Grass clippings are placed in a bin, where they decompose. For \(0 \le t \le 30\), the amount of grass clippings remaining in the bin is modeled by \(A(t) = 6.687(0.931)^{t}\), where \(A(t)\) is measured in pounds and \(t\) is measured in days.

1. (A) Find the average rate of change of \(A(t)\) over the interval \(0 \le t \le 30\). Indicate units of measure.
2. (B) Find the value of \(A'(15)\). Using correct units, interpret the meaning of the value in the context of the problem.
3. (C) Find the time \(t\) for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval \(0 \le t \le 30\).
4. (D) For \(t > 30\), \(L(t)\), the linear approximation to \(A\) at \(t = 30\), is a better model for the amount of grass clippings remaining in the bin. Use \(L(t)\) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

At time \(t = 0\) minutes, a tank contains 100 liters of water. The piecewise-linear graph above shows the rate \(R(t)\), in liters per minute, at which water is pumped into the tank during a 55-minute period.

1. (A) Find \(R'(45)\). Using appropriate units, explain the meaning of your answer in the context of this problem.
2. (B) How many liters of water have been pumped into the tank from time \(t = 0\) to time \(t = 55\) minutes? Show the work that leads to your answer.
3. (C) At time \(t = 10\) minutes, water begins draining from the tank at a rate modeled by the function \(D\), where \(D(t) = 10\,e^{(\sin t)/10}\) liters per minute. Water continues to drain at this rate until time \(t = 55\) minutes. How many liters of water are in the tank at time \(t = 55\) minutes?
4. (D) Using the functions \(R\) and \(D\), determine whether the amount of water in the tank is increasing or decreasing at time \(t = 45\) minutes. Justify your answer.

Water is pumped into a tank at a rate modeled by \(W(t) = 2000\,e^{-t^{2}/20}\) liters per hour for \(0 \le t \le 8\), where \(t\) is measured in hours. Water is removed from the tank at a rate modeled by \(R(t)\) liters per hour, where \(R\) is differentiable and decreasing on \(0 \le t \le 8\). Selected values of \(R(t)\) are shown in the table above. At time \(t = 0\), there are 50,000 liters of water in the tank.

1. (A) Estimate \(R'(2)\). Show the work that leads to your answer. Indicate units of measure.
2. (B) Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of water removed from the tank during the 8 hours. Is this an overestimate or an underestimate of the total amount of water removed? Give a reason for your answer.
3. (C) Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours.
4. (D) For \(0 \le t \le 8\), is there a time \(t\) when the rate at which water is pumped into the tank is the same as the rate at which water is removed from the tank? Explain why or why not.

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height \(h\) feet is given by the function \(A\), where \(A(h)\) is measured in square feet. The function \(A\) is continuous and decreases as \(h\) increases. Selected values for \(A(h)\) are given in the table above.

1. (A) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.
2. (B) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning.
3. (C) The area, in square feet, of the horizontal cross section at height \(h\) feet is modeled by the function \(f\) given by \(f(h) = \dfrac{50.3}{e^{0.2h} + h}\). Based on this model, find the volume of the tank. Indicate units of measure.
4. (D) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure.

People enter a line for an escalator at a rate modeled by the function \(r\) given by \[r(t) = \begin{cases} 44\left(\dfrac{t}{100}\right)^3 \left(1 - \dfrac{t}{300}\right)^7 & \text{for } 0 \le t \le 300 \\ 0 & \text{for } t > 300 \end{cases}\] where \(r(t)\) is measured in people per second and \(t\) is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time \(t = 0\).

1. (A) How many people enter the line for the escalator during the time interval \(0 \le t \le 300\)?
2. (B) During the time interval \(0 \le t \le 300\), there are always people in line for the escalator. How many people are in line at time \(t = 300\)?
3. (C) For \(t > 300\), what is the first time \(t\) that there are no people in line for the escalator?
4. (D) For \(0 \le t \le 300\), at what time \(t\) is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

Researchers on a boat are investigating plankton cells in a sea. At a depth of \(h\) meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by \(p(h) = 0.2 h^2 e^{-0.0025 h^2}\) for \(0 \le h \le 30\) and is modeled by \(f(h)\) for \(h > 30\). The continuous function \(f\) is not explicitly given.

1. (A) Find \(p'(25)\). Using correct units, interpret the meaning of \(p'(25)\) in the context of the problem.
2. (B) Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square meters. To the nearest million, how many plankton cells are in this column of water between \(h = 0\) and \(h = 30\) meters?
3. (C) There is a function \(u\) such that \(0 \le f(h) \le u(h)\) for all \(h \ge 30\) and \(\int_{30}^{\infty} u(h)\,dh = 105\). The column of water in part (b) is \(K\) meters deep, where \(K > 30\). Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to 2000 million.
4. (D) The boat is moving on the surface of the sea. At time \(t \ge 0\), the position of the boat is the point with coordinates \((x(t), y(t))\), where \(x'(t) = 662\sin(5t)\) and \(y'(t) = 880\cos(6t)\). Time \(t\) is measured in hours, and \(x(t)\) and \(y(t)\) are measured in meters. Find the total distance traveled by the boat over the time interval \(0 \le t \le 1\).

The rate at which cars enter a parking lot is modeled by \(E(t) = 30 + 5(t - 2)(t - 5)e^{-0.2t}\). The rate at which cars leave the parking lot is modeled by the differentiable function \(L\). Selected values of \(L(t)\) are given in the table above. Both \(E(t)\) and \(L(t)\) are measured in cars per hour, and time \(t\) is measured in hours after 5 A.M. (\(t = 0\)). Both functions are defined for \(0 \le t \le 12\).

1. (A) What is the rate of change of \(E(t)\) at time \(t = 7\)? Indicate units of measure.
2. (B) How many cars enter the parking lot from time \(t = 0\) to time \(t = 12\)? Give your answer to the nearest whole number.
3. (C) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate \(\displaystyle\int_2^{12} L(t)\,dt\). Using correct units, explain the meaning of \(\displaystyle\int_2^{12} L(t)\,dt\) in the context of this problem.
4. (D) For \(0 \le t < 6\), 5 dollars are collected from each car entering the parking lot. For \(6 \le t \le 12\), 8 dollars are collected from each car entering the parking lot. How many dollars are collected from the cars entering the parking lot from time \(t = 0\) to time \(t = 12\)? Give your answer to the nearest whole dollar.

