AP Calculus AB FRQ Room

Analyzing a Velocity Function with Nested Discontinuities

A particle’s velocity along a line is given by $$v(t)= \frac{(t-1)(t+3)}{(t-1)*\ln(t+2)}$$ for $$t>0

Arithmetic Sequence in Temperature Data and Continuity Correction

A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

Compound Interest and Geometric Series

A bank account accrues interest compounded annually at an annual rate of 10%. The balance after $$n$

Continuity Analysis of a Radical Function

Consider the function $$f(x) = \frac{\sqrt{x+4} - 2}{x}$$. (a) Evaluate $$\lim_{x \to 0} f(x)$$. (b

Continuity and Limit Comparison for Two Particle Paths

Two particles, A and B, travel along the same line. Their position functions are given by $$s_A(t)=

Determining Parameters for Continuity

Consider the function $$f(x)= \begin{cases} 2*x + k, & x < 1 \\ x^2, & x \geq 1 \end{cases}$$, where

Discontinuity in Acceleration Function and Integration

A particle’s acceleration is defined by the piecewise function $$a(t)= \begin{cases} \frac{1-t}{t-2}

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Factoring a Cubic Expression for Limit Evaluation

Consider the function $$f(x)= \frac{x^3 - 8}{x - 2}$$ for x \(\neq\) 2. Answer the following parts.

Graph Analysis of Discontinuities

A graph of a function f(x) shows a jump discontinuity at x = 1 and a removable discontinuity (a hole

Identifying Discontinuities in a Rational Function

Consider the function $$f(x)=\frac{x^2 - 9}{x - 3}$$ defined for $$x \neq 3$$. Answer the following

Intermediate Value Theorem Application

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

Inverse Function and Limit Behavior Analysis

Consider the function $$f(x)= \frac{x-1}{x+2}$$.

Investigating Discontinuities in a Rational Function

Consider the function $$ h(x)=\frac{x^2-4}{x-2} $$ for $$x\ne2$$.

Limit Involving an Exponential Function

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2*x} - 1}{x}$$.

Limits at Infinity and Horizontal Asymptotes

Given the rational function $$ g(x)=\frac{3x^2+5}{2x^2-7} $$, analyze its behavior as $$x\to\infty$$

Limits at Infinity for Non-Rational Functions

Consider the function $$ h(x)=\frac{2*x+3}{\sqrt{4*x^2+7}} $$.

Limits of Absolute Value Functions

Consider the function $$f(x)= \frac{|x-2|}{x-2}$$. Answer the following:

One-Sided Limits and Absolute Value Functions

Let $$f(x) = \frac{|x - 2|}{x - 2}$$. Analyze its behavior as x approaches 2.

One-Sided Limits and Continuity of a Piecewise Function

Consider the piecewise function $$w(x)= \begin{cases} \frac{e^{x}-1}{x} & \text{if } x<0, \\ \frac{\

Piecewise Function Continuity and IVT

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ a*x+b, & x > 1 \end{cases}$$. Determine constants a and

Rational Function Analysis

Consider the rational function $$f(x)=\frac{(x+3)*(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the

Removable Discontinuity and Limit

Consider the function $$ f(x)=\frac{x^2-9}{x-3} $$ for $$ x\ne3 $$, which is not defined at $$ x=3 $

Squeeze Theorem Application

Let $$f(x)=x^2\sin(1/x)$$ for \(x\neq 0\) and define \(f(0)=0\). Use the Squeeze Theorem to complete

Squeeze Theorem Application

Consider the function $$f(x)=x^2\sin(\frac{1}{x})$$ for $$x\neq0$$ and $$f(0)=0$$. Answer the follow

Squeeze Theorem with an Oscillatory Term

Consider the function $$f(x) = x^2 \cdot \cos\left(\frac{1}{x^2}\right)$$ for $$x \neq 0$$, and defi

Trigonometric Limit Computation

Consider the function $$f(x)= \frac{\sin(5*(x-\pi/4))}{x-\pi/4}$$.

Vertical Asymptote Analysis

Consider the function $$f(x)=\frac{x+2}{x-1}$$. Answer the following:

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Analysis of Temperature Change via Derivatives

The temperature in a chemical reactor is modeled by $$T(x)=x^3 - 6*x^2 + 9*x$$, where $$T(x)$$ is in

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

Consider the function $$f(x)= x^2$$.

