Derivatives as the Rate of Change
(i) The average rate of change = Δy\over Δt
(ii) Instantaneous rate of change = \lim_{x \to 0} {Δy\over Δt} = dy\over dt
Marginal Function
(i) Cost Function {C(x)} = FC + V(x)
- Fixed Cost (FC)
- variable Cost {V(x)}
(ii) Revenue Function R(x) = p × x
- Cost per unit (p)
- Number of units (x)
(iii) Profit Function P(x) = R(x) – C(x)
(iv) Break-even point: R(x) – C(x) = 0
(v) Marginal Cost = dC(x)\over dx
(vi) Marginal Revenue = dR(x)\over dx
Tangents and Normal
Let y = f(x) be the curve and P (x1, y1) be the point at which tangent and normal are drawn then,
(i) Slope of the tangent (m1) = dy\over dx
(ii) Slope of the tangent (m2) = – 1\over {dy/dx}
(iii) Eq of tangent: y – y1 = m1(x – x1)
(iv) Eq of Normal: y – y1 = m2(x – x1)
(v) Length of Tangent = y1 \sqrt{1+({dy\over dx})^2} \over {dy\over dx}
(vi) Length of the normal = y1 \sqrt{1+({dy\over dx})^2}
(vii) Angle of the intersection of two curves
tan θ = |{m_1-m_2\over 1+m_1m_2}|
Rolle’s Theorem
If a function f(x) is
(i) continuous in the closed interval [a,b]
(ii) Derivable in the open interval (a,b) and
(iii) f(a) = f(b)
Then there exist at least one real number c in (a,b) such that f’(c) = 0
Lagrange’s Mean Value Theorem
If a function f(x) is
(i) continuous in the closed interval [a,b] and
(ii) Derivable in the open interval (a,b)
then there exists at least one real number c in (a,b) such that
f'(c)= {\frac{f(b) - f(a)}{b - a} }.
Approximations and Errors
1. Let y = f(x) be a given function and Δx denotes a small increment in x, corresponding to which y increases by Δy. Then, for small increments, we assume that
Δy\over Δx ≈dy\over dx
∴ Δy = {dy\over dx}×Δx
For approximations of y, Δy = dy
Then, dy = {dy\over dx}×Δx
Thus y + Δy = f(x + Δx) = f(x) + {dy\over dx}×Δx
2. Let Δx be the error in the measurement of independent variable x and Δy is the corresponding error in the measurement of dependent variable y.
Then, Δy = {dy\over dx}×Δx
Δy =Absolute error in measurement of y
{Δy\over y} = Relative error in measurement of y
{Δy\over y}×100 = Percentage error in measurement of y