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by \(A(t) = 450\sqrt{\sin(0.62t)}\), where \(t\) is the number of hours after 5 A.M. and \(A(t)\) is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.

1. (A) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. \((t = 1)\) to 10 A.M. \((t = 5)\).
2. (B) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. \((t = 1)\) to 10 A.M. \((t = 5)\).
3. (C) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. \((t = 1)\) increasing or decreasing? Give a reason for your answer.
4. (D) A line forms whenever \(A(t) \ge 400\). The number of vehicles in line at time \(t\), for \(a \le t \le 4\), is given by \(N(t) = \displaystyle\int_a^t (A(x) - 400)\,dx\), where \(a\) is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval \(a \le t \le 4\). Justify your answer.

An invasive species of plant appears in a fruit grove at time \(t = 0\) and begins to spread. The function \(C\) defined by \(C(t) = 7.6\,\arctan(0.2t)\) models the number of acres in the fruit grove affected by the species \(t\) weeks after the species appears. It can be shown that \(C'(t) = \dfrac{38}{25 + t^2}\).

Note. Your calculator should be in radian mode.

1. (A) Find the average number of acres affected by the invasive species from time \(t = 0\) to time \(t = 4\) weeks. Show the setup for your calculations.
2. (B) Find the time \(t\) when the instantaneous rate of change of \(C\) equals the average rate of change of \(C\) over the time interval \(0 \le t \le 4\). Show the setup for your calculations.
3. (C) Assume that the invasive species continues to spread according to the given model for all times \(t > 0\). Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.
4. (D) At time \(t = 4\) weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function \(A\), defined by \[A(t) = C(t) - \int_4^{t} 0.1 \cdot \ln(x)\,dx,\] models the number of acres affected by the species over the time interval \(4 \le t \le 36\). At what time \(t\), for \(4 \le t \le 36\), does \(A\) attain its maximum value? Justify your answer.

The graphs of the polar curves \(r = 3\) and \(r = 3 - 2\sin(2\theta)\) are shown in the figure above for \(0 \le \theta \le \pi\).

1. (A) Let \(R\) be the shaded region that is inside the graph of \(r = 3\) and inside the graph of \(r = 3 - 2\sin(2\theta)\). Find the area of \(R\).
2. (B) For the curve \(r = 3 - 2\sin(2\theta)\), find the value of \(\dfrac{dx}{d\theta}\) at \(\theta = \dfrac{\pi}{6}\).
3. (C) The distance between the two curves changes for \(0 < \theta < \dfrac{\pi}{2}\). Find the rate at which the distance between the two curves is changing with respect to \(\theta\) when \(\theta = \dfrac{\pi}{3}\).
4. (D) A particle is moving along the curve \(r = 3 - 2\sin(2\theta)\) so that \(\dfrac{d\theta}{dt} = 3\) for all times \(t \ge 0\). Find the value of \(\dfrac{dr}{dt}\) at \(\theta = \dfrac{\pi}{6}\).

The figure above shows the graph of the polar equation \(r = 2 + \sin(4\theta) + \cos\theta\) for \(0 \le \theta \le \tfrac{\pi}{2}\). The derivative of \(r\) with respect to \(\theta\) is given by \(r'(\theta) = 4\cos(4\theta) - \sin\theta\).

1. (A) Find the area of the region bounded by the graph of \(r\) and the lines \(\theta = 0\) and \(\theta = \tfrac{\pi}{2}\).
2. (B) Find the area of the region in the first quadrant that is outside the graph of \(r = 2 + \sin(4\theta) + \cos\theta\) but inside the graph of the circle of radius 2 centered at the origin.
3. (C) Find the value of \(\theta\) in the interval \(0 \le \theta \le \tfrac{\pi}{2}\) that corresponds to the point on the curve \(r = 2 + \sin(4\theta) + \cos\theta\) with greatest distance from the origin. Justify your answer.

At time \(t\), the position of a particle moving in the \(xy\)-plane is given by the parametric functions \((x(t), y(t))\), where \(\dfrac{dx}{dt} = t^{2} + \sin(3t^{2})\). The graph of \(y\), consisting of three line segments, is shown in the figure above. At \(t = 0\), the particle is at position \((5, 1)\).

1. (A) Find the position of the particle at \(t = 3\).
2. (B) Find the slope of the line tangent to the path of the particle at \(t = 3\).
3. (C) Find the speed of the particle at \(t = 3\).
4. (D) Find the total distance traveled by the particle from \(t = 0\) to \(t = 2\).

The figure above shows the polar curves \(r = f(\theta) = 1 + \sin\theta \cos(2\theta)\) and \(r = g(\theta) = 2\cos\theta\) for \(0 \le \theta \le \dfrac{\pi}{2}\). Let \(R\) be the region in the first quadrant bounded by the curve \(r = f(\theta)\) and the x-axis. Let \(S\) be the region in the first quadrant bounded by the curve \(r = f(\theta)\), the curve \(r = g(\theta)\), and the x-axis.

1. (A) Find the area of \(R\).
2. (B) The ray \(\theta = k\), where \(0 < k < \dfrac{\pi}{2}\), divides \(S\) into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of \(k\).
3. (C) For each \(\theta\), let \(w(\theta)\) be the distance between the points with polar coordinates \((f(\theta), \theta)\) and \((g(\theta), \theta)\). Write an expression for \(w(\theta)\). Find \(w_A\), the average value of \(w(\theta)\) over the interval \(0 \le \theta \le \dfrac{\pi}{2}\).
4. (D) Using the information from part (c), find the value of \(\theta\) for which \(w(\theta) = w_A\). Is the function \(w(\theta)\) increasing or decreasing at that value of \(\theta\)? Give a reason for your answer.

A laser is a device that produces a beam of light. A design, shown above, is etched onto a flat piece of metal using a moving laser. The position of the laser at time \(t\) seconds is represented by \((x(t), y(t))\) in the \(xy\)-plane. Both \(x\) and \(y\) are measured in centimeters, and \(t\) is measured in seconds. The laser starts at position \((0, 0)\) at time \(t = 0\), and the design takes 3.1 seconds to complete. For \(0 \le t \le 3.1\), \(\dfrac{dx}{dt} = 3\cos(t^2)\) and \(\dfrac{dy}{dt} = 4\cos(2.5 t)\).

