1.Find $\frac{dy}{dx}$ if $x^2 + y^2 = 25$ using implicit differentiation.
Sol: Differentiating wrt $x$: $2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$.
2.Evaluate the integral $\int \frac{x}{x^2 + 1} dx$.
Sol: Let $u = x^2 + 1 \implies du = 2x dx$. Then $\int \frac{1}{2u} du = \frac{1}{2} \ln|x^2 + 1| + C$.
3.A right circular cylinder has fixed volume $V$. Find the ratio of height to radius $h/r$ for minimum total surface area.
Sol: $A = 2\pi r^2 + \frac{2V}{r} \implies \frac{dA}{dr} = 4\pi r - \frac{2V}{r^2} = 0 \implies V = 2\pi r^3 \implies \pi r^2 h = 2\pi r^3 \implies h = 2r \implies \frac{h}{r} = 2$.
4.Evaluate $\lim_{x \to 0} \frac{1 - \cos(x)}{x^2}$.
Sol: Using L'Hopital's Rule or identity: $\frac{1-\cos x}{x^2} = \frac{2\sin^2(x/2)}{x^2} = \frac{1}{2} \left(\frac{\sin(x/2)}{x/2}\right)^2 \to \frac{1}{2}$.
5.If $y = e^{ax} \sin(bx)$, prove that $\frac{d^2y}{dx^2} - 2a \frac{dy}{dx} + (a^2 + b^2)y = 0$.
Sol: $y' = a e^{ax}\sin(bx) + b e^{ax}\cos(bx) = ay + b e^{ax}\cos(bx)$. $y'' = a y' + a b e^{ax}\cos(bx) - b^2 e^{ax}\sin(bx) = a y' + a(y' - ay) - b^2 y = 2a y' - (a^2 + b^2)y$. Hence proved!
