Calculus Flashcards
Chain Rule:
f(x) = C
f’(x) = 0
Sum Rule:
f(x) = g(x) + h(x)
f’(x) = g’(x) + h’(x)
Product Rule:
f(x) = g(x)*h(x)
f’(x) = g’(x)*h(x) + g(x)h’(x)
Chain Rule:
f(x) = g(h(x))
f’(x) = g’(h(x)) * h’(x)
Power Rule:
f(x) = x^t
f’(x) = t * x^(t-1)
f(x) = ln(x)
f’(x) = 1/x
f(x) = exp(x)
f’(x) = exp(x)
f(x) = sin(x)
f’(x) = cos(x)
f(x) = cos(x)
f’(x) = -sin(x)
Quotient Rule:
f(x) = g(x)/h(x)
f’(x) = (g’(x)h(x) - g(x)h’(x)) / (h(x)*h(x))
What does the first derivative show?
The gradient
What can f’‘(x) show at f’(x)=0?
Whether point is minima or maxima
What does f’‘(x) > 0 tell us about critical point at x?
It is a (local) minimum
What does f’‘(x) < 0 tell us about critical point at x?
It is a (local) maximum
What does f’‘(x) = 0 tell us about critical point at x?
No conclusion can be made
How do you get partial derivatives of a function f(x,y)?
Differentiate assuming y is contant then differentiate assumming x is constant to get fx(x,y) and fy(x,y) respectively where each is a differentiation in respect to x and y respectively.
What is fxx(x,y)?
The partial derivative of fx(x,y) with respect to x
What is fxy(x,y)?
The partial derivative of fx(x,y) with respect to y
What is fyy(x,y)?
The partial derivative of fy(x,y) with respect to y
What is fyx(x,y)?
The partial derivative of fy(x,y) with respect to x
What link do fxy(x,y) and fyx(x,y) usually have?
They are typically the same
What is the precondition for second derivative testing with 2-variable function f(x,y)?
((fxx)(fyy) - (fxy)^2)(α,β) > 0
What is F(x)?
∫f(x)dx
general rule for integrating:
f(x) = x^t
F(x) = 1/(t+1) * x^(t+1)
Up the power then divide by it as a constant
