Calculus for JEE (Advanced)

Calculus for JEE: core idea

• 1. $\int_0^1 x\ln x\,dx$?

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• 2. Curvature of $y=x^2$ at $0$?

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• 3. Polar area $r=1$, $0\le\theta\le\pi$: area?

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• 4. Rolle: $f(0)=f(1)=0$ for $f(x)=x(x-1)$: $c$ in $(0,1)$ with $f'=0$?

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• 5. $\iint_{[0,1]^2}1\,dA$?

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• 6. DE $y'=y$, $y(0)=1$: $y(1)$?

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• 7. Jensen for $x^2$ on $[0,1]$: $\tfrac{f(0)+f(1)}{2}-f(\tfrac12)$?

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• 8. Find the maximum value of $\dfrac{\ln x}{x}$ for $x>0$.

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• 9. Beta-ish: $\int_0^1 x(1-x)\,dx$?

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