Revision: Section A >> Applications of Derivatives Mathematics ISC (Commerce) Class 12 CISCE

A function f is said to be decreasing at a point c if f '(c) < 0.

x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

Strictly decreasing function:

x₁ < x₂ ⇒ f(x₁) > f(x₂)

f is said to have a maximum value in D if there exists a point x = c in D such that f(c) ≥ f(x) for all x ∈ D. The number f(c) is called the (absolute) maximum value of f in D, and the point c is called the point of maxima of f in D.

f is said to have a local (or relative) maxima at x = c (in D) if there exists a positive real number δ such that f(c) > f(x) for all x in (c − δ, c + δ) x ≠ c i.e. f(c) > f(x) for all x in the immediate neighbourhood of c, and c is called point of local maxima and f(c) is called local maximum value.

f is said to have local (or relative) minima at x = d (in D) if there exists some positive real number δ such that f(d) < f(x) for all x ∈ (d − δ, d + δ) x ≠ d i.e. f(d) < f(x) for all x in the immediate neighbourhood of d, and d is called point of local minima and f(d) is called local minimum value.

A point x = c in the domain of the function f at which either f′(c) = 0 or f is not differentiable i.e. f′(c) does not exist is called a critical point.

A point x = c (in D) is called a stationary point iff f is differentiable at x = c and f′(c) = 0.

The increment δx in x is called the absolute error in x.

Absolute error in x = |δx|

If δx is an error in x, then \[\frac{\delta x}{x}\] is called the relative error in x.

A stationary point x = c (in D) where the function f changes its nature from increasing to decreasing or from decreasing to increasing, i.e. where the function f has local maxima or local minima, is called a turning point.

If δx is an error in x, then \[\frac{\delta x}{x}\] × 100 is called the percentage error in x.

Marginal Cost (MC) is the instantaneous rate of change of total cost with respect to the number of items produced at an instant.

A function f is said to be increasing at a point c if f '(c) > 0.

f is increasing in an interval if

x₁ < x₂ ⇒ f(x₁) ≤  f(x₂)

Strictly increasing function:

x₁​ < x₂​ ⇒ f(x₁​) < f(x₂​)

f is said to have a minimum value in D if there exists a point x = d in D such that f(d) ≤ f(x) for all x ∈ D. The number f(d) is called the (absolute) minimum value of f in D, and the point d is called the point of minima of f in D.

\[\text{Rate of change of}y=\frac{dy}{dx}\times\text{rate of change of}x.\]

y = f(x) at P(x₁,y₁)

\[y-y_1=\left(\frac{dy}{dx}\right)_{x=x_1,y=y_1}(x-x_1)\]

\[\delta y=\frac{dy}{dx}\operatorname{\delta}x\]

slope of tangent at P = \[\left(\frac{dy}{dx}\right)_P\]

\[\text{slope of normal at }P=-\frac{1}{\left(\frac{dy}{dx}\right)_P}\]

If m₁ and m₂ are the slopes of the tangents at the point of intersection, then

\[\tan\theta=\left|\frac{m_1-m_2}{1+m_1m_2}\right|\]

\[\lim_{\delta x\to0}\frac{\delta y}{\delta x}=\lim_{x_2\to x_1}\frac{f(x_2)-f(x_1)}{x_2-x_1}\]

Average rate of change = \[\frac{\delta y}{\delta x}=\frac{f(x_2)-f(x_1)}{x_2-x_1}\]

1. Velocity

\[v=\frac{ds}{dt}\]

2. Acceleration

\[a=\frac{dv}{dt}=\frac{d^2s}{dt^2}\]

3. Jerk

\[j=\frac{da}{dt}=\frac{d^3s}{dt^3}\]

\[f(a+h)\approx f(a)+hf^{\prime}(a)\]

y = f(x) at P(x₁,y₁)

\[y-y_1=-\frac{1}{\left(\frac{dy}{dx}\right)_{x=x_1,y=y_1}}(x-x_1)\]

or

​\[(x-x_1)+\left(\frac{dy}{dx}\right)_{x=x_1,y=y_1}(y-y_1)=0\]

• Step 1: Find critical points in (a, b)

• Step 2: Take end points a and b

• Step 3: Find f(x) at all these points

• Step 4:
  Largest value → Absolute maximum
  Smallest value → Absolute minimum

Let f be twice differentiable at c and f′(c) = 0.

Then:

•  If f′′(c) < 0
  → c is a point of local maxima

• If f′′(c) > 0
  → c is a point of local minima

• If f''(c) = 0
  → Test fails (use first derivative test)

Let f be continuous at a critical point c.

If:

• f′(x) changes from positive to negative as x passes through c
  → c is a point of local maxima

• f′(x) changes from negative to positive as x passes through c
  → c is a point of local minima

• f′(x) does not change sign
  → c is neither a maxima nor a minima (point of inflexion)

\[\frac{dy}{dx}\] > 0 → increasing

\[\frac{dy}{dx}\] < 0 → decreasing

\[\frac{dy}{dx}\] = 0 → tangent parallel to x-axis

\[\frac{dy}{dx}\] does not exist → tangent parallel to y-axis