Analyzing the Derivative of a Trigonometric Function

Consider the function $$f(x)= \sin(x) + \cos(x)$$.

Application of Derivative Rules

Let $$f(x)=7*x^3-5*x+9$$. Using the given rules, answer the following:

Curve Analysis – Increasing and Decreasing Intervals

Given the function $$f(x)= x^3 - 3*x^2 + 2$$, analyze its behavior.

Derivation of $$h(x)= \ln(2*x+3)$$ Using the Chain Rule

Let $$h(x)= \ln(2*x+3)$$, a composition of a logarithmic and a linear function.

Derivative from First Principles

Derive the derivative of the polynomial function $$f(x)=x^3+2*x$$ using the limit definition of the

Derivative from First Principles: The Function $$f(x)=\sqrt{x}$$

Consider the function $$f(x) = \sqrt{x}$$. Use the definition of the derivative to find an expressio

Derivative of a Trigonometric Function

Let \(f(x)=\sin(2*x)\). Answer the following parts.

Derivatives and Optimization in a Real-World Scenario

A company’s profit is modeled by $$P(x)=-2*x^2+40*x-150$$, where $$x$$ represents the number of item

Deriving the Derivative from First Principles for a Reciprocal Square Root Function

Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f

Difference Quotient and Derivative of a Rational Function

Consider the function $$f(x)= \frac{x+2}{x-3}$$.

Differentiability and Continuity

A function is defined piecewise as follows: $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x +

Finding the Derivative Using First Principles

Consider the function $$f(x)= 5*x^3 - 4*x + 7$$. Use the definition of the derivative to find the de

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

Higher-Order Derivatives in Motion

A particle moves along a line with its position given by $$s(t)= t^3 - 6*t^2 + 9*t + 5$$, where $$t$

Highway Traffic Flow Analysis

Vehicles enter a highway ramp at a rate given by $$f(t)=60+4*t$$ (vehicles/min) and exit the highway

Identifying Points of Non-Differentiability

Consider the function $$h(x)= |2*x - 5|$$.

Implicit Differentiation in Demand Analysis

Consider the implicitly defined demand function $$x^2 + x*y + y^2 = 100$$, where x represents the pr

Instantaneous Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

Inverse Function Analysis: Quadratic Transformation

Consider the function $$f(x)=x^2+2*x+2$$ with the domain restricted to $$x\geq -1$$ so that f is one

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

Piecewise Function and Discontinuities

A piecewise function $$f$$ is defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x\ne

Product Rule with Exponential Function

Consider the function $$f(x)= x*e^{x}$$ which exhibits both polynomial and exponential behavior.

Tangent Line and Instantaneous Rate at a Point with a Radical Function

Consider the function $$f(x)= (x+4)^{1/2}$$, which represents a physical measurement (with the domai

Using Derivative Rules on a Trigonometric Function

Consider the function $$f(x)=3*\sin(x)+\cos(2*x)$$. Answer the following questions:

Using the Difference Quotient with a Polynomial Function

Let $$g(x)=2*x^2 - 5*x + 3$$. Answer the following questions:

Advanced Composite Function Differentiation with Multiple Layers

Consider the function $$f(x)= \ln\left(\sqrt{1+e^{3*x}}\right)$$.

Advanced Implicit and Inverse Function Differentiation on Polar Curves

Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,

Chain Rule in an Economic Model

In an economic model, the cost function for producing a good is given by $$C(x)=(3*x+1)^5$$, where $

Composite and Rational Function Differentiation

Let $$P(x)=\frac{x^2}{\sqrt{1+x^2}}$$.

Composite Function and Inverse Analysis via Graph

Consider the function $$f(x)= \sqrt{4*x-1}$$, defined for $$x \geq \frac{1}{4}$$. Analyze the functi

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Composite Function in a Real-World Fuel Consumption Problem

A company models its fuel consumption with the function $$C(t)=\ln(5*t^2+7)$$, where $$t$$ represent

Composite Function Kinematics

A particle moves along a straight line with its position given by $$s(t) = (2*t+3)^4$$. Analyze the

Composite Function Modeling in Finance

A bank models the growth of a savings account by the function $$B(t)= f(g(t))$$, where $$g(t)= \ln(t

Composite Function via Chain Rule in a Financial Context

A company’s profit (in dollars) based on production level (in thousands of units) is modeled by the

Differentiation Involving Exponentials and Inverse Trigonometry

Consider the function $$M(x)=e^{\arctan(x)}\cdot\cos(x)$$.

