A balloon is being filled with air so that the radius is increasing at the rate of \(1\) inch every \(2\) seconds. How fast is the volume changing when the radius is \(2.5\) inches?

A ladder \(10\) feet long is leaning against a wall. If the bottom of the ladder slides away from the wall at \(0.5\) feet/second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is \(6\) feet from the wall?

Discuss the first derivative and its relationship to the graph of \(g(x)=4\).

Assuming the model \(C(t)=\dfrac{7}{5t^2+20}\) to be true, should the manufacturer’s claims be accepted or rejected?

Find, if they exist, all relative and absolute extrema of the function \[f(x)=\frac{1}{x-4}\]

Describe the behavior of the function \[f(t)=-t^3+9t^2 \qquad 0\leq t\leq 8\] in the context of this problem.

Describe the behavior of the function \[f(x)=x^\frac{3}{5}-1\]

Summarize the behavior of \(f(x)=x^3-3x^2+4\) and sketch its graph.

Summarize the behavior of \(f(x)=\dfrac{5x+2}{3x-4}\) and sketch its graph.

Use the second derivative test to find all the relative extrema of the function \[f(x)=x^3+\frac{9}{2}x^2-12x+11\]

The revenue realized by a company on the sale of \(x\) units of its product is given by the revenue function \[R(x)=x^3+4x^2+160x\] Compute and interpret both \(R(70)\) and \(R'(70)\).

The cost to a company to produce \(x\) units of a product is given by the cost function \[C(x)=-0.035x^2+40x+25\] Compute and interpret both \(C(600)\) and \(C'(600)\).

Use the cost function \(C(x)=0.02x^3-0.5x^2+10x+150\) and the revenue function \(R(x)=-1.1x^2+41.5x\). If \(P(x)\) represents the profit on the sale of \(x\) units, compute and interpret \(P(20)\) and \(P'(20)\).