1. (A) Find the speed of the laser at time \(t = 3\) seconds.
2. (B) Find the total distance traveled by the laser from time \(t = 1\) to time \(t = 3\) seconds.
3. (C) The laser is farthest to the right at time \(t = 1.253\) seconds. Find the \(x\)-coordinate of the laser's rightmost position.
4. (D) What is the difference between the \(y\)-coordinates of the laser's highest position and lowest position for \(0 \le t \le 3.1\)? Justify your answer.

For time \(t \ge 0\), a particle moves in the \(xy\)-plane with position \((x(t), y(t))\) and velocity vector \(\bigl\langle (t-1)e^{t^2},\ \sin(t^{1.25})\bigr\rangle\). At time \(t = 0\), the position of the particle is \((-2, 5)\).

1. (A) Find the speed of the particle at time \(t = 1.2\). Find the acceleration vector of the particle at time \(t = 1.2\).
2. (B) Find the total distance traveled by the particle over the time interval \(0 \le t \le 1.2\).
3. (C) Find the coordinates of the point at which the particle is farthest to the left for \(t \ge 0\). Explain why there is no point at which the particle is farthest to the right for \(t \ge 0\).

A particle moving along a curve in the \(xy\)-plane is at position \((x(t), y(t))\) at time \(t > 0\). The particle moves in such a way that \(\dfrac{dx}{dt} = \sqrt{1 + t^2}\) and \(\dfrac{dy}{dt} = \ln(2 + t^2)\). At time \(t = 4\), the particle is at the point \((1, 5)\).

1. (A) Find the slope of the line tangent to the path of the particle at time \(t = 4\).
2. (B) Find the speed of the particle at time \(t = 4\), and find the acceleration vector of the particle at time \(t = 4\).
3. (C) Find the \(y\)-coordinate of the particle's position at time \(t = 6\).
4. (D) Find the total distance the particle travels along the curve from time \(t = 4\) to time \(t = 6\).

For \(0 \le t \le \pi\), a particle is moving along the curve shown so that its position at time \(t\) is \((x(t), y(t))\), where \(x(t)\) is not explicitly given and \(y(t) = 2\sin t\). It is known that \(\dfrac{dx}{dt} = e^{\cos t}\). At time \(t = 0\), the particle is at position \((1, 0)\).

1. (A) Find the acceleration vector of the particle at time \(t = 1\). Show the setup for your calculations.
2. (B) For \(0 \le t \le \pi\), find the first time \(t\) at which the speed of the particle is 1.5. Show the work that leads to your answer.
3. (C) Find the slope of the line tangent to the path of the particle at time \(t = 1\). Find the \(x\)-coordinate of the position of the particle at time \(t = 1\). Show the work that leads to your answers.
4. (D) Find the total distance traveled by the particle over the time interval \(0 \le t \le \pi\). Show the setup for your calculations.

A particle moving along a curve in the \(xy\)-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 \(x'(t) = 8t - t^2\) and \(y'(t) = -t + \sqrt{t^{1.2} + 20}\). At time \(t = 2\) seconds, the particle is at the point \((3, 6)\).

1. (A) Find the speed of the particle at time \(t = 2\) seconds. Show the setup for your calculations.
2. (B) Find the total distance traveled by the particle over the time interval \(0 \le t \le 2\). Show the setup for your calculations.
3. (C) Find the \(y\)-coordinate of the position of the particle at the time \(t = 0\). Show the setup for your calculations.
4. (D) For \(2 \le t \le 8\), the particle remains in the first quadrant. Find all times \(t\) in the interval \(2 \le t \le 8\) when the particle is moving toward the \(x\)-axis. Give a reason for your answer.

Curve \(C\) is defined by the polar equation \(r(\theta) = 2\sin^2\theta\) for \(0 \le \theta \le \pi\). Curve \(C\) and the semicircle \(r = \tfrac{1}{2}\) for \(0 \le \theta \le \pi\) are shown in the \(xy\)-plane.

Note. Your calculator should be in radian mode.

1. (A) Find the rate of change of \(r\) with respect to \(\theta\) at the point on curve \(C\) where \(\theta = 1.3\). Show the setup for your calculations.
2. (B) Find the area of the region that lies inside curve \(C\) but outside the graph of the polar equation \(r = \tfrac{1}{2}\). Show the setup for your calculations.
3. (C) It can be shown that \(\dfrac{dx}{d\theta} = 4\sin\theta\cos^2\theta - 2\sin^3\theta\) for curve \(C\). For \(0 \le \theta \le \tfrac{\pi}{2}\), find the value of \(\theta\) that corresponds to the point on curve \(C\) that is farthest from the \(y\)-axis. Justify your answer.
4. (D) A particle travels along curve \(C\) so that \(\dfrac{d\theta}{dt} = 15\) for all times \(t\). Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where \(\theta = 1.3\). Show the setup for your calculations.

The function \(f\) is defined on the closed interval \([-5, 4]\). The graph of \(f\) consists of three line segments and is shown in the figure above. Let \(g\) be the function defined by \(g(x) = \int_{-3}^{x} f(t)\,dt\).

1. (A) Find \(g(3)\).
2. (B) On what open intervals contained in \(-5 < x < 4\) is the graph of \(g\) both increasing and concave down? Give a reason for your answer.
3. (C) The function \(h\) is defined by \(h(x) = \dfrac{g(x)}{5x}\). Find \(h'(3)\).
4. (D) The function \(p\) is defined by \(p(x) = f(x^2 - x)\). Find the slope of the line tangent to the graph of \(p\) at the point where \(x = -1\).

The figure above shows the graph of the piecewise-linear function \(f\). For \(-4 \le x \le 12\), the function \(g\) is defined by \(g(x) = \int_{2}^{x} f(t)\,dt\).

1. (A) Does \(g\) have a relative minimum, a relative maximum, or neither at \(x = 10\)? Justify your answer.
2. (B) Does the graph of \(g\) have a point of inflection at \(x = 4\)? Justify your answer.
3. (C) Find the absolute minimum value and the absolute maximum value of \(g\) on the interval \(-4 \le x \le 12\). Justify your answers.
4. (D) For \(-4 \le x \le 12\), find all intervals for which \(g(x) \le 0\).

The function \(f\) is differentiable on the closed interval \([-6, 5]\) and satisfies \(f(-2) = 7\). The graph of \(f'\), the derivative of \(f\), consists of a semicircle and three line segments, as shown in the figure above.

