AP Calculus BC FRQ Room

Absolute Value Limit Analysis II

Consider the function $$f(x)=\frac{x}{|x|}$$ for $$x \neq 0$$. Answer the following:

Analysis of a Rational Inflow Function with a Discontinuity

A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$

Continuity Assessment of a Rational Function with a Redefined Value

Consider the function $$r(x)= \begin{cases}\frac{x^2-9}{x-3}, & x \neq 3 \\ 7, & x=3\end{cases}$$.

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Continuity Involving a Radical Expression

Examine the function $$f(x)= \begin{cases} \frac{\sqrt{x+4}-2}{x} & x \neq 0 \\ k & x=0 \end{cases}$

End Behavior Analysis of a Rational Function

Consider the function $$f(x)=\frac{2 * x^3 - 5 * x + 1}{x^3+4 * x^2-x}$$. Answer the following:

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

Investigating Limits at Infinity and Asymptotic Behavior

Given the rational function $$f(x)=\frac{5*x^2-3*x+2}{2*x^2+x-1}$$, answer the following: (a) Evalua

Limits and Continuity of Radical Functions

Examine the function $$f(x)= \frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$.

Limits Involving Trigonometric Functions

Consider the function $$q(x)=\frac{1-\cos(2*x)}{x^2}$$.

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

One-Sided Limits and Jump Discontinuities

Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en

Physical Applications: Temperature Continuity

A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^

Piecewise Function Continuity and Differentiability

Consider the piecewise function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\

Rational Function and Removable Discontinuity

Consider the function $$g(x) = \frac{(x+3)(x-2)}{(x-2)}$$, defined for $$x \neq 2$$, and suppose tha

Related Rates: Changing Shadow Length

A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol

Water Filling a Leaky Tank

A water tank is initially empty. Every minute, 10 liters of water is added to the tank, but due to a

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

Calculating Velocity and Acceleration from a Position Function

A car’s position along a straight road is given by the function $$s(t)= 0.5*t^3 - 3*t^2 + 4*t + 2$$

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Chain Rule Verification with a Power Function

Let $$f(x)= (3*x+2)^4$$.

Chemical Mixing Tank

In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t

Complex Rational Differentiation

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

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

Comprehensive Analysis of $$e^{-x^2}$$

The function $$f(x)=e^{-x^2}$$ is used to model temperature distribution in a material. Provide a co

Drug Concentration in Bloodstream: Differentiation Analysis

A drug's concentration in the bloodstream is modeled by $$C(t)= 50e^{-0.25t} + 5$$, where t is in ho

Hot Air Balloon Altitude Analysis

A hot air balloon’s altitude is modeled by the function $$h(t)=5*\sqrt{t+1}$$, where $$h$$ is in met

Implicit Differentiation: Cost Allocation Model

A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1

Implicit Differentiation: Elliptic Curve

Consider the curve defined by $$2*x^2 + 3*x*y + y^2 = 20$$.

Implicit Differentiation: Mixed Exponential and Polynomial Equation

Consider the curve defined by $$x^3 + e^(y) = 3*x*y$$.

Investment Return Rates: Continuous vs. Discrete Comparison

An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Manufacturing Production Rates

A factory produces items at a rate given by $$P_{in}(t)=\frac{200}{1+e^{-0.3*(t-4)}}$$ items per hou

Motion Along a Line

An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i

Oil Spill Containment

Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat

Optimization Using Derivatives

Consider the quadratic function $$f(x)=-x^2+4*x+5$$. Answer the following:

Piecewise Function and Discontinuity Analysis

Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2 \\ 3 & x = 2 \en

Plant Growth Rate Analysis

A plant’s height (in centimeters) after $$t$$ days is modeled by $$h(t)=0.5*t^3 - 2*t^2 + 3*t + 10$$

River Flow Dynamics

A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$

Secant and Tangent Lines: Analysis of Rate of Change

Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces

Secant and Tangent Slope Analysis

Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:

Secant vs. Tangent: Approximation and Limit Approach

Consider the function $$f(x)= \sqrt{x}$$. Use both a secant line approximation and the limit definit

Secants and Tangents in Profit Function

A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)

Tangent Line Approximation

Consider the function $$g(t)=t^2 - 4*t + 7$$. Answer the following parts to find the equation of the

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Testing Differentiability at a Junction Point

Consider the function $$f(x)= \begin{cases} x+2 & x < 3 \\ 5 & x = 3 \\ 2*x-1 & x > 3 \end{cases}$$.

