Learning these TS Inter 1st Year Maths 1B Formulas Chapter 10 Applications of Derivatives will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1B Applications of Derivatives Formulas

→ If y = f(x) is a differentiable function of x and Δx is a small change in ‘x’ then the

- actual change in y is Δy = f (x + Δx) – f(x)
- the differential of y is dy = f'(x) Δx

→ The approximate value of f(x) in a neighbourhood of Δx is f(x + Δx) – f (x) + f'(x) Δx.

→ If error in x of y = f(x) is Δx then

- Δy is the approximate error in y.
- \(\frac{\Delta \mathrm{y}}{\mathrm{y}}\) is called the relative error in v and
- \(\frac{\Delta \mathrm{y}}{\mathrm{y}}\) × 100 is the percentage error in y.

→ The slope of the curve y = f(x) at the point P(x_{1}, y_{1}) is \(\left(\frac{d y}{d x}\right)_{\left(x_1 \cdot y_1\right)}\) = m = f'(x_{1}).

→ If θ is the angle between the curves at y = f(x) and y = g(x) at the point of intersection P(x_{1}, y_{1}) then tan θ = \(\frac{m_1-m_2}{1+m_1 m_2}\) where m_{1} = f'(x_{1}) and m_{2} = g'(x)

If m_{1} = m_{2}, then the two curves touch each other at (x_{1}, y_{1}) and if m_{1}m_{2} = – 1, the two curves are said to be orthogonal.

→ If m = \(\left(\frac{d y}{d x}\right)_{\left.i x_1, y_1\right)}\) = f'(x,) is the slope of the curve at the point P(x_{1}, y_{1}) on y = f(x) then

- The length of the tangent to the curve at P is \(\frac{y_1 \sqrt{1+\left[f^{\prime}\left(x_1\right)\right]^2}}{f^{\prime}\left(x_1\right)}\)
- The length of the normal to the curve at P is y1\(\sqrt{1+\left[f\left(x_1\right)\right]^2}\)
- The length of the subtangent to the curve at P = \(\left|\frac{y_1}{f^{\prime}\left(x_1\right)}\right|\)
- The length of the subnormal to the curve at P is |y
_{1}f(x_{1})|.

→ The rate of change of the function y = f(x) with respect to ‘t’ is \(\frac{d y}{d x}\)

→ If s = f(t) is the functional relation between the distance ‘s’ and time ‘t’, then the velocity of the body at time ‘t’ is \(\frac{d s}{d x}\) = v and the acceleration of the body at time ‘t’ is \(\frac{d^2 s}{d t^2}=\frac{d v}{d t}\)

→ If a function ‘f’ is increasing and differentiable at a’ ⇔ f'(a) > 0.

- A differentiable function is said to be decreasing at ‘a’ ⇔ f'(a) < 0.
- A differentiable function is said to be stationary at ‘a’ ⇔ f'(a) = 0.

→ A differentiable function f(x) in the interval which has f'(x) and f”(x) at ‘a’ and if

- f’(a) = 0, f”(a) < 0, then f(a) has local maxima.
- f'(a) = 0. f”(a) > 0. then f(a) has local minima.

→ Rolle’s Mean Value Theorem : If a function ‘f defined over [a, b] is such that

- f is continuous over [a, b]
- f is differentiable on (a. b)
- f(a) = f(b). Then ∃ a point c ∈ (a, b) such that f'(c) = 0.

→ Lagrange’s Mean Value Theorem : If a function f is defined over [a, b] is such that

- f is continuous over [a, b] .
- f is differentiable over (a. b) then ∃ a point c ∈ (a, b) such that f’(c) = \(\frac{f(b)-f(a)}{b-a}\)

→ Mensuration fundamentals:

1. If r is the radius, x is the diameter, P is the perimeter and A is the area of the circle then

- A = πr or A = \(\frac{\pi x^2}{4}\).
- P = 2πr = πx.

2. If ‘r’ is the radius, l is the length of the arc and 0 is the angle then

- Area A = \(\frac{1}{2}\) lr = \(\frac{1}{2}\) r
^{2}θ - Perimeter P = l + 2r = r (θ + 2)

3. If r is the radius, h is the height of the cylinder then

- Lateral surface area = 2πrh
- Total surface area S = 2πrh + 2πr
^{2} - Volume V = πr
^{2}h

4. If r is the radius, l is the slant height, h is the height and α is the vertical angle of the cone, then

- l
^{2}= r^{2}+ h^{2} - Lateral surface area = πrl
- Total surface area S = πrl + πr
^{2} - Volume V = \(\frac{1}{3}\)πr
^{2}H

5. If L is the length, T is the period of oscillation of a simple pendulum and g is the acceleration due to gravity then T = 2π\(\sqrt{\frac{l}{g}}\).

6. If r is the radius of sphere then

- Surface area = S = 4 πr
^{2} - Volume V = \(\frac{4}{3}\)πr
^{3}

7. Let x be the side of a cube then surface area of the cube is 6x^{2} and volume of the cube is x^{3}.