Revision: Applications of Derivatives Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

The derivative \[\frac{dy}{dx}\] represents the rate of change of a variable (y) with respect to another variable (x).

In general, if a quantity depends on another, then its derivative gives the instantaneous rate of change of that quantity.

A function f(x) is said to be an increasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)

Strictly Increasing Function:

• If x₁ < x₂ ⇒ f(x₁) < f(x₂)

A function f is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

A function f is said to be constant on I if f(x) = c for every x ∈ I, where c is a constant.

Let x₀ be a point in the domain of a real-valued function f.

The function f is said to be increasing at x₀ if there exists an open interval containing x₀ in which f is increasing.

Similarly, f is said to be decreasing at x₀ if there exists an open interval containing x₀ in which f is decreasing.

A function f(x) is said to be a decreasing function on (a, b) if x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)

Strictly Decreasing Function:

• If x₁ < x₂ ⇒ f(x₁) > f(x₂)

Let f be a function defined on an interval I.

Maximum value: f has a maximum value at c ∈ I if \[ \boxed{f(c) \geq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the maximum value and c is called a point of maximum.

Minimum value: f has a minimum value at c ∈ I if \[ \boxed{f(c) \leq f(x) \quad \text{for all } x \in I} \]

The value f(c) is called the minimum value and c is called a point of minimum.

Extreme value: A maximum or minimum value of f is called an extreme value.

 
           Maximum value                          Minimum value

A point in the domain of a function is called a critical point if either the derivative is zero there or the derivative does not exist there. Critical points are checked while locating possible maxima or minima.

The points where a function changes from decreasing to increasing or from increasing to decreasing are called turning points.

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.

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.

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​)

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.

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

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

y = f(x) at P(x₁,y₁)

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

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

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

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

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

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

Statement:

If a function f(x):

1. Is continuous on the closed interval [a,b]

2. Is differentiable on the open interval (a,b)

3. Satisfies f(a) = f(b)

Then there exists at least one $c\in (a,b)$ such that:

\[f^{\prime}(c)=0\]

Statement: 

If a function f(x):

1. Is continuous on the closed interval [a,b]

2. Is differentiable on the open interval (a,b)

Then there exists at least one number c ∈ (a,b) such that:

\[f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}\]

Assume f'(c) = 0 and the second derivative exists at c:

• Local Maximum: f''(c) < 0

• Local Minimum: f''(c) > 0

• Test Fails: f''(c) = 0. If this happens, you must go back and use the First Derivative Test to check if it is a maxima, minima, or point of inflection.

Let c be a critical point of a continuous function f:

• Local Maximum: If f'(x) changes sign from positive to negative as x passes through c, then cc is a point of local maximum.

• Local Minimum: If f'(x) changes sign from negative to positive then c is a point of local minimum.

• Point of Inflection: f'(x) does not change sign as x passes through c (it is neither a maxima nor a minima).

• Increasing means output does not decrease as input increases.
• Strictly increasing means output always increases.
• Decreasing means output does not increase as input increases.
• Monotonic means either increasing or decreasing on an interval.
• f′(x) > 0 implies increasing, f′(x) < 0 implies decreasing, and f′(x) = 0 on an interval implies constant behaviour.
• If \[ f'(x) = 0 \] throughout an interval, the function is constant on that interval.
• A single point where \[ f'(x) = 0 \] does not necessarily make the function constant.
• To find intervals of increase or decrease, find the zeros of f'(x), divide the domain into intervals, and check the sign of f'(x).
• A function that is increasing or decreasing on an interval is called monotonic on that interval.
• A function may be increasing on one interval and decreasing on another; in that case it is not monotonic on its entire domain.

• Maxima and minima are extreme values of a function.

• Critical points occur where \(f'(x)=0\) or \(f'(x)\) is not defined.

• If \(f'(x)\) changes from positive to negative, the function has a local maximum.

• If \(f'(x)\) changes from negative to positive, the function has a local minimum.

• If \(f''(c) < 0\), there is a local maximum at \(x=c\).

• If \(f''(c) > 0\), there is a local minimum at \(x=c\).

• For absolute extrema on \([a,b]\), compare values at critical points and endpoints.

• Not every critical point gives a maximum or minimum.

• The second derivative test is quick, but the first derivative test is often more reliable in detailed reasoning.

• 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