Velocity Function from a Cubic Position Function

An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a

Chain Rule Combined with Inverse Trigonometric Differentiation

Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:

Chain Rule in a Trigonometric Light Intensity Model

A light sensor records the intensity of light according to the function $$I(x) = \cos(\sqrt{3*x + 2}

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

Composite Function with Exponential and Radical

Consider the function $$ f(x)= \sqrt{e^{5*x}+x^2} $$.

Composite Functions in a Biological Growth Model

A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an

Composite Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Continuity and Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca

Differentiation of a Nested Trigonometric Function

Let $$h(x)= \sin(\arctan(2*x))$$.

Differentiation of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Differentiation of an Arctan Composite Function

For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Implicit Differentiation and Concavity of a Logarithmic Curve

The curve defined implicitly by $$y^3 + x*y - \ln(x+y) = 5$$ is given. Use implicit differentiation

Implicit Differentiation and Inverse Challenges

Consider the implicit relation $$x^2+ x*y+ y^2=10$$ near the point (2,2).

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Implicit Differentiation on a Trigonometric Curve

Consider the curve defined implicitly by $$\sin(x+y) = x^2$$.

Implicit Differentiation with Trigonometric Components

Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti

Inverse Analysis of a Composite Exponential-Trigonometric Function

Let $$f(x)=e^x+\cos(x)$$. Analyze the behavior of its inverse function under appropriate domain rest

Inverse Function Differentiation with a Cubic Function

Let $$f(x)= x^3+ x + 1$$ be a one-to-one function, and let $$g$$ be its inverse function. Answer the

Inverse of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases} x^2 & x \ge 0 \\ -x & x < 0 \end{cases}$$. Anal

Inverse of a Radical Function with Domain Restrictions

Consider the function $$f(x)=\sqrt{1-x^2}$$. Analyze its invertibility.

Inverse Trigonometric Function Differentiation

Let $$y=\arctan(\sqrt{3*x+1})$$. Answer the following parts:

Investment Growth and Rate of Change

An investor makes monthly deposits that increase according to an arithmetic sequence. The deposit am

Lake Water Level Dynamics: Seasonal Variation

A lake's water inflow is modeled by the composite function $$I(t)=p(q(t))$$, where $$q(t)=0.5*t-1$$

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Physics Lab: Logarithmic Chain Rule in a Kinetics Experiment

In a kinetics experiment, the reactant concentration is modeled by $$C(t)=\ln(3*e^{2*t}+4)$$, where

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Rate of Change in a Biochemical Process Modeled by Composite Functions

The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,

Related Rates: Temperature Change in a Moving Object

An object moves along a path where its temperature is given by $$T(x)= \ln(3*x + 2)$$ and its positi

Tangent Line to a Circle via Implicit Differentiation

Consider the circle defined by $$x^2 + y^2 = 25$$. At the point $$(3, -4)$$, determine the slope of

Taylor/Maclaurin Polynomial Approximation for a Logarithmic Function

Let $$f(x) = \ln(1+3*x)$$. Develop a second-degree Maclaurin polynomial, determine its radius of con

Temperature Control: Heating Element Dynamics

A room's temperature is controlled by a heater whose output is given by the composite function $$H(t

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

Air Conditioning Refrigerant Balance

An air conditioning system is charged with refrigerant at a rate given by $$I(t)=12-0.5t$$ (kg/min)

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

Approximating Changes with Differentials

Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch

Conical Tank Filling

A conical water tank has a height of $$10$$ m and a top radius of $$4$$ m. The water in the tank for

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

Critical Points and Inflection Analysis

Consider the function $$f(x)= (x-2)^2*(x+5)$$ which models a physical quantity. Answer the following

Curvature Analysis in the Design of a Bridge

A bridge's vertical profile is modeled by $$y(x)=100-0.5*x^2+0.05*x^3$$, where $$y$$ is in meters an

Estimating Rates from Experimental Position Data

The table below represents experimental measurements of the position (in meters) of a moving particl

Firework Trajectory Analysis

A firework is launched and its height (in meters) is modeled by the function $$h(t)=-4.9t^2+30t+5$$,

Implicit Differentiation on an Ellipse

An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo

Inflating Spherical Balloon

A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1

L'Hôpital's Analysis

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

L'Hospital's Rule for Indeterminate Limits

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ using L'Hospita

Linearization of Trigonometric Implicit Function

Consider the implicit equation $$\tan(x + y) = x - y$$, which implicitly defines $$y$$ as a function

