What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

How is the derivative interpreted as a rate of change?

\(\dfrac{dy}{dx}\) is the instantaneous rate at which \(y\) changes per unit change in \(x\); related rates link several such derivatives.

When is a function increasing on an interval?

A differentiable function is increasing where \(f'(x)>0\) and strictly increasing if \(f'(x)>0\) throughout the interval.

How is the slope of the tangent found?

The slope of the tangent to \(y=f(x)\) at a point is \(f'(x)\) evaluated there; the normal slope is its negative reciprocal.

State the first-derivative test for a local maximum.

If \(f'\) changes from positive to negative as \(x\) increases through \(c\), then \(f\) has a local maximum at \(c\).

State the second-derivative test.

If \(f'(c)=0\) and \(f''(c) 0\), \(c\) is a local minimum.

What is a critical point?

A point in the domain where \(f'(x)=0\) or \(f'(x)\) fails to exist; extrema can occur only at such points (or endpoints).

How do you find the equation of a tangent?

At \((x_0,y_0)\), the tangent is \(y-y_0=f'(x_0)(x-x_0)\).

What is the second-derivative test inconclusive case?

When \(f'(c)=0\) and \(f''(c)=0\); then the test fails and one falls back on the first-derivative test or higher derivatives.

Outline how an optimisation problem is solved.

Express the quantity as a function of one variable, differentiate, set the derivative to zero to find critical points, and use a derivative test to identify the maximum or minimum.

Give the normal line equation at a point.

At \((x_0,y_0)\) with \(f'(x_0)\neq0\), the normal is \(y-y_0=-\dfrac{1}{f'(x_0)}(x-x_0)\).

Application of Derivatives

Class 12 Mathematics · Mathematics Part I · Chapter 6

163 members in the CBSE community

View all

CBSE Class 12 · Mathematics

Every Part I question from 2025 + the SQP

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
35 marks: Calculus, shared with Part II
10 marks: Algebra, Ch 3–4
Common traps that cost marks, listed
6 principal branches of inverse trig
Every derivative rule with its condition

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

CBSE Class 12 · Mathematics

Integrals: 35 marks across the four papers

Marks booster + cheatsheet: key concepts in 2 lines each
23 marks in the four papers: Vector Algebra
Integral tables with properties P0–P7
3 methods for differential equations
8 marks: Probability, with Bayes' theorem
Patterns the board repeats, by chapter

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

View all 46 amazing Marks Booster kits →

Summary

Derivatives measure rates of change and the local behaviour of functions, and this chapter applies them widely. The derivative as a rate of change links related quantities (for example, how a sphere's volume changes with its radius). A function is increasing where \(f'(x)>0\) and decreasing where \(f'(x)<0\). The derivative also gives the slope of a tangent, so equations of tangents and normals follow at once. Maxima and minima are located using the first-derivative test and the second-derivative test: at a local maximum \(f'(x)=0\) and \(f''(x)<0\), while at a local minimum \(f'(x)=0\) and \(f''(x)>0\). These ideas drive optimisation problems — maximising area, minimising cost or material — that appear throughout applied mathematics.

Rate of change of quantitiesIncreasing and decreasing functionsTangents and normalsMaxima and minimaOptimisation problems

Key terms

Rate of change

\(\dfrac{dy}{dx}\) gives the instantaneous rate of change of \(y\) with respect to \(x\).

Increasing function

A function with \(f'(x)>0\) on an interval.

Decreasing function

A function with \(f'(x)<0\) on an interval.

Tangent slope

The slope of the tangent at a point equals \(f'(x)\) there.

Critical point

A point where \(f'(x)=0\) or \(f'(x)\) does not exist.

Second-derivative test

At a critical point, \(f''<0\) gives a maximum and \(f''>0\) a minimum.

Extra questions & answers

Try one from this chapter

A differentiable function is increasing on an interval where:

Flashcards & interactive practice