1. (A) Find the values of \(f(-6)\) and \(f(5)\).
2. (B) On what intervals is \(f\) increasing? Justify your answer.
3. (C) Find the absolute minimum value of \(f\) on the closed interval \([-6, 5]\). Justify your answer.
4. (D) For each of \(f''(-5)\) and \(f''(3)\), find the value or explain why it does not exist.

The graph of the continuous function \(g\), the derivative of the function \(f\), is shown above. The function \(g\) is piecewise linear for \(-5 \le x < 3\), and \(g(x) = 2(x - 4)^2\) for \(3 \le x \le 6\).

1. (A) If \(f(1) = 3\), what is the value of \(f(-5)\)?
2. (B) Evaluate \(\int_{1}^{6} g(x)\,dx\).
3. (C) For \(-5 < x < 6\), on what open intervals, if any, is the graph of \(f\) both increasing and concave up? Give a reason for your answer.
4. (D) Find the x-coordinate of each point of inflection of the graph of \(f\). Give a reason for your answer.

Let \(f\) be the function defined by \[f(x) = \begin{cases} \sqrt{9 - x^2} & \text{for }-3 \le x \le 0 \\ -x + 3\cos\!\left(\dfrac{\pi x}{2}\right) & \text{for }0 < x \le 4. \end{cases}\]

1. (A) Find the average rate of change of \(f\) on the interval \(-3 \le x \le 4\).
2. (B) Write an equation for the line tangent to the graph of \(f\) at \(x = 3\).
3. (C) Find the average value of \(f\) on the interval \(-3 \le x \le 4\).
4. (D) Must there be a value of \(x\) at which \(f(x)\) attains an absolute maximum on the closed interval \(-3 \le x \le 4\)? Justify your answer.

Let \(f\) be a continuous function defined on the closed interval \(-4 \le x \le 6\). The graph of \(f\), consisting of four line segments, is shown above. Let \(G\) be the function defined by \(G(x) = \displaystyle\int_0^x f(t)\,dt\).

1. (A) On what open intervals is the graph of \(G\) concave up? Give a reason for your answer.
2. (B) Let \(P\) be the function defined by \(P(x) = G(x)\cdot f(x)\). Find \(P'(3)\).
3. (C) Find \(\displaystyle\lim_{x \to 2} \dfrac{G(x)}{x^2 - 2x}\).
4. (D) Find the average rate of change of \(G\) on the interval \([-4, 2]\). Does the Mean Value Theorem guarantee a value \(c\), \(-4 < c < 2\), for which \(G'(c)\) is equal to this average rate of change? Justify your answer.

Let \(f\) be a differentiable function with \(f(4) = 3\). On the interval \(0 \le x \le 7\), the graph of \(f'\), the derivative of \(f\), consists of a semicircle and two line segments, as shown in the figure above.

1. (A) Find \(f(0)\) and \(f(5)\).
2. (B) Find the \(x\)-coordinates of all points of inflection of the graph of \(f\) for \(0 < x < 7\). Justify your answer.
3. (C) Let \(g\) be the function defined by \(g(x) = f(x) - x\). On what intervals, if any, is \(g\) decreasing for \(0 \le x \le 7\)? Show the analysis that leads to your answer.
4. (D) For the function \(g\) defined in part (c), find the absolute minimum value on the interval \(0 \le x \le 7\). Justify your answer.

The function \(f\) is defined on the closed interval \([-2, 8]\) and satisfies \(f(2) = 1\). The graph of \(f'\), the derivative of \(f\), consists of two line segments and a semicircle, as shown in the figure.

1. (A) Does \(f\) have a relative minimum, a relative maximum, or neither at \(x = 6\)? Give a reason for your answer.
2. (B) On what open intervals, if any, is the graph of \(f\) concave down? Give a reason for your answer.
3. (C) Find the value of \(\displaystyle\lim_{x \to 2} \dfrac{6f(x) - 3x}{x^2 - 5x + 6}\), or show that it does not exist. Justify your answer.
4. (D) Find the absolute minimum value of \(f\) on the closed interval \([-2, 8]\). Justify your answer.

The graph of the differentiable function \(f\), shown for \(-6 \le x \le 7\), has a horizontal tangent at \(x = -2\) and is linear for \(0 \le x \le 7\). 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.

1. (A) The function \(g\) is defined by \(g(x) = \displaystyle\int_0^{x} f(t)\,dt\). Find the values of \(g(-6)\), \(g(4)\), and \(g(6)\).
2. (B) For the function \(g\) defined in part (a), find all values of \(x\) in the interval \(0 \le x \le 6\) at which the graph of \(g\) has a critical point. Give a reason for your answer.
3. (C) The function \(h\) is defined by \(h(x) = \displaystyle\int_{-6}^{x} f'(t)\,dt\). Find the values of \(h(6)\), \(h'(6)\), and \(h''(6)\). Show the work that leads to your answers.

The continuous function \(f\) is defined on the closed interval \(-6 \le x \le 12\). The graph of \(f\), consisting of two semicircles and one line segment, is shown in the figure. Let \(g\) be the function defined by \(g(x) = \displaystyle\int_6^{x} f(t)\,dt\).

1. (A) Find \(g'(8)\). Give a reason for your answer.
2. (B) Find all values of \(x\) in the open interval \(-6 < x < 12\) at which the graph of \(g\) has a point of inflection. Give a reason for your answer.
3. (C) Find \(g(12)\) and \(g(0)\). Label your answers.
4. (D) Find the value of \(x\) at which \(g\) attains an absolute minimum on the closed interval \(-6 \le x \le 12\). Justify your answer.

Train A runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function \(v_A(t)\), where time \(t\) is measured in minutes. Selected values for \(v_A(t)\) are given in the table above.

1. (A) Find the average acceleration of train A over the interval \(2 \le t \le 8\).
2. (B) Do the data in the table support the conclusion that train A's velocity is \(-100\) meters per minute at some time \(t\) with \(5 < t < 8\)? Give a reason for your answer.
3. (C) At time \(t = 2\), train A's position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time \(t = 12\). Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time \(t = 12\).
4. (D) A second train, train B, travels north from the Origin Station. At time \(t\) the velocity of train B is given by \(v_B(t) = -5t^2 + 60t + 25\), and at time \(t = 2\) the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time \(t = 2\).