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Logarithmic Transformation and Derivative Limits

Consider the function $$f(x)=\ln\left(\frac{e^{3x}+1}{1+e^{-x}}\right)$$. Answer the following:

Motion with Non-Uniform Acceleration

A particle moves along a straight line and its position is given by $$s(t)= 2*t^3 - 9*t^2 + 12*t + 3

Optimizing Factory Production with Log-Exponential Model

A factory's production is modeled by $$P(x)=200x^{0.3}e^{-0.02x}$$, where x represents the number of

Particle Motion Along a Line with Polynomial Velocity

A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Production Cost Analysis

A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa

Related Rates in a Circular Pool

A circular pool is being filled such that the surface area increases at a constant rate of $$10$$ ft

Revenue Function and Marginal Revenue

A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)

Series Approximation in Population Dynamics

A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans

Spherical Balloon Inflation

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

Temperature Change in Coffee Cooling

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$, where $$T(t)$$ is in °F a

Vehicle Motion on a Curved Path

A vehicle moving along a straight road has its position given by $$s(t)= 4*t^3 - 24*t^2 + 36*t + 5$$

Volume Measurement Inversion

The volume of a sphere is given by $$f(x)=\frac{4}{3}*\pi*x^3$$, where $$x$$ is the radius. Analyze

Water Tank Filling: Related Rates

A cylindrical tank has a fixed radius of 2 m. The volume of water in the tank is given by $$V=\pi*r^

Analysis of a Decay Model with Constant Input

Consider the concentration function $$C(t)= 30\,e^{-0.5t} + \ln(t+1)$$, where t is measured in hours

Analysis of a Logarithmic Function

Consider the function $$q(x)=\ln(x)-\frac{1}{2}*x$$ defined on the interval [1,8]. Answer the follow

Analysis of a Rational Function and Its Inverse

Consider the function $$f(x)= \frac{2*x+3}{x-1}$$ defined for $$x \neq 1$$. Answer the following par

Analyzing a Function with Implicit Logarithmic Differentiation

Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima

Bank Account Growth and Instantaneous Rate

A bank account balance is modeled by the function $$B(t) = 1000*e^{0.05*t}$$, where t (in years) rep

Concavity and Inflection Points Analysis

Consider the function \( f(x)=\ln(x) - x \) where \( x > 0 \). Answer the following parts:

Economic Production Optimization

A company’s cost function is given by $$C(x) = 0.5*x^3 - 3*x^2 + 4*x + 200$$, where x represents the

Elasticity Analysis of a Demand Function

The demand function for a product is given by $$Q(p) = 100 - 5*p + 0.2*p^2$$, where p (in dollars) i

Exponential Decay in Velocity

A particle’s velocity is modeled by the function $$v(t)=10e^{-0.5*t}-3$$ (in m/s) for $$t\ge0$$.

Implicit Differentiation and Tangent Slope

Consider the curve defined implicitly by $$x^2 + x*y + y^2 = 7$$. Answer the following parts:

Intervals of Increase and Decrease in Vehicle Motion

A vehicle’s position along a straight road is given by the function $$s(t) = t^3 - 6*t^2 + 9*t + 10$

Inverse Function in a Physical Context

Suppose $$f(t)= t^3 + 2*t + 1$$ represents the displacement (in meters) of an object over time t (in

Logarithmic-Quadratic Combination Inverse Analysis

Consider the function $$f(x)= \ln(x^2+1)$$ for $$x \ge 0$$. Answer the following parts.

Mean Value Theorem with a Trigonometric Function

Let $$f(x)=\sin(x)$$ be defined on the interval $$[0,\pi]$$. Answer the following parts:

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Optimizing Fencing for a Field

A farmer has $$100$$ meters of fencing to construct a rectangular field that borders a river (no fen

Optimizing Fencing for a Rectangular Garden

A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee

Particle Motion on a Curve

A particle moves along a straight-line path with its position given by \( s(t)=t^3 - 6*t^2 + 9*t + 1

Projectile Motion Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Projectile Trajectory: Parametric Analysis

A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Region Area and Volume: Polynomial and Linear Function

A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo

Retirement Savings with Diminishing Deposits

Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th

Taylor Polynomial for $$\ln(x)$$ about $$x=1$$

For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,

Taylor Series for $$\arcsin(x)$$

Derive the Maclaurin series for $$f(x)=\arcsin(x)$$ up to the $$x^5$$ term, determine the radius of

Taylor Series for $$\ln(1+3*x)$$

Let $$f(x)=\ln(1+3*x)$$. Develop its Maclaurin series up to the 3rd degree, determine the radius of

Taylor Series for $$\sqrt{1+x}$$

Consider the function $$f(x)=\sqrt{1+x}$$. In this problem, compute its 3rd degree Maclaurin polynom

Temperature Change in a Weather Balloon

A weather balloon’s temperature and altitude are related by the implicit equation $$T*e^{z} + z = 50

Volume Using Cylindrical Shells

The region bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is revolved about the y-axis to form a solid.

