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

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(x₁, y₁) is \(\left(\frac{d y}{d x}\right)_{\left(x_1 \cdot y_1\right)}\) = m = f'(x₁).

→ If θ is the angle between the curves at y = f(x) and y = g(x) at the point of intersection P(x₁, y₁) then tan θ = \(\frac{m_1-m_2}{1+m_1 m_2}\) where m₁ = f'(x₁) and m₂ = g'(x)
If m₁ = m₂, then the two curves touch each other at (x₁, y₁) and if m₁m₂ = – 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(x₁, y₁) 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 |y₁f(x₁)|.

→ 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 = \(\frac{1}{2}\) r²θ
• 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 + 2πr²
• Volume V = πr²h

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

• l² = r² + h²
• Lateral surface area = πrl
• Total surface area S = πrl + πr²
• Volume V = \(\frac{1}{3}\)πr²H

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 πr²
• Volume V = \(\frac{4}{3}\)πr³

7. Let x be the side of a cube then surface area of the cube is 6x² and volume of the cube is x³.