Johanna jogs along a straight path. For \(0 \le t \le 40\), Johanna's velocity is given by a differentiable function \(v\). Selected values of \(v(t)\), where \(t\) is measured in minutes and \(v(t)\) is measured in meters per minute, are given in the table above.

1. (A) Use the data in the table to estimate the value of \(v'(16)\).
2. (B) Using correct units, explain the meaning of the definite integral \(\int_{0}^{40} |v(t)|\,dt\) in the context of the problem. Approximate the value of \(\int_{0}^{40} |v(t)|\,dt\) using a right Riemann sum with the four subintervals indicated in the table.
3. (C) Bob is riding his bicycle along the same path. For \(0 \le t \le 10\), Bob's velocity is modeled by \(B(t) = t^3 - 6t^2 + 300\), where \(t\) is measured in minutes and \(B(t)\) is measured in meters per minute. Find Bob's acceleration at time \(t = 5\).
4. (D) Based on the model \(B\) from part (c), find Bob's average velocity during the interval \(0 \le t \le 10\).

The height of a tree at time \(t\) is given by a twice-differentiable function \(H\), where \(H(t)\) is measured in meters and \(t\) is measured in years. Selected values of \(H(t)\) are given in the table above.

1. (A) Use the data in the table to estimate \(H'(6)\). Using correct units, interpret the meaning of \(H'(6)\) in the context of the problem.
2. (B) Explain why there must be at least one time \(t\), for \(2 < t < 10\), such that \(H'(t) = 2\).
3. (C) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval \(2 \le t \le 10\).
4. (D) The height of the tree, in meters, can also be modeled by the function \(G\), given by \(G(x) = \dfrac{100x}{1 + x}\), where \(x\) is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure above. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height \(h\) of the water in the barrel with respect to time \(t\) is modeled by \(\dfrac{dh}{dt} = -\dfrac{1}{10}\sqrt{h}\), where \(h\) is measured in feet and \(t\) is measured in seconds. (The volume \(V\) of a cylinder with radius \(r\) and height \(h\) is \(V = \pi r^2 h\).)

1. (A) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
2. (B) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
3. (C) At time \(t = 0\) seconds, the height of the water is 5 feet. Use separation of variables to find an expression for \(h\) in terms of \(t\).

The density of a bacteria population in a circular petri dish at a distance \(r\) centimeters from the center of the dish is given by an increasing, differentiable function \(f\), where \(f(r)\) is measured in milligrams per square centimeter. Values of \(f(r)\) for selected values of \(r\) are given in the table above.

1. (A) Use the data in the table to estimate \(f'(2.25)\). Using correct units, interpret the meaning of your answer in the context of this problem.
2. (B) The total mass, in milligrams, of bacteria in the petri dish is given by the integral expression \(2\pi\displaystyle\int_0^4 r\,f(r)\,dr\). Approximate the value of \(2\pi\displaystyle\int_0^4 r\,f(r)\,dr\) using a right Riemann sum with the four subintervals indicated by the data in the table.
3. (C) Is the approximation found in part (b) an overestimate or underestimate of the total mass of bacteria in the petri dish? Explain your reasoning.
4. (D) The density of bacteria in the petri dish, for \(1 \le r \le 4\), is modeled by the function \(g\) defined by \(g(r) = 2 - 16\bigl(\cos(1.57\sqrt{r})\bigr)^3\). For what value of \(k\), \(1 < k < 4\), is \(g(k)\) equal to the average value of \(g(r)\) on the interval \(1 \le r \le 4\)?

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function \(r\), where \(r(t)\) is measured in centimeters and \(t\) is measured in days. The table above gives selected values of \(r'(t)\), the rate of change of the radius, over the time interval \(0 \le t \le 12\).

1. (A) Approximate \(r''(8.5)\) using the average rate of change of \(r'\) over the interval \(7 \le t \le 10\). Show the computations that lead to your answer, and indicate units of measure.
2. (B) Is there a time \(t\), \(0 \le t \le 3\), for which \(r'(t) = -6\)? Justify your answer.
3. (C) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of \(\displaystyle\int_0^{12} r'(t)\,dt\).
4. (D) The height of the cone decreases at a rate of 2 centimeters per day. At time \(t = 3\) days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time \(t = 3\) days. (The volume \(V\) of a cone with radius \(r\) and height \(h\) is \(V = \tfrac{1}{3}\pi r^2 h\).)

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function \(f\), where \(f(t)\) is measured in gallons per second and \(t\) is measured in seconds since pumping began. Selected values of \(f(t)\) are given in the table.

1. (A) Using correct units, interpret the meaning of \(\displaystyle\int_{60}^{135} f(t)\,dt\) in the context of the problem. Use a right Riemann sum with the three subintervals \([60, 90]\), \([90, 120]\), and \([120, 135]\) to approximate the value of \(\displaystyle\int_{60}^{135} f(t)\,dt\).
2. (B) Must there exist a value of \(c\), for \(60 < c < 120\), such that \(f'(c) = 0\)? Justify your answer.
3. (C) The rate of flow of gasoline, in gallons per second, can also be modeled by \(g(t) = \left(\dfrac{t}{500}\right)\cos\!\left(\left(\dfrac{t}{120}\right)^2\right)\) for \(0 \le t \le 150\). Using this model, find the average rate of flow of gasoline over the time interval \(0 \le t \le 150\). Show the setup for your calculations.
4. (D) Using the model \(g\) defined in part (c), find the value of \(g'(140)\). Interpret the meaning of your answer in the context of the problem.

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 \(0 \le t \le 12\), selected values of \(C(t)\) are given in the table shown.

1. (A) Approximate \(C'(5)\) using the average rate of change of \(C\) over the interval \(3 \le t \le 7\). Show the work that leads to your answer and include units of measure.
2. (B) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of \(\displaystyle\int_0^{12} C(t)\,dt\). Interpret the meaning of \(\dfrac{1}{12}\displaystyle\int_0^{12} C(t)\,dt\) in the context of the problem.
3. (C) For \(12 \le t \le 20\), the rate of change of the temperature of the coffee is modeled by \(C'(t) = \dfrac{-24.55\,e^{0.01t}}{t}\), 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.
4. (D) For the model defined in part (c), it can be shown that \(C''(t) = \dfrac{0.2455\,e^{0.01t}(100 - t)}{t^2}\). 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.