Accumulation Function from a Rate Function

The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for

Accumulation Function in an Investment Model

An investment has an instantaneous rate of return given by $$r(t)=0.05*t+0.02$$ (per year). The accu

Antiderivative Application in Crop Growth

A crop field grows at a rate modeled by the function $$G'(t)=4*t-3$$ (in square meters per week). Th

Antiderivatives and the Fundamental Theorem

Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i

Approximating Energy Consumption Using Riemann Sums

A household’s power consumption (in kW) is recorded over an 8‐hour period. The following table shows

Area Between Two Curves

Given the functions $$f(x)= x^2$$ and $$g(x)= 4*x$$, determine the area of the region bounded by the

Chemical Reaction Rates

A chemical reaction in a vessel occurs at a rate given by $$R(t)= 8*e^{-t/2}$$ mmol/min. Determine t

Cost Function Accumulation

A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x

Determining Antiderivatives and Initial Value Problems

Suppose that $$F(x)$$ is an antiderivative of the function $$f(x)=5*x^4 - 2*x + 3$$, and that it is

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Differentiation and Integration of a Power Series

Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.

Error Bound Analysis for the Trapezoidal Rule

For the function $$f(x)=\ln(x)$$ on the interval $$[1,2]$$, the error bound for the trapezoidal rule

Evaluating an Integral Involving an Exponential Function

Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Fuel Consumption Estimation with Midpoint Riemann Sums

A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The

Integration of a Piecewise-Defined Function

Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

Interpreting Area Under a Curve from a Graph

A graph displays the function $$f(x)=0.5*x+1$$ over the interval $$[0,6]$$.

Marginal Cost and Production

A factory's marginal cost function is given by $$MC(x)= 4 - 0.1*x$$ dollars per unit, where $$x$$ is

Midpoint Approximation Analysis

Let $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Answer the following:

Minimizing Material for a Container

A company wants to design an open-top rectangular container with a square base that must have a volu

Riemann Sum Estimation from Tabular Data

The following table lists values of a function $$f(x)$$ at selected points, which are used to approx

Signal Energy through Trigonometric Integration

A signal is described by $$f(t)=3*\sin(2*t)+\cos(2*t)$$. The energy of the signal over one period

Trapezoidal Approximation of a Definite Integral from Tabular Data

The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra

U-Substitution in Accumulation Functions

In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe

U-Substitution Integration Challenge

Evaluate the integral $$\int_0^2 (2*x+1)\,(x^2+x+3)^5\,dx$$ using an appropriate u-substitution.

Vehicle Motion and Inverse Time Function

A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se

Water Flow and the Trapezoidal Rule

Water flows into a reservoir at a rate given by $$R(t)$$ (in m³/hour) as provided in the table below

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Analysis of an Inverse Function from a Differential Equation Solution

Suppose a differential equation is solved to give an increasing function $$f(x)=\ln(2*x+3)$$ defined

Chemical Reaction and Separable Differential Equation

In a particular chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to t

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Depreciation Model of a Vehicle

A vehicle's value depreciates continuously over time according to the differential equation $$\frac{

Euler's Method Approximation

Consider the initial value problem $$\frac{dy}{dt}=t\sqrt{y}$$ with $$y(0)=1$$. Use Euler's method w

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

Exponential Growth with Variable Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the

Exponential Population Growth and Doubling Time

A certain population is modeled by the differential equation $$\frac{dP}{dt} = k*P$$. This equation

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

FRQ 10: Cooling of a Metal Rod

A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th

FRQ 11: Linear Differential Equation via Integrating Factor

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

Growth and Decay in a Bioreactor

In a bioreactor, the concentration of a chemical P (in mg/L) evolves according to the differential e

Inverse Function Analysis Derived from a Differential Equation Solution

Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e

Investment Growth Model

An investment account grows continuously at a rate proportional to its current balance. The balance

Logistic Equation with Harvesting

A fish population in a lake follows a logistic growth model with the addition of a constant harvesti

