Unit 2B Practice Exam B

A particle has velocity $v(t)=t^3-6t^2+8t$. Determine direction and speeding behavior on $[0,6]$, then describe a compatible position graph.

A damped oscillator has $s(t)=8e^{-t/5}\cos(2t)$. Find velocity and interpret the two contributions to it.

Use differentials to estimate the percentage volume error of a sphere when radius has $0.4\%$ error.

Water enters a cone of height $15$ and radius $5$ at $3$ m$^3$/min. Derive $dh/dt$ as a function of $h$.

A response curve is $R(d)=100d/(d+12)$. Analyze monotonicity, concavity, limiting response, and the practical meaning of diminishing returns.

Perform a complete curve analysis of $f(x)=x/(x^2-1)$.

Minimize material for a closed cylinder with fixed volume $500\pi$, and interpret the relationship between optimal height and radius.

A cable must connect a point offshore to a point on land, with underwater installation costing four times as much per meter. Build and optimize a one-variable cost model.

For $Q=12K^{0.4}L^{0.6}$, interpret labor elasticity and explain diminishing marginal output with capital fixed.

Evaluate $\lim_{x\to0^+}x^{\sin x}$.

Evaluate $\lim_{x\to\infty}x^4/e^x$, and explain the growth-rate conclusion.

Show that ordinary Newton iteration for $f(x)=x^3-2x+2$ cycles from $x_0=0$, then explain what this teaches about algorithms.

Prove that any critical point of a differentiable convex function is a global minimum.

Write a brief critique of one modeling assumption in the stopping-distance or medication model from this unit.