AP Calculus BC Quiz: Estimating Derivatives Of A Function

The height of a plant, $H(t)$, in centimeters, is a differentiable function of time $t$ in days. Measurements are recorded as follows: $H(10) = 20$, $H(12) = 24$, and $H(15)=33$.

Which of the following is the best estimate of the growth rate $H'(12)$ in centimeters per day?

The temperature $T$, in degrees Celsius, of a chemical reaction is a differentiable function of time $t$, in minutes. At time $t=5$ minutes, the temperature is $40^{\circ}C$. At time $t=5.5$ minutes, the temperature is $48^{\circ}C$.

Based on this information, which of the following is the best estimate for the rate of change of the temperature, in degrees Celsius per minute, at time $t=5$ minutes?

The depth of water in a reservoir, $D(t)$, is a differentiable function of time $t$. On a certain day, at $t=12$ hours, the depth is 50 feet. Two hours prior, the depth was 50.8 feet, and two hours later, the depth is 49.6 feet.

What is the best estimate for $D'(12)$, the rate of change of the depth at $t=12$ hours, in feet per hour?

The derivative of a function $f$ at $x=c$, denoted $f'(c)$, is best approximated by the slope of a secant line through two points on the graph of $f$. For the most accurate approximation, the two points should be:

The velocity of a particle, $v(t)$, in meters per second, is a differentiable function of time $t$ in seconds. At $t=2$ seconds, the velocity is $v(2) = 10$ m/s. At $t=2.1$ seconds, the velocity is $v(2.1)=10.5$ m/s.

Based on this information, what is a possible estimate for the acceleration of the particle at $t=2$ seconds, in m/s$^2$?

A tank is being filled with water. The volume $V(t)$ of water in the tank, in gallons, is a differentiable function of time $t$ in hours. At $t=2$ hours, the volume is 300 gallons. A half-hour earlier, at $t=1.5$ hours, the volume was 280 gallons.

Based on this information, what is the best estimate for the rate at which the volume is changing at $t=2$ hours, in gallons per hour?

A function $g(x)$ is differentiable. Selected values of $g(x)$ are $g(1.9) = 4.2$, $g(2.0) = 4.5$, $g(2.1) = 4.9$, and $g(2.2) = 5.4$.

Which of the following is the best estimate for $g'(2)$?

Let $f(x) = x^2 \cos(\sqrt{x})$. It is given that $f(3.99) \approx 6.431$ and $f(4.01) \approx 6.277$.

Which of the following is the best approximation for $f'(4)$?

For a differentiable function $h(t)$, the following values are known: $h(4.8) = 10.2$, $h(5.0) = 10.8$, and $h(5.2) = 11.6$.

Which of the following is the best estimate for $h'(5.0)$?

The population of a town, $P(t)$, in thousands, is a differentiable function of time $t$ in years since the beginning of 2010. It is known that $P(8) = 150$ and $P(10) = 162$.

Which of the following is the best estimate for the rate at which the population was growing at the beginning of 2019 ($t=9$), in thousands of people per year?

Let $f$ be a differentiable function with $f(3.4) = 7.1$ and $f(3.8) = 6.5$.

Which of the following is the best approximation for $f'(3.6)$?

A differentiable function $g$ has values $g(4.98) = 1.25$ and $g(5.02) = 1.35$.

Using the secant line between $x=4.98$ and $x=5.02$ as an approximation to the tangent line at $x=5$, what is the estimated value of $g'(5)$?

Let $f(x)$ be a differentiable function. The expression $D(h) = \frac{f(c+h) - f(c-h)}{2h}$ is used to approximate $f'(c)$.

If $f(x) = x^3$, which of the following is the value of $D(h)$ used to approximate $f'(c)$?

To estimate the value of $f'(2)$ for the function $f(x) = x^3 - x$, an analyst uses the slope of the secant line connecting the points on the graph of $f$ at $x=1$ and $x=3$. What is the value of this estimate?

The amount of a certain drug in a patient's bloodstream, $A(t)$, in milligrams (mg), is a differentiable function of time $t$ in hours. At $t=2$ hours, $A(2) = 15$ mg. At $t=2.5$ hours, $A(2.5) = 12$ mg.

Using this data, what is the approximate rate of change of the amount of the drug in the bloodstream at $t=2.25$ hours, in mg per hour?

A function $f$ is differentiable for all real numbers. If $f(2.9) = 5.8$ and $f(3.1) = 6.2$, which of the following is the best estimate for $f'(3)$?

The quantity $Q = \frac{g(a+0.01) - g(a)}{0.01}$ is calculated for a differentiable function $g$. Which of the following statements about $Q$ is the most accurate?

Let $k$ be a differentiable function. For a small positive value $h$, it is known that $k(3) - k(3-h) \approx 2h$. Which of the following is the best estimate for $k'(3)$?

Let $f$ be a differentiable function. Given $f(x_0 - \Delta x) = y_1$ and $f(x_0 + \Delta x) = y_2$, where $\Delta x > 0$. Which expression represents the best estimate for $f'(x_0)$?

Let $f$ be a differentiable function. The average rate of change of $f$ on the interval $[c, c+h]$ is given by the expression $4 - 2h$. Which of the following is the value of $f'(c)$?