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 \(x\).

1. (A) For \(x \ge 0\), the function \(h\) is defined by \(h(x) = \displaystyle\int_0^{x} \sqrt{1 + (f'(t))^2}\,dt\). Find the value of \(h'(\pi)\). Show the work that leads to your answer.
2. (B) What information does \(\displaystyle\int_0^{\pi} \sqrt{1 + (f'(x))^2}\,dx\) provide about the graph of \(f\)?
3. (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.
4. (D) Find \(\displaystyle\int (t + 5)\cos\!\left(\dfrac{t}{4}\right) dt\). Show the work that leads to your answer.

A student starts reading a book at time \(t = 0\) minutes and continues reading for the next 10 minutes. The rate at which the student reads is modeled by the differentiable function \(R\), where \(R(t)\) is measured in words per minute. Selected values of \(R(t)\) are given in the table shown.

1. (A) Approximate \(R'(1)\) using the average rate of change of \(R\) over the interval \(0 \le t \le 2\). Show the work that leads to your answer. Indicate units of measure.
2. (B) Must there be a value \(c\), for \(0 < c < 10\), such that \(R(c) = 155\)? Justify your answer.
3. (C) Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate the value of \(\displaystyle\int_0^{10} R(t)\,dt\). Show the work that leads to your answer.
4. (D) A teacher also starts reading at time \(t = 0\) minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function \(W\) defined by \(W(t) = -\tfrac{3}{10}t^2 + 8t + 100\), where \(W(t)\) is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

In a national park, the population of mountain lions grows over time. At time \(t = 0\), where \(t\) is measured in years, the population is found to be 20 mountain lions.

1. (A) One zoologist suggests a population model \(P\) that satisfies the differential equation \(\dfrac{dP}{dt} = \dfrac{1}{4}(220 - P)\). Use separation of variables to solve this differential equation for \(P\) with the initial condition \(P(0) = 20\).
2. (B) A second zoologist suggests a population model \(Q\) that satisfies \(\dfrac{dQ}{dt} = \dfrac{1}{500}Q(220 - Q)\). Find the value of \(\dfrac{dQ}{dt}\) at the time when \(Q\) grows most rapidly.
3. (C) For the population model \(Q\) introduced in part (b), use Euler's method, starting at \(t = 0\) with two steps of equal size, to approximate \(Q(10)\). Show the computations that lead to your answer.

Consider the differential equation \(\dfrac{dy}{dx} = x^{2} - \dfrac{1}{2}y\).

1. (A) Find \(\dfrac{d^{2}y}{dx^{2}}\) in terms of \(x\) and \(y\).
2. (B) Let \(y = f(x)\) be the particular solution to the given differential equation whose graph passes through the point \((-2, 8)\). Does the graph of \(f\) have a relative minimum, a relative maximum, or neither at the point \((-2, 8)\)? Justify your answer.
3. (C) Let \(y = g(x)\) be the particular solution to the given differential equation with \(g(-1) = 2\). Find \(\displaystyle\lim_{x \to -1} \dfrac{g(x) - 2}{3(x + 1)^{2}}\). Show the work that leads to your answer.
4. (D) Let \(y = h(x)\) be the particular solution to the given differential equation with \(h(0) = 2\). Use Euler's method, starting at \(x = 0\) with two steps of equal size, to approximate \(h(1)\).

At time \(t = 0\), a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius (\(^\circ\)C) at time \(t = 0\), and the internal temperature of the potato is greater than \(27^\circ\)C for all times \(t > 0\). The internal temperature of the potato at time \(t\) minutes can be modeled by the function \(H\) that satisfies the differential equation \(\dfrac{dH}{dt} = -\dfrac{1}{4}(H - 27)\), where \(H(t)\) is measured in degrees Celsius and \(H(0) = 91\).

1. (A) Write an equation for the line tangent to the graph of \(H\) at \(t = 0\). Use this equation to approximate the internal temperature of the potato at time \(t = 3\).
2. (B) Use \(\dfrac{d^2 H}{dt^2}\) to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time \(t = 3\).
3. (C) For \(t < 10\), an alternate model for the internal temperature of the potato at time \(t\) minutes is the function \(G\) that satisfies the differential equation \(\dfrac{dG}{dt} = -(G - 27)^{2/3}\), where \(G(t)\) is measured in degrees Celsius and \(G(0) = 91\). Find an expression for \(G(t)\). Based on this model, what is the internal temperature of the potato at time \(t = 3\)?

During a chemical reaction, the function \(y = f(t)\) models the amount of a substance present, in grams, at time \(t\) seconds. At the start of the reaction (\(t = 0\)), there are 10 grams of the substance present. The function \(y = f(t)\) satisfies the differential equation \(\dfrac{dy}{dt} = -0.02 y^2\).

1. (A) Use the line tangent to the graph of \(y = f(t)\) at \(t = 0\) to approximate the amount of the substance remaining at time \(t = 2\) seconds.
2. (B) Using the given differential equation, determine whether the graph of \(f\) could resemble the following graph. Give a reason for your answer.
3. (C) Find an expression for \(y = f(t)\) by solving the differential equation \(\dfrac{dy}{dt} = -0.02 y^2\) with the initial condition \(f(0) = 10\).
4. (D) Determine whether the amount of the substance is changing at an increasing or a decreasing rate. Explain your reasoning.

Let \(y = f(x)\) be the particular solution to the differential equation \(\dfrac{dy}{dx} = y\cdot(x\ln x)\) with initial condition \(f(1) = 4\). It can be shown that \(f''(1) = 4\).

1. (A) Write the second-degree Taylor polynomial for \(f\) about \(x = 1\). Use the Taylor polynomial to approximate \(f(2)\).
2. (B) Use Euler's method, starting at \(x = 1\) with two steps of equal size, to approximate \(f(2)\). Show the work that leads to your answer.
3. (C) Find the particular solution \(y = f(x)\) to the differential equation \(\dfrac{dy}{dx} = y\cdot(x\ln x)\) with initial condition \(f(1) = 4\).

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function \(M\) models the temperature of the milk at time \(t\), where \(M(t)\) is measured in degrees Celsius (\(^\circ\)C) and \(t\) is the number of minutes since the bottle was placed in the pan. \(M\) satisfies the differential equation \(\dfrac{dM}{dt} = \tfrac{1}{4}(40 - M)\). At time \(t = 0\), the temperature of the milk is \(5^\circ\)C. It can be shown that \(M(t) < 40\) for all values of \(t\).

