Unit 2B Cumulative Applications Practice

Interpret $R'(200)=45$ and $R"(200)=-0.3$ if $R$ is revenue in dollars and $q$ is units.

For $s(t)=t^3-6t^2+9t$ on $[0,5]$, find velocity, acceleration, rest times, direction intervals, and speeding-up intervals.

Sketch a position graph whose velocity is positive and decreasing.

Estimate first and second derivatives at a central table entry using equally spaced data.

A Ferris-wheel height is $h(t)=22-18\cos(\pi t/20)$. Find vertical velocity at $t=10$ and interpret it.

Linearize $\sqrt{x}$ at $x=25$ and estimate $\sqrt{25.4}$.

Linearize $\ln x$ at $x=1$ and estimate $\ln1.03$.

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

Use Newton's method from $x_0=2$ to approximate $\sqrt5$ with two iterations.

Explain one reason Newton's method can fail even when a real root exists.

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

Water enters a cone of height $15$ m and radius $5$ m at $3$ m$^3$/min. Find the water-depth rate at depth $6$ m.

A runner passes the closest point to a camera $40$ m away at $8$ m/s. Find the angular rate at the closest point.

Two vehicles are $6$ km east and $8$ km north of an intersection. One moves east at $50$ km/h and the other south at $40$ km/h. Find the separation rate.

Design a related-rate problem involving an expanding sphere and solve it with units.

Find absolute extrema of $x^3-3x$ on $[-2,2]$.

Verify Rolle's Theorem for $\sin x$ on $[0,\pi]$ and find the guaranteed point.

Apply the Mean Value Theorem to $\sqrt{x}$ on $[1,4]$.

Analyze increase, decrease, local extrema, concavity, and inflection points for $x^3-3x^2$.

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

Prove from the Mean Value Theorem that $|f'(x)|\le M$ implies $|f(b)-f(a)|\le M|b-a|$.

Maximize the area of a rectangle with perimeter $60$.

A farmer has $600$ m of fence for three sides of a rectangle along a river. Find maximum area dimensions.

Minimize surface area of a closed cylinder with fixed volume $500\pi$.

Find the closest point on $y=\sqrt{x}$ to $(3,0)$.

For demand $p(q)=80-0.05q$ and cost $C(q)=2000+20q$, maximize feasible profit.

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

Evaluate $\lim_{x\to\infty}x^3/e^x$.

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

Give an example where L'Hopital's Rule is unnecessary because direct substitution already works.