Differentiation of a Log-Exponential-Trigonometric Composite

Consider the function $$f(x)= \ln\left(e^(\cos(x)) + x^2\right)$$. Solve the following:

Differentiation of Inverse Trigonometric Composite Function

Given the function $$y = \arctan(\sqrt{x})$$, answer the following parts.

Graph Analysis of a Composite Motion Function

A displacement function representing the motion of an object is given by $$s(t)= \ln(2*t+3)$$. The g

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

Implicit Differentiation in a Hyperbola

Consider the hyperbola defined by $$x*y=10$$. Answer the following parts.

Implicit Differentiation in a Population Growth Model

Consider the model $$e^{x*y} + x - y = 5$$ that relates time \(x\) to a population scale value \(y\)

Implicit Differentiation in an Ellipse

Consider the ellipse defined by $$4*x^2+9*y^2=36$$.

Implicit Differentiation in an Elliptical Orbit

The orbit of a satellite is given by the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Answer the

Implicit Differentiation in an Elliptical Orbit

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$, which can model the orbit of a satellite.

Implicit Differentiation in an Exponential Context

Consider the equation $$e^{x*y}+x=y$$. Answer the following:

Implicit Differentiation in Circular Motion

Given the circle defined by $$x^2 + y^2 = 16$$, analyze its differential properties.

Implicit Differentiation Involving Sine

Consider the equation $$\sin(x*y)+x-y=0$$.

Implicit Differentiation with Exponential and Trigonometric Functions

Consider the equation $$e^{y}\cos(x)+ x*y=1$$. Answer the following:

Implicit Differentiation with Mixed Terms

Consider the equation $$x*y + y^2 = 10$$. Answer the following parts.

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Implicit Differentiation with Product Rule

Consider the equation $$x*e^{y} + y*\ln(x)=5$$. Answer the following:

Implicit Differentiation with Trigonometric Terms

Consider the implicit equation $$\sin(x*y)+x^2=y^2$$. Find the derivative $$\frac{dy}{dx}$$.

Inverse Derivative of a Sum of Exponentials and Linear Terms

Let $$f(x)= e^(x)+ x$$ and let g be its inverse function satisfying $$g(f(x))= x$$. Answer the follo

Inverse Function Derivative for a Logarithmic Function

Let $$f(x)=\ln(x+1)-\sqrt{x}$$, which is one-to-one on its domain.

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ and assume it is invertible. Answer the following:

Inverse Function Differentiation Involving a Polynomial

Let $$f(x)= x^3 + 2*x + 1$$. Analyze its invertibility and the derivative of its inverse function.

Projectile Motion and Composite Function Analysis

A projectile is launched and its height $$h(t)$$ (in meters) is recorded at various times t (in seco

Analyzing Position Data with Table Values

A moving object’s position, given by $$x(t)$$ in meters, is recorded in the table below. Use the dat

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is given by $$C(t)=100e^{-0.05*t}$$ mg/L, wh

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Designing a Flower Bed: Optimal Shape

A landscape designer wants to create a rectangular flower bed with a fixed area of 200 square meters

Economic Efficiency in Speed

A vehicle’s fuel consumption per mile (in dollars) is modeled by the function $$C(v)=0.05*v^2 - 3*v

Economics: Marginal Revenue Analysis

A firm’s revenue function is given by $$R(x)=\frac{100x}{x+5}$$ (in dollars), where $$x$$ represents

Estimating Instantaneous Rates from Discrete Data

In a laboratory experiment, the concentration of a chemical (in molarity, M) is recorded over time (

Estimating Small Changes using Differentials

In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame

Expanding Oil Spill: Related Rates Problem

An oil spill forms a circular patch on the water with area $$A = \pi r^2$$. The area is increasing a

Exponential Decay in Radioactive Material

A radioactive substance decays according to $$M(t)=M_0e^{-0.07t}$$, where $$M(t)$$ is the mass remai

FRQ 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

FRQ 10: Chemical Kinetics Analysis

In a chemical reaction, the concentration of reactant A, denoted by [A], and time t (in minutes) are

FRQ 14: Optimizing Box Design with Fixed Volume

A manufacturer wants to design an open-top box with a fixed volume of $$V = x^2*y = 32$$ cubic units

FRQ 18: Chemical Reaction Concentration Changes

During a chemical reaction, the concentrations of reactants A and B are related by $$[A]^2 + 3*[A]*[

Inflating Balloon Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

Inverse Function Analysis in a Real-World Model

Consider the function $$f(x)=x^3+1$$ which models a certain transformation of measurable quantities.

