Tiêu đề 6. APPLICATIONS OF DERIVATIVES.pdf ✅

TOPIC-1 Rate of Change of Bodies Concepts Covered:  Use of differentiation as a measure of change in rate. Revision Notes Interpretation of dy dx as a rate measure: z If two variables x and y are varying with respect to another variables say t, i.e., if x = f(t), then by the Chain Rule, we have dy dx dy dt dx dt dx dt = ≠ / / , 0 [Board 2023, 20] z Thus, the rate of change of y with respect to x can be calculated by using the rate of change of y and that of x both with respect to t. z Also, if y is a function of x and they are related as y = f(x) then, f'(a), i.e., represents the rate of change of y with respect to x at the instant when x = a. Example-1 Find the rate of change of the area of a circle per second with respect to its radius r when r = 5 cm. [NCERT] Solution: Given, Area of a circle, A = πr2 . Therefore, the rate of change of the area A w.r.t its radius r is given by dA dr = d dr (πr2 ) = 2πr When r = 5 cm, dA dr = 10π Thus, the area of the circle is changing at the rate of 10π cm2 /s. SUBJECTIVE TYPE QUESTIONS Very Short Answer Type Questions (1 mark each) 1. The radius of a circle is increasing at the uniform rate of 3 cm/s. At the instant the radius of the circle is 2 cm, then at what rate area increases. U [Delhi Set-1, 2020] Topper's Answer, 2020 Sol. 1 2. The total expenditure (in `) required for providing the cheap edition of a book for poor and deserving students is given R(x) = 3x2 + 36x, where x is the number of set of books. If the marginal expenditure is defined as d dx R , write the marginal expenditure required for 1200 such sets. U [OEB] 3. The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in `) received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5. U [OEB] Concept Applied Rate of change of total revenue R(x) w.r.t. x is Marginal revenue. APPLICATIONS OF DERIVATIVES LEARNING OBJECTIVES After going through this Chapter, the student would be able to learn:  The uses of derivatives of the functions.  Optimization Problems: Learn how to use derivatives to find the maximum or minimum values of a function.  Graphical Analysis: Explore the relationship between the graph of a function and its derivatives.  Applied Problems: Solve problems related to motion, velocity, acceleration and other physical phenomena using derivatives. 6 CHAPTER LIST OF TOPICS Topic-1: Rate of Change of Bodies Topic-2: Increasing/Decreasing Functions Topic-3: Maxima and Minima This Question is for practice and its solution is given at the end of the chapter.
Applications of Derivatives Sol. Total revenue is given by R(x) = 3x2 + 36x + 5 Marginal revenue dR dx = 6x + 36 At x = 5, dR dx x       =5 = 6 × 5 + 36 = ` 66 1 Short Answer Type Questions-I (2 marks each) 1. A particle moves along the curve 3y = ax3 + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a. [Outside Delhi Set-1, 2023] Sol. 3y = ax3 + 1 Given: dy dt = 2 dx dt       at x = 1 3y = ax3 + 1 3dy dt = 3 0 2 x a dx dt + dy dt = ax dx dt 2 2 dx dt       = a dx dt ( ) 1 2 \ a = 2 2. If the circumference of circle is increasing at the constant rate, prove that rate of change of area is directly proportional to its radius. Ap [OD Set-2, 2023] Sol. Let, the radius of a circle be r. We have, C = 2πr and dC dt = k ...(i) Now, A = πr2 ⇒ dA dt = 2πr dr dt ...(ii) and dC dt = 2π dr dt ⇒ k = 2π dr dt ...from (i) ⇒ dr dt = k 2p ...