Unit 2B Practice Exam A

A particle has $s(t)=t^3-6t^2+5t$. Find rest times, direction intervals, acceleration, and speeding-up intervals on $[0,6]$.

Use linearization at $a=25$ to estimate $\sqrt{25.4}$ and discuss whether the estimate is high or low.

A measured sphere radius is $12.0\pm0.05$ cm. Estimate maximum volume error and percentage error.

Use two Newton iterations to approximate the positive root of $x^3+x-4=0$, starting from a sensible integer.

A $10$-foot ladder slides away from a wall at $1$ ft/s. Find the rate of the top when the bottom is $6$ feet from the wall.

A radar station is $3$ km from the point below a rising balloon. The balloon rises at $0.2$ km/min. Find the angle rate when the balloon is $4$ km high.

Verify the Mean Value Theorem for $f(x)=1/x$ on $[1,4]$ and find $c$.

Find intervals of increase, decrease, concavity, and all extrema and inflection points for $f(x)=x^3-3x^2$.

A rectangle is inscribed under $y=16-x^2$, symmetric about the $y$-axis. Find maximum area.

A manufacturer's average cost is $A(q)=1500/q+12+0.01q$. Find the output minimizing average cost and interpret $A'(100)$.

Evaluate $\lim_{x\to0}(e^{2x}-1-2x)/x^2$.

Evaluate $\lim_{x\to\infty}(\sqrt{x^2+8x}-x)$.

A concentration model is $C(t)=30te^{-0.4t}$. Find peak time, peak concentration, and the interval of increase.

Explain using a theorem why a differentiable function with $|f'|\le3$ on $[1,5]$ cannot satisfy $|f(5)-f(1)|>12$.