Kinematics on a Straight Line

A particle moves along a straight line with a position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, wher

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

L'Hôpital's Rule in Chemical Kinetics

In a chemical kinetics experiment, the reaction rate is modeled by the function $$f(x)=\frac{\ln(1+3

Limit Evaluation Using L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 4x^2 + 1}{7x^3 + 2x - 6}$$.

Linear Approximation in Estimating Function Values

Let $$f(x)= \ln(x)$$. Analyze its linear approximation.

Linear Approximation of Natural Logarithm

Estimate $$\ln(1.05)$$ using linear approximation for the function $$f(x)=\ln(x)$$ at $$a=1$$.

Linearization and Differentials

Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.

Linearization for Function Estimation

Use linear approximation to estimate the value of $$\ln(4.1)$$. Let the function be $$f(x)=\ln(x)$$

Linearization of a Machine Component's Length

A machine component's length is modeled by $$L(x)=x^4$$, where x is a machine setting in inches. Use

Logarithmic Profit Optimization

A company’s profit is modeled by $$P(x) = 50x \ln(x) - 100x$$, where $$x$$ (in thousands) is the num

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Motion Analysis of a Particle on a Line

A particle’s position is modeled by $$s(t)=3t^3-6t^2+2t+1$$, where s is in meters and t in seconds.

Population Growth Rate Analysis

A town's population is modeled by the exponential function $$P(t) = 500e^{0.03t}$$, where $$t$$ is i

Related Rates: Expanding Circular Ripple

A ripple in a still pond expands in the shape of a circle. The area of the ripple is given by $$A=\p

Related Rates: Inflating Balloon

A spherical balloon is being inflated such that its volume increases at a rate of $$15\;cm^3/s$$. Th

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

Vehicle Position and Acceleration

A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where

Analyzing Continuity and Discontinuity in a Function with a Square Root

Consider the function $$ f(x) = \begin{cases} \frac{\sqrt{x+4}-2}{x}, & x < 0, \\ 1 + \sqrt{1+x}, &

Analyzing Increasing/Decreasing Behavior of a Cubic Polynomial

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Analyze the function's behavior in terms of i

Application of Rolle's Theorem

Let $$f(x)$$ be a function that is continuous on $$[0,5]$$ and differentiable on $$(0,5)$$ with $$f(

Application of the Mean Value Theorem on a Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x < 2, \\ 4x - 4, & x \ge 2. \end{cases} $$ A

Average Value of a Function and Mean Value Theorem for Integrals

Consider the function $$f(x)= e^{-x}$$ on the interval $$[0, 2]$$. Answer the following:

Bacterial Culture Growth: Identifying Critical Points from Data

A microbiologist records the population of a bacterial culture (in millions) at different times (in

Behavior Analysis of a Logarithmic Function

Consider the function $$f(x)= \frac{\ln(x)}{x}$$ for $$x>0$$. Analyze the critical points and concav

Composite Function with Piecewise Exponential and Logarithmic Parts

Consider the function $$ f(x) = \begin{cases} e^{x}-1, & x < 2, \\ \ln(x+1), & x \ge 2. \end{cases}

Evaluating Rate of Change in Electric Current Data

An electrical engineer recorded the current (in amperes) in a circuit over time. The table below sho

Garden Fence Optimization Problem

A rectangular garden is to be built adjacent to a building. Fencing is required on only three sides

Inverse Analysis of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$. Perform the following analysis regarding its inver

Investigating a Piecewise Function with a Vertical Asymptote

Consider the function $$ f(x) = \begin{cases} \frac{x^2-1}{x-1}, & x < 1, \\ 3, & x = 1, \\ 2x+1, &

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

Consider the function $$ f(x) = \begin{cases} \frac{x^2-9}{x-3}, & x < 3, \\ \frac{x^2-9}{x-3} + 4,

