The temperature of coffee in a cup at time \(t\) minutes is modeled by a decreasing differentiable function \(C\), where \(C(t)\) is measured in degrees Celsius. For \(0 \leq t \leq 12\), selected values of \(C(t)\) are given in the table shown.

\(t\) (minutes)	0	3	7	12
\(C(t)\) (degrees Celsius)	100	85	69	55

(a) Approximate \(C'(5)\) using the average rate of change of \(C\) over the interval \(3 \leq t \leq 7\). Show the work that leads to your answer and include units of measure. (b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of \(\displaystyle\int_{0}^{12} C(t) d t\). Interpret the meaning of \(\dfrac{1}{12} \displaystyle\int_{0}^{12} C(t) d t\) in the context of the problem. (c) For \(12 \leq t \leq 20\), the rate of change of the temperature of the coffee is modeled by \(C'(t) = \dfrac{-24.55 e^{0.01 t}}{t}\), where \(C'(t)\) is measured in degrees Celsius per minute. Find the temperature of the coffee at time \(t = 20\). Show the setup for your calculations. (d) For the model defined in part (c), it can be shown that \(C''(t) = \dfrac{0.2455 e^{0.01 t} (100 - t)}{t^2}\). For \(12 < t < 20\), determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

A particle moving along a curve in the \(x y\)-plane has position \((x(t), y(t))\) at time \(t\) seconds, where \(x(t)\) and \(y(t)\) are measured in centimeters. It is known that \(x'(t) = 8 t - t^2\) and \(y'(t) = -t + \sqrt{t^1.2 + 20}\). At time \(t = 2\) seconds, the particle is at the point \((3, 6)\). (a) Find the speed of the particle at time \(t = 2\) seconds. Show the setup for your calculations. (b) Find the total distance traveled by the particle over the time interval \(0 \leq t \leq 2\). Show the setup for your calculations. (c) Find the \(y\)-coordinate of the position of the particle at the time \(t = 0\). Show the setup for your calculations. (d) For \(2 \leq t \leq 8\), the particle remains in the first quadrant. Find all times \(t\) in the interval \(2 \leq t \leq 8\) when the particle is moving toward the \(x\)-axis. Give a reason for your answer.

The depth of seawater at a location can be modeled by the function \(H\) that satisfies the differential equation \(\dfrac{d H}{d t} = \dfrac{1}{2} (H - 1) \cos\left(\dfrac{t}{2}\right)\), where \(H(t)\) is measured in feet and \(t\) is measured in hours after noon \((t = 0)\). It is known that \(H(0) = 4\). (a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, \(y = H(t)\), through the point \((0, 4)\). (b) For \(0 < t < 5\), it can be shown that \(H(t) > 1\). Find the value of \(t\), for \(0 < t < 5\), at which \(H\) has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your answer. (c) Use separation of variables to find \(y = H(t)\), the particular solution to the differential equation \(\dfrac{d H}{d t} = \dfrac{1}{2} (H - 1) \cos\left(\dfrac{t}{2}\right)\) with initial condition \(H(0) = 4\).

The graph of the differentiable function \(f\), shown for \(-6 \leq x \leq 7\), has a horizontal tangent at \(x = -2\) and is linear for \(0 \leq x \leq 7\). Let \(R\) be the region in the second quadrant bounded by the graph of \(f\), the vertical line \(x = -6\), and the \(x\)- and \(y\)-axes. Region \(R\) has area \(12\). (a) The function \(g\) is defined by \(g(x) = \displaystyle\int_{0}^{x} f(t) d t\). Find the values of \(g(-6)\), \(g(4)\), and \(g(6)\). (b) For the function \(g\) defined in part (a), find all values of \(x\) in the interval \(0 \leq x \leq 6\) at which the graph of \(g\) has a critical point. Give a reason for your answer. (c) The function \(h\) is defined by \(h(x) = \displaystyle\int_{-6}^x f'(t) d t\). Find the values of \(h(6)\), \(h'(6)\), and \(h''(6)\). Show the work that leads to your answers.

The function \(f\) is twice differentiable for all \(x\) with \(f(0) = 0\). Values of \(f'\), the derivative of \(f\), are given in the table for selected values of \(x\).

\(x\)	0	\(\pi\)	\(2 \pi\)
\(f'(x)\)	5	6	0

(a) For \(x \geq 0\), the function \(h\) is defined by \(h(x) = \displaystyle\int_{0}^{x} \sqrt{1 + (f'(t))^2} d t\). Find the value of \(h'(\pi)\). Show the work that leads to your answer. (b) What information does \(\displaystyle\int_{0}^{2 \pi} \sqrt{1 + (f'(x))^2} d x\) provide about the graph of \(f\)? (c) Use Euler's method, starting at \(x = 0\) with two steps of equal size, to approximate \(f(2 \pi)\). Show the computations that lead to your answer. (d) Find \(\int (t + 5) \cos\left(\dfrac{t}{4}\right) d t\). Show the work that leads to your answer.

The Maclaurin series for a function \(f\) is given by \(\displaystyle\sum_{n=1}^\infty \dfrac{(n + 1) x^n}{n^2 \cdot 6^n}\) and converges to \(f(x)\) for all \(x\) in the interval of convergence. It can be shown that the Maclaurin series for \(f\) has a radius of convergence of \(6\). (a) Determine whether the Maclaurin series for \(f\) converges or diverges at \(x = 6\). Give a reason for your answer. (b) It can be shown that \(f(-3) = \displaystyle\sum_{n=1}^\infty \dfrac{(n + 1) (-3)^n}{n^2 \cdot 6^n} = \displaystyle\sum_{n=1}^\infty \dfrac{n + 1}{n^2} \left(-\dfrac{1}{2}\right)^n\) and that the first three terms of this series sum to \(S_3 = -\dfrac{125}{144}\). Show that \(|f(-3) - S_3| < \dfrac{1}{50}\). (c) Find the general term of the Maclaurin series for \(f'\), the derivative of \(f\). Find the radius of convergence of the Maclaurin series for \(f'\). (d) Let \(g(x) = \displaystyle\sum_{n=1}^\infty \dfrac{(n + 1) x^{2 n}}{n^2 \cdot 3^n}\). Use the ratio test to determine the radius of convergence of the Maclaurin series for \(g\).