Application of Derivatives

Application of Derivatives uses the tools of differentiation to analyze the behavior of functions and solve problems related to rates of change and optimization. The core idea is to interpret the derivative, dy/dx, not just as a formula, but as the instantaneous rate of change of y with respect to x, or as the slope of the tangent to the curve y = f(x). The chapter covers several key applications: finding the rate of change of quantities, determining where functions are increasing or decreasing, and finding the equations of tangents and normals to curves.

A significant part of the chapter is dedicated to finding maxima and minima. This involves using the first and second derivative tests to locate the highest and lowest points of a function, both locally and over a given interval. These concepts are then applied to solve practical optimization problems. The chapter also introduces the Mean Value Theorems (Rolle's and Lagrange's), which provide important theoretical results about differentiable functions. The main difficulty for students often lies in setting up the word problems for maxima and minima correctly. A solid understanding of the chapter on Differentiation is essential before starting this one.

Formulas for Application of Derivatives

Rate of change of y w.r.t. time: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$ Used when y is a function of x, and x is a function of time t.
Equation of the tangent at a point (x₁, y₁): $y - y_1 = m(x - x_1)$ m is the slope, found by calculating the derivative f'(x) at the point (x₁, y₁).
Equation of the normal at a point (x₁, y₁): $y - y_1 = -\frac{1}{m}(x - x_1)$ The normal is perpendicular to the tangent with slope m, provided m is not zero.
Condition for an increasing function: $f'(x) > 0$ The function f(x) is strictly increasing in an interval if this holds for all x in it.
Condition for a decreasing function: $f'(x) < 0$ The function f(x) is strictly decreasing in an interval if this holds for all x in it.
Condition for a critical point: $f'(c) = 0 \text{ or } f'(c) \text{ is undefined}$ c is a critical point if it is in the domain of f. Extrema occur at critical points.
Second derivative test for local maximum: $f'(c) = 0 \text{ and } f''(c) < 0$ This indicates a local maximum at x = c.
Second derivative test for local minimum: $f'(c) = 0 \text{ and } f''(c) > 0$ This indicates a local minimum at x = c.
Conclusion of Rolle's Theorem: $f'(c) = 0$ For some c in (a, b), if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b).
Conclusion of Lagrange's Mean Value Theorem: $f'(c) = \frac{f(b) - f(a)}{b - a}$ For some c in (a, b), if f is continuous on [a, b] and differentiable on (a, b).

One of these questions, worked through

JEE Advanced 2012

let $f: IR \to IR$ be defined as $f(x) = |x| + |x^{2} - 1|$ . The total number of points at which $f$ attains either a local maximum or a local minimum is

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let f: IR → IR be defined as f(x) = |x| + |x² - 1|. The total number of points at which f attains either a local maximum or a local minimum is

$f(x)=|x|+|x^2-1|$

$\text{To find: total number of local maxima/minima of } f$

$|x|=0 \Rightarrow x=0;\quad x^2-1=0 \Rightarrow x=\pm 1$

$\text{breakpoints: } -1,\ 0,\ 1$

f(x)= x²-x-1, x<-1 -x²-x+1, -1≤ x<0 -x²+x+1, 0≤ x<1 x²+x-1, x≥ 1

$\text{Plan: } f' \text{ on each piece; check sign change at } -1,0,1$

$x<-1:\ f'=2x-1;\qquad -1<x<0:\ f'=-2x-1$

$0<x<1:\ f'=-2x+1;\qquad x>1:\ f'=2x+1$

$x<-1:\ f'=2x-1<0;\qquad x>1:\ f'=2x+1>0$

$-1<x<0:\ f'=-2x-1=0 \Rightarrow x=-\dfrac{1}{2},\quad f''<0 \Rightarrow \text{local max}$

$0<x<1:\ f'=-2x+1=0 \Rightarrow x=\dfrac{1}{2},\quad f''<0 \Rightarrow \text{local max}$

$x=-1:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$x=0:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$x=1:\ f'\ \text{changes } -\to+ \Rightarrow \text{local min}$

$\text{maxima}=2,\qquad \text{minima}=3 \;\Rightarrow\; \text{total}=5$

$\text{Total number of local extrema} = 5$

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Past-paper questions on Application of Derivatives

JEE Advanced 33 questions