Jump Discontinuity in a Piecewise Linear Function

Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end

Logarithmic Transformation of Data

A scientist models an exponential relationship between variables by the equation $$y= A*e^{k*x}$$. T

Logistic Population Model Analysis

Consider the logistic model $$P(t)= \frac{500}{1+ 9e^{-0.4t}}$$, where $$t$$ is in years. Answer the

Mean Value Theorem Applied to Exponential Functions

Consider the function $$f(x)= e^{x}$$ on the interval \([0, 1]\). Answer the following:

Mean Value Theorem for a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x \le 2, \\ 4x - 4, & x > 2, \end

Modeling Disease Spread with an Exponential Model

In an epidemic, the number of infected individuals is modeled by $$I(t)= I_0 * e^{r*t}$$, where $$t$

Optimization of an Open-Top Box

A company is designing an open-top box with a square base. The volume of the box is modeled by the f

Optimizing a Box with a Square Base

A company is designing an open-top box with a square base of side length $$x$$ and height $$h$$. The

Piecewise Function and the Mean Value Theorem

Consider the piecewise function $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x - 1 & \text{

Trigonometric Function Behavior

Consider the function $$f(x)= \sin(x) + \cos(2*x)$$ defined on the interval $$[0,2\pi]$$. Analyze it

Using Derivatives to Solve a Rate-of-Change Problem

A particle’s displacement is given by $$s(t) = t^3 - 9*t^2 + 24*t$$ (in meters), where \( t \) is in

Accumulated Bacteria Growth

A laboratory observes a bacterial colony whose rate of growth (in bacteria per hour) is modeled by t

Accumulation and Total Change in a Population Model

A population grows at a rate given by $$r(t)=0.2*t^2 - t + 5$$ (in thousands per year), where t is i

Antiderivatives and the Constant of Integration in Modelling

A moving car has its velocity modeled by $$v(t)= 5 - 2*t$$ (in m/s). Answer the following parts to o

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

Application of the Fundamental Theorem of Calculus

A particle moves along a straight line with an instantaneous velocity given by $$v(t)=3*t^2+2*t$$ (i

Area Under a Parabola

Consider the quadratic function $$f(x)=x^2 - 4*x + 3$$. Analyze the function on the interval $$[1,4]

Area Under a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for

Average Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Chemical Production via Integration

The production rate of a chemical in a reactor is given by $$r(t)=5*(t-2)^3$$ (in kg/hr) for $$t\ge2

Comparing Riemann Sum and the Fundamental Theorem

Let $$f(x)=3*x^2$$ on the interval $$[1,4]$$.

Consumer Surplus and Definite Integrals in Economics

The demand function for a product is given by $$p(q)= 100 - 2*q$$, where $$p$$ is the price in dolla

Cumulative Solar Energy Collection

A solar panel's power output (in Watts) is recorded during a sunny day at various times. Use the dat

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Economic Cost Function Analysis

A company's marginal cost (in dollars per unit) is recorded at various production levels. Use the da

Electric Charge Accumulation

An electrical circuit records the current (in amperes) at various times during a brief experiment. U

Estimating Accumulated Water Inflow Using Riemann Sums

A water tank fills at varying rates. The table below shows the inflow rate in liters per second at d

Estimating the Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$ and its values are listed in the table belo

Evaluating Total Rainfall Using Integral Approximations

During a storm, the rainfall rate (in inches per hour) was recorded at several times. The table belo

FRQ5: Inverse Analysis of a Non‐Elementary Integral Function

Consider the function $$ P(x)=\int_{0}^{x} e^{t^2}\,dt $$ for x ≥ 0. Answer the following parts.

FRQ12: Inverse Analysis of a Temperature Accumulation Function

The cumulative temperature above freezing over the morning is modeled by $$ T(t)=\int_{0}^{t} (0.8*t

General Antiderivatives and the Constant of Integration

Given the function $$f(x)= 4*x^3$$, address the following questions about antiderivatives.

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Mixed Method Approximation of an Integral

A function $$f(t)$$ that represents a biological rate is recorded over time. Use the table below to

Motion Analysis with Variable Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=8-2*t$$ (in m/s²). The part

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=4*t - 2$$ (in m/s²). Giv

Population Accumulation in a Lake

A researcher is studying a fish population in a lake. The rate of change of the fish population is m

Riemann Sum Approximation for Sin(x)

Consider the function $$f(x)=\sin(x)$$ on the interval $$[0,\pi]$$. Answer the following parts.

