1 Monotonicity and extrema / 4 pts
Let \(f(x)=2x^3-3x^2-12x+4\). (a) Compute \(f^{\prime}\) and find the critical points. (b) Give the intervals where \(f\) increases or decreases. (c) Find the local extrema.
2 Concavity / 3 pts
For the same function \(f\): (a) Find the intervals of concavity. (b) Find the inflection point. (c) Confirm the nature of the extrema with the second derivative test.
3 Mean Value Theorem / 3 pts
Let \(f(x)=\sqrt x\) on \([1,9]\). Verify the hypotheses of the Mean Value Theorem and find the guaranteed value \(c\).
4 Two pens / 4 pts
A rancher has 240 m of fencing for a rectangular field divided into two equal pens by one interior fence parallel to a width \(w\). The length is \(L\). (a) Write \(L\) in terms of \(w\). (b) Find \(w\) and \(L\) that maximize the total area, and the area.
5 Oil spill / 3 pts
A circular oil slick has radius growing at 0.5 ft/min. How fast is its area increasing when the radius is 15 ft? Give the exact value and a decimal approximation.
6 Limits and linearization / 3 pts
(a) Compute \(\displaystyle\lim_{x\to0}\dfrac{e^{3x}-1}{\sin 2x}\). (b) Use the linear approximation of \(\ln(1+x)\) at \(0\) to estimate \(\ln1.05\).
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