Riemann Sums & Definite Integrals (College Board AP® Calculus BC): Free Response Questions

Let $f$ be a continuous function defined on the closed interval $-4\le x\le 4$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function defined by $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$.

On what open intervals is the graph of $g$ concave up? Give a reason for your answer.

$f$ is a differentiable function. The table shown gives values of the function $f$ and its first derivative at selected values of $x$.

Let $g$ be the function defined by $g\left(x\right)=2x^3+{\int }_0^x{f}^{\text{'}}\left(t\right)dt$. Find $g\left(3\right)$. Show the work that leads to your answer.

Is the function $g$ defined in part (a) increasing, decreasing, or neither at $x=3$? Justify your answer.

The density of algae on the surface of a circular garden pond at a distance of $r$ meters from the center of the pond is given by an increasing, differentiable function $f$, where $f\left(r\right)$ is measured in kilograms per square meter. Values of $f\left(r\right)$ for selected values of $r$ are given in the table above.

The total mass, in kilograms, of algae in the pond is given by the integral expression $2\pi {\int }_0^{3}rf\left(r\right)dr$. Approximate the value of $2\pi {\int }_0^{3}rf\left(r\right)dr$ using a right Riemann sum with the four subintervals indicated by the data in the table.

Is the approximation found in part (a) an overestimate or underestimate of the total mass of algae in the pond? Explain your reasoning.

The temperature of coffee in a mug at time $t$ is modeled by a strictly decreasing, twice-differentiable function $M$, where $M\left(t\right)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t=0$, the temperature of the coffee is $212°\mathrm{F}$. The mug of coffee is then left to cool, beginning at time $t=0$. Values of $M\left(t\right)$ at selected times $t$ for the first 15 minutes are given in the table above.

Use the data in the table to evaluate ${\int }_0^{15}{M}^{\text{'}}\left(t\right)dt$. Using correct units, interpret the meaning of ${\int }_0^{15}{M}^{\text{'}}\left(t\right)dt$ in the context of this problem.

For $0\le t\le 15$, the average temperature of the water in the mug is $\frac{1}{15}{\int }_0^{15}M\left(t\right)dt$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac{1}{15}{\int }_0^{15}M\left(t\right)dt$. Does this approximation overestimate or underestimate the average temperature of the coffee over these 15 minutes? Explain your reasoning.

For $15\le t\le 20$, the function $M$ that models the coffee temperature has first derivative given by $M^{\text{'}}\left(t\right)=-18.45e^{-0.125t}$. Based on the model, what is the temperature of the coffee at time $t=20$?

Let $f$ be the function defined by $f\left(x\right)={\int }_1^x{\left(\frac{1}{t}+t\right)}^{2}dt$ for $x\ge 1$.

Find $f\left(3\right)$, $f\text{'}\left(3\right)$ and $f\text{'}\text{'}\left(3\right)$.

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

Find $f\left(0\right)$ and $f\left(5\right)$.

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 $G\left(x\right)={\int }_0^{x}f\left(t\right)\text{d}t$.

On what open intervals is the graph of $G$ concave up? Give a reason for your answer.

Let $P$ be the function defined by $P\left(x\right)=G\left(x\right)·f\left(x\right)$. Find $P\text{'}\left(3\right)$.

$f$ is a continuous function which is increasing on the interval $\left[0,40\right]$. The table shown gives values of the function $f$ at selected values of $x$.

Using relevant approximations with all the data in table, find an interval of shortest width which contains the exact value of ${\int }_0^{40}f\left(x\right)dx$. Justify why your interval is valid.

The function $f$ is defined on the closed interval $\left[0,9\right]$ and satisfies $f\left(3\right)=2$. The graph of $f^{\text{'}}$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure above.

Find the absolute minimum value of $f$ on the closed interval $\left[0,9\right]$. Justify your answer.

The graph of the differentiable function $f$, shown for $-3\le x\le 4$, is linear for $0\le x\le 4$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x=-3$ and the $x$- and $y$- axes. The area of region $R$ is 12.

The function $g$ is defined by $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$. Find the values of $g\left(-3\right)$, $g\left(2\right)$ and $g\left(3\right)$.

The function $h$ is defined by $h\left(x\right)={\int }_{-3}^{x}f\text{'}\left(t\right)dt$. Find the values of $h\left(3\right)$, $h\text{'}\left(3\right)$ and $h\text{'}\text{'}\left(3\right)$.

A person is on a roller coaster ride which rises vertically rapidly. The height of the person from the ground is modeled by the twice-differentiable function $h\left(t\right)$ for $0\le t\le 50$, where $t$ is measured in seconds and $h\left(t\right)$ is measured in feet. Values of $h\text{'}\left(t\right)$ at selected values of $t$ are shown in the table above. Initially, the person is 1 foot above the ground.

Use a trapezoidal sum with five subintervals indicated by the table to estimate the height of the person at time $t=50$ seconds.

It is known that the function $h\text{'}\text{'}\left(t\right)$ is increasing on the interval $0\le t\le 50$.

Is your approximation in part (a) greater than or less than the height of the person from the ground, as predicted by the model, at time $t=50$ seconds? Give a reason for your answer.

Let $f$ be the function defined by $f\left(x\right)=\frac{4x^2+x+3}{2x^2}$ for $x>0$.

Evaluate ${\int }_1^{3}f\left(x\right)dx$.

Let $g$ be the function defined by $g\left(x\right)={\int }_1^{x}f\text{'}\text{'}\left(t\right)dt$ for $x>0$.

Evaluate ${\int }_1^{3}g\left(x\right)dx$.

The continuous function $f$ is defined on the closed interval $-6\le x\le 5$. The figure below shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $(5,3)$. It is known that the point $\left(3,3-\sqrt{5}\right)$ is on the graph of $f$.

If ${\int }_{-6}^{5}f\left(x\right)dx=7$, find the value of ${\int }_{-6}^{-2}f\left(x\right)dx$. Show the work that leads to your answer.

Evaluate ${\int }_3^5\left(2f\text{'}\left(x\right)+4\right)dx$.

The function $g$ is given by $g\left(x\right)={\int }_{-2}^{x}f\left(t\right)dt$. Find the absolute maximum value of $g$ on the interval $-2\le x\le 5$. Justify your answer.

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

If $f\left(1\right)=3$, what is the value of $f(-5)$?

Evaluate ${\int }_1^{6}g\left(x\right) dx$.

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

Find the values of $f(-6)$ and $f\left(5\right)$.

The figure above shows the graph of the piecewise-linear function $f$. For $-4\le x\le 12$, the function $g$ is defined by

$g\left(x\right)={\int }_2^{x}f\left(t\right) dt$

Does $g$ have a relative minimum, a relative maximum, or neither at $x=10$? Justify your answer.

Does the graph of $g$ have a point of inflection at $x=4$? Justify your answer.

Find the absolute minimum value and the absolute maximum value of $g$ on the interval $-4\le x\le 12$. Justify your answers.

For $-4\le x\le 12$, find all intervals for which $g\left(x\right)\le 0$.

The graph of the differentiable function $f$, shown below 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.

The function $g$ is defined by $g\left(x\right)={\int }_0^{x}f\left(t\right) \text{d}t$. Find the values of $g(-6)$, $g\left(4\right)$, and $g\left(6\right)$.

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.

The function $h$ is defined by $h\left(x\right)={\int }_{-6}^{x}f\text{'}\left(t\right) \text{d}t$. Find the values of $h\left(6\right)$, $h\text{'}\left(6\right)$, and $h\text{'}\text{'}\left(6\right)$. Show the work that leads to your answers.