Logistic Growth in Population Dynamics

The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l

Mixing Problem with Constant Flow Rate

A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen

Mixing Problem with Time-Dependent Inflow

A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa

Modeling Ambient Temperature Change

The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

Motion Under Gravity with Air Resistance

An object falling under gravity experiences air resistance proportional to its velocity. Its motion

Newton's Law of Cooling

An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling, which is modeled by the differential equation

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Population Growth with Logistic Differential Equation

A population $$P(t)$$ is modeled by the logistic differential equation $$\frac{dP}{dt} = r\,P\left(1

Solving a Separable Differential Equation

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

Temperature Control in a Chemical Reaction Vessel

In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Arc Length in Polar Coordinates

Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Area Between a Rational Function and Its Asymptote

Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter

Area Between Curves: Enclosed Region

The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:

Average Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Temperature Calculation

A city's temperature during a day is modeled by $$T(t)=10+5*\sin\left(\frac{\pi*t}{12}\right)$$ for

Average Temperature Over a Day

A function modeling the temperature (in °F) throughout a day is given by $$T(t)= 3*\sin\left(\frac{\

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Average Value of a Velocity Function

A particle moves along a line with its velocity given by $$v(t)= 2*\cos(t) + \sin(t)$$ for $$t \in [

Bacterial Decay Modeled by a Geometric Series

A bacterial culture is treated with an antibiotic that reduces the bacterial population by 20% each

Center of Mass of a Nonuniform Rod

A thin rod extends from $$x=0$$ to $$x=3$$ and has a linear density given by $$\delta(x)=1+x$$ (in k

Chemical Mixing in a Tank

A tank initially contains 100 liters of water. A chemical solution with a concentration of 0.5 g/l f

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Moment of Inertia of a Thin Plate

A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=3*t-4$$ (in m/s²). At time

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

Volume about a Vertical Line using Two Methods

A region in the first quadrant is bounded by $$y=x$$, $$y=0$$, and $$x=2$$. This region is rotated a

Volume by Revolution: Washer Method

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$. When this region is rotated about

Volume of a Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of a Solid by the Disc Method

Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Volume of a Solid using the Shell Method

Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, $$x=1$$, and $$x=4$$. When this region is ro

Volume of a Solid Using the Shell Method

The region in the first quadrant bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is rotated about the y-axi

Volume of a Solid via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Volume of a Solid with Square Cross Sections

The base of a solid is the region in the plane bounded by $$y=x$$ and $$y=x^2$$ (with $$x$$ between

Acceleration Analysis in a Vector-Valued Function

Consider a particle whose position is given by $$ r(t)=\langle \sin(2*t),\; \cos(2*t) \rangle $$ for

Analysis of Vector Trajectories

A particle in the plane follows the path given by $$\mathbf{r}(t)=\langle \ln(t+1), \sqrt{t} \rangle

Area Between Two Polar Curves

Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of

Continuity Analysis of a Discontinuous Parametric Curve

Consider the parametric curve defined by $$x(t)= \begin{cases} t^2, & t < 1 \\ 2*t - 1, & t \ge 1 \

Curvature of a Vector-Valued Function

Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given

Designing a Parametric Curve for a Cardioid

A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.

Designing a Race Track with Parametric Equations

An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:

Distance Traveled in a Turning Curve

A curve is defined by the parametric equations $$x(t)=4*\sin(t)$$ and $$y(t)=4*\cos(t)$$ for $$0\le

Intersection Points of Polar Curves

Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Modeling Periodic Motion with a Vector Function

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle \cos(2*t),\;

Modeling with Polar Data

A researcher collects the following polar coordinate data for a phenomenon.

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Parametric Curve Intersection

Two curves are defined parametrically as follows: For curve C1, $$x(t) = t^2$$ and $$y(t) = 2*t + 1$

Parametric Curve with a Loop and Tangent Analysis

Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An

Periodic Motion: A Vector-Valued Function

A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$

Polar to Parametric Conversion and Arc Length

A curve is defined in polar coordinates by $$r= 1+\sin(\theta)$$. Convert and analyze the curve.

Sensitivity Analysis and Linear Approximation using Implicit Differentiation

The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.

Symmetry and Self-Intersection of a Parametric Curve

Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \

Tangent Line to a 3D Vector-Valued Curve

Let $$\textbf{r}(t)= \langle t^2, \sin(t), \ln(t+1) \rangle$$ for $$t \in [0,\pi]$$. Answer the foll

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*