P2 Calculus Flashcards
Differentiate sin(kx)
k cos(kx)
Differentiate cos(kx)
-k sin(kx)
Differentiate e^(kx)
k e^(kx)
Differentiate ln(x)
1/x
Differentiate a^(kx)
(a^(kx))(k)(ln(a))
What is the chain rule?
dy/dx = (dy/du) × (du/dx)
How to get from dx/dy to dy/dx?
dy/dx = 1 / (dx/dy)
What is the product rule?
If y = uv, then dy/dx = u(dv/dx) + v(du/dx)
What is the quotient rule?
If y = u/v, then dy/dx = (v(du/dx) - u(dv/dx)) / v²
Differentiate tan(kx)
k sec²(kx)
Differentiate cosec(kx)
-k cosec(kx) cot(kx)
Differentiate sec(kx)
k sec(kx) tan(kx)
Differentiate cot(kx)
-k cosec²(kx)
For parametric functions, how do you get dy/dx?
(dy/dt) / (dx/dt)
How to do implicit differentiation?
Differentiate y like x, then multiply it by dy/dx
f(x) is concave if f’‘(x) is what?
≤ 0
f(x) is convex if f’‘(x) is what?
≥ 0
What is the point called where a curve changes from concave to convex or vice versa?
The point of inflection
How is the point of inflection seen mathematically?
f’‘(x) changes signs
Integration of x^n
(x^(n+1)) / (n+1)
Integration of e^x
e^x
Integration of 1/x
ln(x)
Integration of cos(x)
sin(x)
Integration of sin(x)
-cos(x)
Integration of sec²(x)
tan(x)
Integration of cosec(x)cot(x)
-cosec(x)
Integration of cosec²(x)
-cot(x)
Integration of sec(x)tan(x)
sec(x)
Integration of f’(ax + b)
(1/a) * f(ax + b)
Formula for integration by parts
∫u * (dv/dx) = uv - ∫v * (du/dx)
Trapezium rule formula
∫y between a and b = (h/2)(y₀ + 2(y₁ + y₂ + y₃ … + Yn₋₁) + yn)
When dy/dx = f(x)g(y), how can this be rewritten?
∫(1/g(y)) = ∫f(x)
Integration of the limit of a sum for the integration of f(x) between a and b
limit of dx → 0 for ∑f(x) when x = a and b on the top
