Overview of Applications of Derivatives

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

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

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

y = f(x) at P(x₁,y₁)

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

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

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

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

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

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

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

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.

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

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 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.

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

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.

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.

\[\frac{dy}{dx}\] > 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

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)

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)

• 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

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

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

1. Velocity

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

2. Acceleration

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

3. Jerk

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

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