1. (A) A slope field for the differential equation \(\dfrac{dM}{dt} = \tfrac{1}{4}(40 - M)\) is shown. Sketch the solution curve through the point \((0, 5)\).
2. (B) Use the line tangent to the graph of \(M\) at \(t = 0\) to approximate \(M(2)\), the temperature of the milk at time \(t = 2\) minutes.
3. (C) Write an expression for \(\dfrac{d^2 M}{dt^2}\) in terms of \(M\). Use \(\dfrac{d^2 M}{dt^2}\) to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of \(M(2)\). Give a reason for your answer.
4. (D) Use separation of variables to find an expression for \(M(t)\), the particular solution to the differential equation \(\dfrac{dM}{dt} = \tfrac{1}{4}(40 - M)\) with initial condition \(M(0) = 5\).

The depth of seawater at a location can be modeled by the function \(H\) that satisfies the differential equation \(\dfrac{dH}{dt} = \dfrac{1}{2}(H - 1)\cos\!\left(\dfrac{t}{2}\right)\), where \(H(t)\) is measured in feet and \(t\) is measured in hours after noon (\(t = 0\)). It is known that \(H(0) = 4\).

1. (A) A portion of the slope field for the differential equation is provided. Sketch the solution curve, \(y = H(t)\), through the point \((0, 4)\).
2. (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.
3. (C) Use separation of variables to find \(y = H(t)\), the particular solution to the differential equation \(\dfrac{dH}{dt} = \dfrac{1}{2}(H - 1)\cos\!\left(\dfrac{t}{2}\right)\) with initial condition \(H(0) = 4\).

Let \(y = f(x)\) be the particular solution to the differential equation \(\dfrac{dy}{dx} = (3 - x)y^2\) with initial condition \(f(1) = -1\).

1. (A) Find \(f''(1)\), the value of \(\dfrac{d^2 y}{dx^2}\) at the point \((1, -1)\). Show the work that leads to your answer.
2. (B) Write the second-degree Taylor polynomial for \(f\) about \(x = 1\).
3. (C) The second-degree Taylor polynomial for \(f\) about \(x = 1\) is used to approximate \(f(1.1)\). Given that \(|f'''(x)| \le 60\) for all \(x\) in the interval \(1 \le x \le 1.1\), use the Lagrange error bound to show that this approximation differs from \(f(1.1)\) by at most \(0.01\).
4. (D) Use Euler's method, starting at \(x = 1\) with two steps of equal size, to approximate \(f(1.4)\). Show the work that leads to your answer.

The Taylor series for a function \(f\) about \(x = 1\) is given by \[\sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{2^{n}}{n} (x - 1)^{n}\] and converges to \(f(x)\) for \(|x - 1| < R\), where \(R\) is the radius of convergence of the Taylor series.

1. (A) Find the value of \(R\).
2. (B) Find the first three nonzero terms and the general term of the Taylor series for \(f'\), the derivative of \(f\), about \(x = 1\).
3. (C) The Taylor series for \(f'\) about \(x = 1\), found in part (b), is a geometric series. Find the function \(f'\) to which the series converges for \(|x - 1| < R\). Use this function to determine \(f\) for \(|x - 1| < R\).

The Maclaurin series for a function \(f\) is given by \(\dfrac{x}{3} + \dfrac{x^2}{4} + \dfrac{x^3}{5} + \cdots + \dfrac{x^n}{n + 2} + \cdots\).

1. (A) Use the ratio test to find the interval of convergence of the Maclaurin series for \(f\).
2. (B) Let \(g\) be the function given by \(g(x) = f(-2x)\). Find the first three terms and the general term of the Maclaurin series for \(g\).
3. (C) The first two terms of the Maclaurin series for \(f\) are used to approximate \(f(0.1)\). Given that \(|f'''(x)| \le 2\) for \(0 \le x \le 0.1\), use the Lagrange error bound to show that this approximation differs from \(f(0.1)\) by at most \(\dfrac{1}{3000}\).

The function \(f\) has a Taylor series about \(x = 1\) that converges to \(f(x)\) for all \(x\) in the interval of convergence. It is known that \(f(1) = 1\), \(f'(1) = -\dfrac{1}{2}\), and the \(n\)th derivative of \(f\) at \(x = 1\) is given by \[f^{(n)}(1) = (-1)^{n} \dfrac{(n-1)!}{2^{n}}\] for \(n \ge 2\).

1. (A) Write the first four nonzero terms and the general term of the Taylor series for \(f\) about \(x = 1\).
2. (B) The Taylor series for \(f\) about \(x = 1\) has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer.
3. (C) The Taylor series for \(f\) about \(x = 1\) can be used to represent \(f(1.2)\) as an alternating series. Use the first three nonzero terms of the alternating series to approximate \(f(1.2)\).
4. (D) Show that the approximation found in part (c) is within 0.001 of the exact value of \(f(1.2)\).

A function \(f\) has derivatives of all orders for \(-1 < x < 1\). The derivatives of \(f\) satisfy the conditions \[f(0) = 0, \quad f'(0) = 1, \quad f^{(n+1)}(0) = -n \cdot f^{(n)}(0)\text{ for all }n \ge 1.\] The Maclaurin series for \(f\) converges to \(f(x)\) for \(|x| < 1\).

1. (A) Show that the first four nonzero terms of the Maclaurin series for \(f\) are \(x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4}\), and write the general term of the Maclaurin series for \(f\).
2. (B) Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at \(x = 1\). Explain your reasoning.
3. (C) Write the first four nonzero terms and the general term of the Maclaurin series for \(g(x) = \displaystyle\int_0^{x} f(t)\,dt\).
4. (D) Let \(P_n\!\left(\dfrac{1}{2}\right)\) represent the \(n\)th-degree Taylor polynomial for \(g\) about \(x = 0\) evaluated at \(x = \dfrac{1}{2}\), where \(g\) is the function defined in part (c). Use the alternating series error bound to show that \(\left|P_4\!\left(\dfrac{1}{2}\right) - g\!\left(\dfrac{1}{2}\right)\right| < \dfrac{1}{500}\).

