Rate of Change of Quantities
Rate of Change: The change in one quantity with respect to time is known as rate of change of quantity.
* If a quantity y varies with another quantity x, satisfying y = ƒ(x), then \frac{dy}{dx} represents the rate of change quantity with respect to x.
* If x = (t) and y = ɡ(t), then \frac{dy}{dx} = \frac{dy}{dt} × \frac{dt}{dx}.
* The value of \frac{dy}{dx} at x = x_{0} i.e. ( \frac{dy}{dx} ) X = X0
* Change of y with respect to x at x = x_{0}.
Increasing and Decreasing Functions On An Open Interval
* Let I be an open interval contained in the domain of real valued function y = ƒ(x).
Then ƒ(x) is said to be:
- Increasing on I if x_{1} < x_{2} ⇒ ƒ ( x_{1} ) ≤ ƒ ( x_{2} ), ∀ x_{1} , x_{2} ∈ I
- Strictly increasing on I if x_{1} < x_{2} ⇒ ƒ ( x_{1} ) < ƒ ( x_{2} ), ∀ x_{1} , x_{2} ∈ I
- Decreasing on I if x_{1} < x_{2} ⇒ ƒ ( x_{1} ) ≥ ƒ ( x_{2} ), ∀ x_{1} , x_{2} ∈ I
- Strictly Decreasing on I if x_{1} < x_{2} ⇒ ƒ ( x_{1} ) > ƒ ( x_{2} ), ∀ x_{1} , x_{2} ∈ I
Increasing and Decreasing Functions at a Point
* Function ƒ is increasing in [a, b] if ƒ’(x) ≥ 0, ∀ x ∈ (a, b).
* Function ƒ is strictly increasing in (a, b) if ƒ’(x) > 0, ∀ x ∈ (a, b).
* Function ƒ is decreasing in [a, b] if ƒ’(x) ≤ 0, ∀ x ∈ (a, b).
* Function ƒ is strictly decreasing in if ƒ’(x) < 0, ∀ x ∈ (a, b).
* Function ƒ is a constant function in [a, b] if f’(x) = 0, ∀ x ∈ (a, b).
Tangent and Normal Curve
* The slope of the tangent to the curve, y = ƒ(x), at a point P(a, b) is ( \frac{dy}{dx} _{p} ) = ƒ’ (a).
* If the tangent line to the curve, y = ƒ(x), makes an angle ‘∅’ with the positive direction of x-axis, then the slope of the tangent = \frac{dy}{dx} = tan ∅.
* The equation of the tangent to the curve, y = ƒ(x), at a point P(a, b) is (y – b) = ƒ’(a) (x – a).
* If ( \frac{dy}{dx} _{p} ) = 0, then the tangent at P to the curve, y = ƒ(x), y = b.
* If ( \frac{dy}{dx} _{p} ) does not exists, then the tangent will be a vertical line, and the equation of the tangent at P to the curve, y = ƒ(x), is x = a.
* A normal is a line perpendicular to a tangent at the point of contact.
* The equation of a normal to the curve, y = ƒ(x), at point P(a, b) is (y – b) = – \frac{1}{ƒ' (a)} )
* If ( \frac{dy}{dx} _{p} ) = ƒ’ (a) does not exist, then the normal at P(a, b) is parallel to X-axis, and its equation is y = b.
* If ( \frac{dy}{dx} _{p} )= ƒ’ (a) = 0, then the normal at P(a, b) is parallel to the Y-axis, and it equation is x = a.
Approximations
If y = ƒ(x) be any function. Let ∆x be any increment in x, then ∆y is the corresponding increment in y.
∆y = ƒ(x + ∆x) – ƒ(x)
dy ≈ ∆y
dx = ∆x
∆y = \frac{dy}{dx} )∆x
ƒ(x + ∆x) = \frac{dy}{dx} ) ∆x + ƒ(x)
Maxima and Minima
* A function is said to have a maximum value at a point ‘a’ in its domain D if (a) ≥ ƒ(x), ∀ x ∈ D.
* (a) is called the maximum value of ƒ in D and point ‘a’ is called the point of a maximum value of ƒ in D.
* A function is said to have a minimum value at a point ‘a’ in its domain D if (a) ≤ ƒ(x), ∀ x ∈ D.
* (a) is called the minimum value of f in D and point ‘a’ is called the point of minimum value of ƒ in D.
* The function ƒ(x) is said to have an extreme value at a point ‘a’ in its domain D if
(a) is either a maximum value or minimum value of f in D.
* The number (a) is called an extreme value of ƒ in D, and point ‘a’ is called an extreme point.
* Every continuous function on a closed interval has a maximum and a minimum value.
* Let ƒ be a real valued function defined on the domain D. Let c ∈ D. Then:
- C is called a point of local maxima if there is an h > 0 such that ƒ(c) ≥ ƒ(x), ∀ x ∈ (c – h, c + h). The value ƒ(c) is called the local maximum value of ƒ.
- C is called a point of local minima if there is an h > 0 such that ƒ(c) ≤ ƒ(x), ∀ x ∈ (c – h, c + h). The value ƒ(c) is called the local minimum value of ƒ.
First Derivative Test
Let ƒ be a continuous function defined on an open interval I. Also, let ‘a’ be a critical point in open interval I.
Local Maxima:
If ƒ’(x) changes sign from positive to negative as ‘x’ increases through ‘a’, i.e. if f’(x) > 0 at every point sufficiently close to and to the
left of ‘a’, and ƒ’(x) < 0 at every point sufficiently close to and to the right of ‘a’, then ‘a’ is the point of local maxima.
Local minimum value = ƒ(a).
Second Derivative Test
If ƒ be a function defined on an interval I:
- If ƒ’(a) = 0 and ƒ’’(a) < 0, then a is the point of local maxima and ƒ(a) is local maximum value of ƒ.
- If ƒ’(a) = 0 and ƒ’’(a) > 0, then a is the point of local minima and ƒ(a) is local minimum value of ƒ.
- If ƒ’(a) = 0 and ƒ’’(a) = 0, then the second derivative test fails.
Maximum and Minimum Value of a Function in a Closed Interval
Absolute minimum value:
- A function ƒ has absolute minimum value at point c ∈ I if ƒ(c) ≤ ƒ(x), ∀ x ∈ I, where I is the domain of ƒ.
- The number ƒ(c) is called the absolute minimum value of ƒ on I.
Absolute maximum value:
- A function ƒ has absolute maximum value at point c ∈ I if ƒ(c) ≥ ƒ(x), ∀ x ∈ I, where I is the domain of ƒ.
- The number ƒ(c) is called the absolute maximum value of ƒ on I.