Riemann Sum Approximation from a Table

The table below gives values of a function $$f(x)$$ at selected points: | x | 0 | 2 | 4 | 6 | 8 | |

Roller Coaster Work Calculation

An amusement park engineer recorded the force applied by a roller coaster engine (in Newtons) at var

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

Temperature Change Over Time

A region experiences a temperature change over time that is modeled by the derivative function $$T'(

Trapezoidal Approximation for a Changing Rate

The following table represents the flow rate (in L/min) of water entering a tank at various times:

Trigonometric Integral with U-Substitution

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(t)\tan(t)\,dt$$.

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

Volume of a Solid of Revolution Using the Disk/Washer Method

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane bounded by the curves $$y=x$$ and $$y=x^2$$. Cross sections perpe

Volume of a Solid: Exponential Rotation

Consider the region bounded by the curve $$y=e^{-x}$$, the x-axis, and the vertical lines $$x=0$$ an

Water Flow in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t+2$$ (in liters per minute) for $$0 \le t \le 6

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=3*x^{2} - 2*x + 1$$ (in newtons) fo

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Applying the SIPPY Method to $$dy/dx = \frac{4x}{y}$$

Solve the differential equation $$\frac{dy}{dx}=\frac{4x}{y}$$ with the initial condition $$y(0)=5$$

Bacterial Growth under Logistic Model

A bacterial culture grows according to the logistic differential equation $$\frac{dB}{dt}=rB\left(1-

Bernoulli Differential Equation via Substitution

Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ

Chemical Reaction Rate and Concentration Change

The rate of a chemical reaction is described by the differential equation $$\frac{dC}{dt}=-0.3*C^2$$

Chemical Reactor Mixing

In a continuously stirred chemical reactor, a reactant is added at a rate that results in an inflow

Drug Concentration with Continuous Infusion

A drug is administered intravenously such that its blood concentration $$C(t)$$ (in mg/L) follows th

Environmental Contaminant Dissipation in a Lake

A lake has a pollutant concentration $$C(t)$$ (in mg/L) that evolves according to $$\frac{dC}{dt}=-0

Epidemic Spread Modeling

An epidemic in a closed population of 1000 individuals is modeled by the logistic equation $$\frac{d

Implicit Solution of a Differential Equation

The differential equation $$\frac{dy}{dx} = \frac{2x}{1+y^2}$$ requires an implicit solution.

Integrating Factor Initial Value Problem

Solve the initial value problem $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ for $$x>0$$ with $$y(1)=3$$.

Integrating Factor Method

Consider the differential equation $$\frac{dy}{dx} + 2y = e^{-x}$$ with the initial condition $$y(0)

Inverse Function Analysis of a Differential Equation Solution

Consider the function $$f(x)=\sqrt{4*x+9}$$, which arises as a solution to a differential equation i

Investment Account with Continuous Withdrawals

An investment account grows continuously at an annual rate of 5% and experiences continuous withdraw

Logistic Growth Model

A population is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\frac{P}{

Logistic Population Growth

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\left(1

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

Mixing Problem with Time-Dependent Inflow Rate

A tank initially holds 200 L of water with 10 kg of salt. Brine containing 0.2 kg/L of salt flows in

Modeling Cooling with Newton's Law

An object is cooling in a room where the ambient temperature remains constant at $$20^\circ C$$. The

Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=(1-y^2)\cos(x)$$ with the initial condition $$y(0

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-kN$$. If the

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

Salt Mixing in a Tank

A tank initially contains 100 L of water with 5 kg of salt dissolved. Brine with a concentration of

Separable Differential Equation with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\frac{3x^2}{1+\sin(y)}$$ with the initial condition

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

Sketching Solution Curves on a Slope Field

Consider the differential equation $$\frac{dy}{dx}=x-y$$. A slope field for this equation is provide

Slope Field Interpretation for $$\frac{dy}{dx} = y-x$$

A slope field for the differential equation $$\frac{dy}{dx}=y-x$$ is provided in the stimulus. Use t

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Area Between a Function and Its Tangent

A function $$f(x)$$ and its tangent line at $$x=a$$, given by $$L(x)=m*x+b$$, are considered on the

