts Limiting behaviour for small and large x Stationary points Singular points and vertical asymptotes
As x tends to infinity, order these: e^x, ln(x) and x^n in terms of which approaches ∞ the fastest
e^x, x^n, ln(x)
where n is constant and >0
What is proportional to the width of distribution for Normal distribution?
1/a from e^(-ax^2)
What is sinc(x)?
sinc(x) = sin(x)/x
L’Hopital’s rule
Provided that lim as x->x(0) (f’/g’) exists
lim as x->x(0) f(x)/g(x) = lim as x->x(0) (f’/g’)
If lim as x->x(0) f(x) = A and g(x) = B, what arises from this through addition, multiplication and division?
Addition: lim as x->x(0) [f(x) + g(x)] = A + B
Multiplication: lim as x->x(0) [f(x)g(x)] = AB
Division: lim as x->x(0) [f(x)/g(x)] = A/B for B =/= 0
lim as x->x(0) [f(x)/g(x)] doesn’t exist for A=/= 0 and B = 0
lim as x->x(0) [f(x)/g(x)] may exist for A=B=0
lim as x->x(0) [f(x)/g(x)] may exist for A=B=infinity
What is the condition for absolute convergence?
Sum between ∞ and k=0 for |u(k)| converges
u(k) can be complex
What is the condition for conditional convergence?
If the sum between ∞ and k=0 for u(k) converges, but the sum between ∞ and k=0 for |u(k)| does not converge
What is the sum between n and k=0 for r^k equal to?
(1-r^(n+1))/(1-r)
What is the sum between ∞ and k=0 for r^k equal to?
1/(1-r) where |r|<1
What is the harmonic series?
Sum between infinity and k=1 for 1/k
Series diverges
How can the comparison test be used to show a series converges?
If u(k) =< v(k) for all k >= some K, then sum between ∞ and k=0 for u(k) converges if the sum between ∞ and k=0 for v(k) also converges
How can the comparison test be used to show a series converges?
If u(k) >= v(k) for all k > some K, then sum between ∞ and k=0 for u(k) diverges if the sum between ∞ and k=0 for v(k) also diverges
How does the ratio test work?
Given a positive series sum of u(k) if:
lim as k->∞ u(k+1)/u(k) < 1 then sum of u(k) converges
lim as k->∞ u(k+1)/u(k) > 1 then sum of u(k) diverges
lim as k->∞ u(k+1)/u(k) = 1 then sum of u(k) may converge
What is the Leibnitz criterion?
If the series alternates and terms monotonically decrease in magnitude as n increases, series converges
What is the form of a power series?
f(x) = sum between ∞ and k=0 for a(k)x^k = a(0) + a(1)x + a(2)x^2 + …
defines the function f(x) provided series converges
The general form of a Taylor series
f(x) = f(0) + xf’(0) + (x^2)/2! f’‘(0) + (x^3)/3! f’’‘(0) + …
The general form of a Maclaurin series
f(x) = f(a) + (x-a)f’(a) + ((x-a)^2)/2! f’‘(a) + …
What does ln(1+x) equal?
ln(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + …
Newton-Raphson root-finding method
x(1) = x(0) - f(x(0))/f’(x(0))
