2017 Free Response

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.

When a certain grocery store opens, it has 50 pounds of bananas on a display table. Customers remove bananas from the display table at a rate modeled by

where $f(t)$ is measured in pounds per hour and $t$ is the number of hours after the store opened. After the store has been open for three hours, store employees add bananas to the display table at a rate modeled by

where $g(t)$ is measured in pounds per hour and $t$ is the number of hours after the store opened.

[Graph of $f'$, labeled "Graph of $f'$"; consists of a semicircle and three line segments. x-axis $x$, y-axis $y$. A line segment starts at the labeled point $(-6, 2)$ and decreases linearly to the point $(-2, 0)$ on the x-axis. From $(-2, 0)$ to $(0, 0)$ the graph is a semicircle dipping below the x-axis (minimum around $(-1, -1)$) and returning to the x-axis at $(0,0)$. From $(0,0)$ the graph rises linearly to the labeled point $(3, 2)$. From $(3,2)$ the graph decreases linearly to the point $(5, 0)$ on the x-axis. Gridlines/tick marks are shown at integer x-values and at $y=1$.]

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.

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 ($°C$) at time $t = 0$, and the internal temperature of the potato is greater than $27°C$ 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 $H(0) = 91$.

Two particles move along the $x$-axis. For $0 \le t \le 8$, the position of particle $P$ at time $t$ is given by $x_P(t) = \ln\!\left(t^2 - 2t + 10\right)$, while the velocity of particle $Q$ at time $t$ is given by $v_Q(t) = t^2 - 8t + 15$.

Particle $Q$ is at position $x = 5$ at time $t = 0$.

Let $f$ be the function defined by $f(x) = \cos(2x) + e^{\sin x}$.

Let $g$ be a differentiable function. The table above gives values of $g$ and its derivative $g'$ at selected values of $x$.

Let $h$ be the function whose graph, consisting of five line segments, is shown in the figure above.

[Graph of $h$, labeled "Graph of $h$", on a gridded coordinate plane with axes $x$ and $y$; gridlines at each integer, with $1$ marked on both axes near the origin. The graph consists of five line segments: starting at approximately $(-5, 3)$, rising to a flat peak spanning roughly $(-3.5, 4)$ to $(-3, 4)$, then decreasing linearly, crossing the x-axis slightly to the right of the y-axis (around $x = 0.5$), continuing to decrease to a local minimum at approximately $(2, -1)$, then rising steeply and linearly to approximately $(4, 5)$.]