Area Between Curves with Variable Limits

Consider two functions, $$f(x)$$ and $$g(x)$$, whose values are tabulated below. The functions inter

Average Speed from a Velocity Function

A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$

Average Value Calculation for a Polynomial Function

Consider the function $$f(x)=2*x^2-3*x+1$$ defined on the interval $$[0,5]$$. Compute the average va

Average Value of a Deposition Rate Function

During a sediment deposition experiment, the deposition rate (in mm/hr) was recorded over a 10-hour

Consumer Surplus Calculation

The demand and supply for a product are given by $$p_d(x)=20-0.5*x$$ and $$p_s(x)=10+0.2*x$$ respect

Designing a Water Slide

A water slide is designed along the curve $$y=-0.1*x^2+2*x+3$$ (in meters) over the interval $$[0,10

Determining a Function from Its Derivative

A function $$F(x)$$ has a derivative given by $$F'(x)= 2*x - 4$$. Given that $$F(1)=3$$, determine $

Electric Charge Accumulation

The current flowing into a capacitor is defined by $$I(t)=\frac{10}{1+e^{-2*(t-3)}}$$ (in amperes) f

Estimating Instantaneous Velocity from Position Data

A car's position along a straight road is recorded over a 10-second interval as shown in the table b

Exponential Decay Function Analysis

A lab experiment models the decay of a chemical concentration with the function $$f(t)=8*e^{-0.5*t}$

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$. Answer the following parts assuming the point $$(3,4)

Interpreting Integrated Quantities in a Changing System

A system is modeled by a rate function given by $$R(t)=t^2-4*t+6$$, where $$t$$ is in minutes. The c

Investment Compound Interest

An investment account starts with an initial deposit of $$1000$$ dollars and earns $$5\%$$ interest

Motion along a Straight Path

A particle moving along the x-axis has its acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t \g

Net Change and Total Distance in Particle Motion

A particle has acceleration $$a(t)=12-8*t$$ (in $$m/s^2$$) for $$t \ge 0$$, with initial velocity $$

Particle Motion Along a Straight Line

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t \

Particle Motion Analysis

A particle moving along a straight line has an acceleration given by $$a(t)=6-0.5*t$$ (in m/s²) for

Pipeline Installation Cost Analysis

The cost to install a pipeline along a route is given by $$C(x)=100+5*\sin(x)$$ (in dollars per mete

Position Analysis of a Particle with Piecewise Acceleration

A particle moving along a straight line experiences a piecewise constant acceleration given by $$a(

Probability from a Density Function

Let a continuous random variable $$X$$ be defined on $$[0,20]$$ with the probability density functio

Solid of Revolution: Water Tank

A water tank is formed by rotating the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and t

Tank Draining with Variable Flow Rates

A water tank is undergoing simultaneous inflow and outflow. The inflow rate is given by $$I(t)=10+2\

Tank Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Voltage and Energy Dissipation Analysis

The voltage across an electrical component is modeled by $$V(t)=12*e^{-0.1*t}*\ln(t+2)$$ (in volts)

Volume by Discs: Revolved Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ between their intersection points. T

Volume of a Solid of Revolution Rotated about a Line

Consider the region bounded by $$y=x^2$$ and $$y=x$$ for $$x\in [0,1]$$. This region is rotated abou

Volume of a Solid Using the Disc Method

Consider the region in the xy-plane bounded by $$y = \sqrt{x}$$ and $$y=0$$ for $$0 \le x \le 9$$. T

Volume of a Solid Using the Washer Method

Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev

Volume of a Solid with Rectangular Cross Sections

A solid has a base on the x-axis from $$x=0$$ to $$x=3$$. The cross-sectional areas (in m²) perpendi

Volume of a Solid with Square Cross-Sections

A solid has a base in the xy-plane bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. Every cro

Washer Method with Logarithmic and Exponential Curves

Consider the region bounded by the curves $$f(x)=\ln(x+1)$$ and $$g(x)=e^{-x}$$ on the interval $$[0

Water Tank Volume and Average Cross-Sectional Area

A water tank has a shape where the horizontal cross-sectional area at a depth $$x$$ (in feet) from t

Work Calculation from an Exponential Force Function

An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen

Work Done by a Variable Force

A variable force acting along a straight line is described by $$F(x)=3*x^2$$ Newtons, where $$x$$ is