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

State the condition for continuity of \(f\) at \(x=c\).

\(f\) is continuous at \(c\) if \(\lim_{x\to c}f(x)\) exists and equals \(f(c)\).

Does differentiability imply continuity?

Yes; every differentiable function is continuous, but a continuous function need not be differentiable (e.g. \(|x|\) at \(0\)).

State the chain rule.

If \(y=f(u)\) and \(u=g(x)\), then \(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\).

Differentiate \(y=\sin^{-1}x\).

\(\dfrac{dy}{dx}=\dfrac{1}{\sqrt{1-x^2}}\) for \(-1<x<1\).

Differentiate \(y=e^x\) and \(y=\log x\).

\(\dfrac{d}{dx}e^x=e^x\) and \(\dfrac{d}{dx}\log x=\dfrac{1}{x}\) for \(x>0\).

When is logarithmic differentiation used?

For functions of the form \(y=u(x)^{v(x)}\), or complicated products/quotients; take \(\log\) of both sides first.

State Rolle's Theorem.

If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a)=f(b)\), then there exists \(c\in(a,b)\) with \(f'(c)=0\).

State Lagrange's Mean Value Theorem.

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there exists \(c\in(a,b)\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\).

Find \(\dfrac{dy}{dx}\) for parametric \(x=f(t),\,y=g(t)\).

\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\), provided \(\dfrac{dx}{dt}\neq0\).

Why is \(f(x)=|x|\) not differentiable at \(0\)?

Its left-hand derivative is \(-1\) and right-hand derivative is \(+1\); the two do not match, so \(f'(0)\) does not exist.

Continuity and Differentiability

Q: Differentiate \(y=e^x\) and \(y=\log x\).

\(\dfrac{d}{dx}e^x=e^x\) and \(\dfrac{d}{dx}\log x=\dfrac{1}{x}\) for \(x>0\).

Class 12 Mathematics · Mathematics Part I · Chapter 5

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CBSE Class 12 · Mathematics

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Summary

This chapter formalises continuity and differentiability. A function \(f\) is continuous at \(x=c\) if \(\lim_{x\to c}f(x)=f(c)\); it is continuous on an interval if continuous at every point. Differentiability is stronger: \(f'(c)\) exists if the limit defining the derivative exists, and differentiability implies continuity (but not conversely). The chapter develops rules for differentiating composite functions (chain rule), implicit functions, inverse trigonometric functions, exponential and logarithmic functions, and functions given in parametric form. Logarithmic differentiation handles products, quotients and variable exponents. Second-order derivatives \(\dfrac{d^2y}{dx^2}\) are introduced, and the chapter closes with the Mean Value Theorems of Rolle and Lagrange, which connect the derivative to the behaviour of the function on an interval.

Continuity of functionsDifferentiability and the chain ruleDerivatives of inverse trig, exponential and log functionsLogarithmic and parametric differentiationMean Value Theorems

Key terms

Continuity at a point

\(\lim_{x\to c}f(x)=f(c)\).

Differentiability

Existence of \(f'(c)=\lim_{h\to0}\dfrac{f(c+h)-f(c)}{h}\).

Chain rule

\(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\) for composite functions.

Implicit differentiation

Differentiating an equation in \(x\) and \(y\) without solving for \(y\).

Logarithmic differentiation

Taking \(\log\) before differentiating, useful for \(y=u^v\).

Mean Value Theorem

There is \(c\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\) (Lagrange).

Extra questions & answers

Try one from this chapter

\(f\) is continuous at \(x=c\) if:

Flashcards & interactive practice