Consider the integral expression \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin(2 x) e^{\cos(2 x)} d x\). If \(u = \cos 2 x\), then which integral below is equivalent to the given integral?

Let \(f(x) = \dfrac{1}{x}\) and \(k > 1\). If the area between the x-axis and the graph of \(f(x)\) in the closed interval \(k \leq x \leq k + 1\) is \(0.125\) where \(k > 1\), then what is the value of \(k\)?

[Calculator] Shampoo drips from a crack in the side of a plastic bottle at a rate modeled by \(Y(t) = \dfrac{t}{\sqrt{1 + t^{\dfrac{3}{2}}}}\), where \(Y(t)\) is in ounces per minute. If there are \(32\) ounces in the bottle at \(t = 0\), how many ounces are left in the bottle after \(5\) minutes?

[Calculator] Consider the function \(f(x) = x^3 + 2\) in the closed interval \(0 < a \leq c \leq 2\). If the value guaranteed by the Mean Value Theorem in the closed interval is \(c = 1.720\), then what is the value of \(a\)?

Let \(h(x) = x g(x)\), where \(g(x) = f^{-1}(x)\). Use the table of values below to find \(h'(5)\). Table: \(f(2) = 4\), \(f'(2) = -1\); \(f(3) = 5\), \(f'(3) = 2\); \(f(5) = 1\), \(f'(5) = 3\).

[Calculator] Let \(f(x) = \sin x\) and \(g(x) = p \ln x\) in the closed interval \(0 \leq x \leq \dfrac{\pi}{2}\). For what value of \(p\) will the tangents to the curves at their points of intersection be perpendicular?

The height of a conical sand pile is always twice the radius. If sand is being added to the pile at a rate of \(30 \pi\) cm\(^3\)/min, how fast is the height of the pile increasing when the circumference of the base of the sand pile is \(120 \pi\) cm? \(\left(V_{\text{cone}} = \dfrac{\pi}{3} r^2 h\right)\)

[Calculator] The number of home fires each day in a certain city increases as the temperature drops. The rate of home fires is modeled by \(F(t) = 4 \cos\left(\dfrac{t}{58} - 2\right) + 6\), for \(0 \leq t \leq 365\) days, where midnight on January 1st corresponds to \(t = 0\). Which of the following is closest to the approximate number of fires in the first quarter of the year?

If \(f(x)\) and \(g(x)\) are differentiable functions with values \(f(1)=3\), \(g(1)=4\), \(f'(1) = \dfrac{2}{3}\), \(g'(1) = -\dfrac{5}{2}\); \(f(2)=4\), \(g(2)=2\), \(f'(2) = \dfrac{4}{3}\), \(g'(2) = -\dfrac{3}{2}\); \(f(4)=8\), \(g(4)=1\), \(f'(4) = \dfrac{8}{3}\), \(g'(4) = \dfrac{1}{2}\), and \(k(x) = f(g(x^2))\), what is \(k'(2)\)?

[Calculator] The price of a newly issued stock varies sinusoidally during the first \(10\) days after its initial offering and is modeled by \(P(t) = \log(2 t + 1) \sin t + 20\), where \(t\) is in days. To the nearest cent, what is the price of the stock when the price of the stock is decreasing most rapidly in the interval \(0 \leq t \leq 10\)?