(iii) 1 Put the value of dr dt from equation (iii) in (ii) ⇒ dA dt = 2πr × k 2p = kr ⇒ dA dr ∝ r Hence Proved 1 Commonly Made Error Instead of taking constant rate of increment in circumference, few students take the circumference constant. Read the question properly to understand what is asked? Answering Tip 3. A man 1.6 m tall walks at the rate of 0.3 m/s away from a street light is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening? U&A [OEB] Sol. Let AB represent the height of the street light from the ground. At any time t seconds, let the man represented as ED of height 1.6 m be at a distance of x m from AB and the length of his shadow EC by y m. Using similarity of triangles, we have ⇒ AB ED = AC EC 4 1.6 = x y y + Þ 3y = 2x 1⁄2 Differentiating both sides w.r.t. t, we get 3 2 dy dx dt dt = dy dt = 2 3 × ⇒ 0 3. . = 0 2 dy dt 1⁄2 At any time t seconds, the tip of his shadow is at a distance of (x + y) m from AB. The rate at which the tip of his shadow moving = dx dy dt dt   +     m/s = 0.5 m/s 1⁄2 The rate at which his shadow is lengthening = dy dt m/s = 0.2 m/s 1⁄2 4. If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/s, how fast is the area of triangle increasing at an instant when all sides become equal ? A [Delhi Set-3, 2023] Concept Applied Area of triangle = 1 2 (base × height)

Applications of Derivatives Then, A = 4pr2 \ dA dt = 8 dr r dt π (Differentiating both sides w.r.t. 't') Given, dA dt = 2 cm2 /s \ 2 = 8 dr r dt π Þ dr dt = 2 8 1 π π r r 4 = ...(i) 1 Now, volume of spherical bottom V = 4 3 3 πr \ dV dt = 4 2 (3 ) 3 dr r dt π (Differentiating both sides w.r.t. 't') or, dV dt = 4 1 4 2 π π r r × [from eq (i)] or, dV dt = r or, dV dt = 8 cm3 /s [Given r = 8 cm] 2 Therefore, the volume of the balloon is increasing at the rate of 8 cm3 /s. Long Answer Type Questions (5 marks each) 1. The median of an equilateral triangle is increasing at the rate of 2 3 cm/s. Find the rate at which its side is increasing. U&Ap [Delhi & Outside Delhi Set-1, 2023] Sol. Let the length of the median be x cm. 1 and side of equilateral triangle be y cm In DABD AB2 = AD2 + BD2 (∠D = 90°) y2 = x2 + y x 2 2  2      + 3 4 2 y = x2 y2 = 4 3 2 x y = 2 3 x 2 d dt ( ) y = 2 3 d dt ( ) x dy dt = 2 3 × dx dt dy dt = 2 3 × 2 3 = 4 cm/s 2 Hence, side of equilateral triangle increases at 4cm/s. Commonly Made Error Students sometimes ignore the basic properties of equilateral triangle. Draw the necessary diagram to make the calculation easier. Answering Tip 2. Water is dripping out from a conical funnel of semi- vertical angle 4 π at the uniform rate of 2 cm2 /s in the surface, through a tiny hole at the vertex of bottom. When the slant height of water level is 4 cm, find the rate of decrease of the slant height of the water. U&Ap [OEB] TOPIC-2 Increasing/Decreasing Functions Concepts Covered:  Increasing function  Decreasing function  Constant function  Monotonic function Revision Notes 1. A function f(x) is said to be an increasing function in [a, b], if as x increases, f(x) also increases i.e., if a, b Î [a, b] and a > b, f(a) > f(b). If f’(x) 3 0 lies in (a, b), then f(x) is an increasing function in [a, b], provided f(x) is continuous at x = a and x = b. 2. A function f(x) is said to be a decreasing function in [a, b], if, as x increases, f(x) decreases i.e., if a, b Î [a, b] and a > b Þ f(a) < f(b). If f’(x) £ 0 lies in (a, b), then f(x) is a decreasing function in [a, b] provided f(x) is continuous at x = a and x = b. [Board 2017] z A function f(x) is a constant function in [a, b] if f’(x) = 0 for each x Î (a, b). z By monotonic function f(x) in interval I, we mean that f is either only increasing in I or only decreasing in I. 3. Finding the intervals of increasing and/or decreasing of a function: ALGORITHM STEP 1: Consider the function y = f(x). STEP 2: Find f’(x). STEP 3: Put f’(x) = 0 and solve to get the critical point(s). STEP 4: The value(s) of x for which f’(x) > 0, f(x) is increasing; and the value(s) of x for which f’(x) < 0, f(x) is decreasing. [Board 2023, 20 18, 17] This Question is for practice and its solution is given at the end of the chapter.