Let \(f\) be a function having derivatives of all orders for all real numbers. Selected values of \(f\) and its first four derivatives are shown in the table above.

1. (A) Write the second-degree Taylor polynomial for \(f\) about \(x = 0\) and use it to approximate \(f(0.2)\).
2. (B) Let \(g\) be a function such that \(g(x) = f(x^3)\). Write the fifth-degree Taylor polynomial for \(g'\), the derivative of \(g\), about \(x = 0\).
3. (C) Write the third-degree Taylor polynomial for \(f\) about \(x = 1\).
4. (D) It is known that \(\left|f^{(4)}(x)\right| \le 300\) for \(0 \le x \le 1.125\). The third-degree Taylor polynomial for \(f\) about \(x = 1\), found in part (c), is used to approximate \(f(1.1)\). Use the Lagrange error bound along with the information about \(f^{(4)}(x)\) to find an upper bound on the error of the approximation.

Consider the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}(x - 3)^n}{5^n \cdot n^p}\), where \(p\) is a constant and \(p > 0\).

1. (A) For \(p = 3\) and \(x = 8\), does the series converge absolutely, converge conditionally, or diverge? Explain your reasoning.
2. (B) For \(p = 1\) and \(x = 8\), does the series converge absolutely, converge conditionally, or diverge? Explain your reasoning.
3. (C) When \(x = -2\), for what values of \(p\) does the series converge? Explain your reasoning.
4. (D) When \(p = 1\) and \(x = 3.1\), the series converges to a value \(S\). Use the first two terms of the series to approximate \(S\). Use the alternating series error bound to show that this approximation differs from \(S\) by less than \(\dfrac{1}{300{,}000}\).

The function \(g\) has derivatives of all orders for all real numbers. The Maclaurin series for \(g\) is given by \[g(x) = \sum_{n=0}^{\infty} \dfrac{(-1)^n x^n}{2e^n + 3}\] on its interval of convergence.

1. (A) State the conditions necessary to use the integral test to determine convergence of the series \(\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{e^n}\). Use the integral test to show that \(\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{e^n}\) converges.
2. (B) Use the limit comparison test with the series \(\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{e^n}\) to show that the series \(g(1) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{2e^n + 3}\) converges absolutely.
3. (C) Determine the radius of convergence of the Maclaurin series for \(g\).
4. (D) The first two terms of the series \(g(1) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{2e^n + 3}\) are used to approximate \(g(1)\). Use the alternating series error bound to determine an upper bound on the error of the approximation.

The function \(f\) is defined by the power series \[f(x) = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \cdots + \dfrac{(-1)^n x^{2n+1}}{2n+1} + \cdots\] for all real numbers \(x\) for which the series converges.

1. (A) Using the ratio test, find the interval of convergence of the power series for \(f\). Justify your answer.
2. (B) Show that \(\left| f\!\left(\tfrac{1}{2}\right) - \tfrac{1}{2} \right| < \tfrac{1}{10}\). Justify your answer.
3. (C) Write the first four nonzero terms and the general term for an infinite series that represents \(f'(x)\).
4. (D) Use the result from part (c) to find the value of \(f'\!\left(\tfrac{1}{6}\right)\).

The function \(f\) has derivatives of all orders for all real numbers. It is known that \(f(0) = 2\), \(f'(0) = 3\), \(f''(x) = -f(x^2)\), and \(f'''(x) = -2x\cdot f'(x^2)\).

1. (A) Find \(f^{(4)}(x)\), the fourth derivative of \(f\) with respect to \(x\). Write the fourth-degree Taylor polynomial for \(f\) about \(x = 0\). Show the work that leads to your answer.
2. (B) The fourth-degree Taylor polynomial for \(f\) about \(x = 0\) is used to approximate \(f(0.1)\). Given that \(\left|f^{(5)}(x)\right| \le 15\) for \(0 \le x \le 0.5\), use the Lagrange error bound to show that this approximation is within \(\dfrac{1}{10^5}\) of the exact value of \(f(0.1)\).
3. (C) Let \(g\) be the function such that \(g(0) = 4\) and \(g'(x) = e^x f(x)\). Write the second-degree Taylor polynomial for \(g\) about \(x = 0\).

The Maclaurin series for a function \(f\) is given by \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(n+1)x^n}{n^2\,6^n}\) 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 \(6\).

1. (A) Determine whether the Maclaurin series for \(f\) converges or diverges at \(x = 6\). Give a reason for your answer.
2. (B) It can be shown that \(f(-3) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(n+1)(-3)^n}{n^2\,6^n} = \displaystyle\sum_{n=1}^{\infty} \dfrac{n+1}{n^2}\left(-\dfrac{1}{2}\right)^n\) and that the first three terms of this series sum to \(S_3 = -\dfrac{125}{144}\). Show that \(|f(-3) - S_3| < \dfrac{1}{50}\).
3. (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 \(f'\).
4. (D) Let \(g(x) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(n+1)x^{2n}}{n^2\,3^n}\). Use the ratio test to determine the radius of convergence of the Maclaurin series for \(g\).

The Taylor series for a function \(f\) about \(x = 4\) is given by \[\sum_{n=1}^{\infty} \frac{(x-4)^{n+1}}{(n+1)\,3^n} = \frac{(x-4)^2}{2 \cdot 3} + \frac{(x-4)^3}{3 \cdot 3^2} + \frac{(x-4)^4}{4 \cdot 3^3} + \cdots + \frac{(x-4)^{n+1}}{(n+1)\,3^n} + \cdots\] and converges to \(f(x)\) on its interval of convergence.

1. (A) Using the ratio test, find the interval of convergence of the Taylor series for \(f\) about \(x = 4\). Justify your answer.
2. (B) Find the first three nonzero terms and the general term of the Taylor series for \(f'\), the derivative of \(f\), about \(x = 4\).
3. (C) The Taylor series for \(f'\) described in part B is a geometric series. For all \(x\) in the interval of convergence of the Taylor series for \(f'\), show that \(f'(x) = \dfrac{x - 4}{7 - x}\).
4. (D) It is known that the radius of convergence of the Taylor series for \(f\) about \(x = 4\) is the same as the radius of convergence of the Taylor series for \(f'\) about \(x = 4\). Does the Taylor series for \(f'\) described in part B converge to \(f'(x) = \dfrac{x - 4}{7 - x}\) at \(x = 8\)? Give a reason for your answer.