AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

Consider the function $$f(x)=\frac{|x-3|}{x-3}$$. Answer the following:

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

Continuity Analysis of a Piecewise Function

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

Continuity Conditions for a Piecewise-Defined Function

Consider the function defined by $$ f(x)= \begin{cases} 2*x+1, & x < 3 \\ ax^2+ b, & x \ge 3 \end{c

Continuity in Composition of Functions

Let $$g(x)=\frac{x^2-4}{x-2}$$ for x ≠ 2 and undefined at x = 2, and let f(x) be a continuous functi

Continuity of Log‐Exponential Function

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - 1}{x}$$ for $$x \neq 0$$, with $$f(0)=c$$. Dete

Determining Limits for a Function with Absolute Values and Parameters

Consider the function $$ f(x)= \begin{cases} \frac{|x-2|}{x-2}, & x \neq 2 \\ c, & x = 2 \end{cases

End Behavior and Horizontal Asymptote Analysis

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

Estimating Limits from Tabulated Data

A function $$g(x)$$ is experimentally measured near $$x=2$$. Use the following data to estimate $$\l

Evaluating a Limit with Algebraic Manipulation

Examine the function $$g(x)= \frac{\sqrt{x+9}-3}{x}$$ for $$x \neq 0$$.

Evaluating a Logarithmic Limit

Given the limit $$\lim_{x \to 2} \frac{\ln(x-1)}{x^2-4} = k$$, find the value of $$k$$ using algebra

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Factorable Discontinuity Analysis

Let $$q(x)=\frac{x^2-x-6}{x-3}.$$ Answer the following:

Graphical Analysis of a Continuous Polynomial Function

Consider the function $$f(x)=2*x^3-5*x^2+x-7$$ and its graph depicted below. The graph provided accu

Intermediate Value Theorem Application with a Cubic Function

A function f(x) is continuous on the interval [-2, 2] and its values at certain points are given in

Interplay of Polynomial Growth and Exponential Decay

Consider the function $$s(x)= x\cdot e^{-x}$$.

Investigating a Function with a Removable Discontinuity

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

Limits Involving Exponential Functions

Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.

Limits Involving Infinity and Vertical Asymptotes

Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following:

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

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

One-Sided Limits and Jump Discontinuity Analysis

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

Piecewise Function Critical Analysis

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

Population Growth and Limits

The population $$P(t)$$ of a small town is recorded every 10 years as shown in the table below. Assu

Resistor Network Convergence

A resistor network is constructed by adding resistors in a ladder configuration. The resistance adde

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Squeeze Theorem with Oscillatory Behavior

Examine the function $$s(x)=x^2*\sin(1/x)$$ for x ≠ 0.

Taylor Series Expansion for $$\arctan(x)$$

Consider the function $$f(x)=\arctan(x)$$ and its Taylor series about $$x=0$$.

Telecommunications Signal Strength

A telecommunications tower emits a signal whose strength decreases by $$20\%$$ for every additional

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

Analyzing a Function with an Oscillatory Component

Consider the function $$f(x)= x*\sin(x)$$. Answer the following:

Analyzing a Function's Rate of Change from Graphical Data

A function has been experimentally measured and its values are represented by the following graph. U

Average and Instantaneous Growth Rates in a Bacterial Culture

A bacterial population is modeled by the function $$P(t)= e^{0.3*t} + 10$$, where $$t$$ is measured

Chain Rule Verification with a Power Function

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

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=10 - 2*\ln(t+1)$$, wher

Compound Exponential Rate Analysis

Consider the function $$f(t)=\frac{e^{2*t}}{1+t}$$, which arises in compound growth models. Analyze

Derivative Using Limit Definition

Let $$f(x)=\frac{1}{x+2}$$. Using the definition of the derivative, find $$f'(x)$$.

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Differentiation in Polar Coordinates

Consider the polar curve defined by $$r(\theta)= 1+\cos(\theta).$$ (a) Use the formula for polar

Economic Model Rate Analysis

A company models its cost variations with respect to price $$p$$ using the function $$C(p)=e^{-p}+\l

Efficiency Ratio Rate Change Using the Quotient Rule

An efficiency ratio is modeled by $$E(x) = \frac{x^2+2}{3*x-1}$$, where x represents an input variab

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Higher Order Derivatives: Concavity and Inflection Points

Consider the function $$f(x)= x^4 - 4*x^3+6*x^2.$$ (a) Find the first derivative \(f'(x)\) and th

Higher-Order Derivatives

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

Icy Lake Evaporation and Refreezing

An icy lake gains water from melting ice at a rate of $$M_{in}(t)=5+0.2*t$$ liters per hour and lose

Implicit Differentiation in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2 + y^2 = 25$$.

Instantaneous vs. Average Rate of Change

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

Logarithmic Differentiation: Equating Powers

Consider the equation $$y^x = x^y$$ that relates $$x$$ and $$y$$ implicitly.

Market Price Rate Analysis

The market price of a product (in dollars) has been recorded over several days. Use the table below

Particle Motion on a Straight Line: Average and Instantaneous Rates

A particle moving along a straight line has its position given by $$s(t)=t^3 - 6*t^2 + 9*t + 4$$ for

Pharmacokinetics: Drug Concentration Analysis

The concentration of a drug in the bloodstream is modeled by the function $$C(t)=10*\ln(t+2)*e^{-0.3

Quotient Rule in a Chemical Concentration Model

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{t+2}{t^2+1}$$ (in mg/L), w

Radius of Convergence of a Power Series for e^x

Consider the power series representation $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$, known to represent

Rational Function Derivative Using Quotient Rule

Consider the function $$g(x)=\frac{5*x-7}{x+2}$$. Find its derivative and analyze its critical featu

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

Trigonometric Function Differentiation

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

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

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

Water Treatment Plant Simulator

A water treatment plant receives contaminated water at a rate of $$R_{in}(t)=50e^{-0.1*t}$$ liters p

Analysis of a Piecewise Function with Discontinuities

Consider the piecewise function $$ f(x) = \begin{cases} 2*x+1, & x < 1, \\ 3, & 1 \le x \le 2, \\ \s

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Analyzing a Composite Function with Nested Radicals

Consider the function $$h(x)=\sqrt{1+\sqrt{2+3x}}$$. Answer the following parts:

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Chain Rule Combined with Inverse Trigonometric Differentiation

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

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))

Derivative of an Inverse Function with a Quadratic

Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of

Enzyme Kinetics in a Biochemical Reaction

In an enzymatic reaction, the substrate concentration $$S(t)$$ and the product concentration $$P(t)$

Implicit Differentiation in an Economic Model

A company’s production is modeled by the implicit relationship $$x*y^2 + \ln(x+y) = 10$$, where $$x$

Implicit Differentiation with Logarithmic Equation

Consider the curve defined by $$\ln(x*y) + x^2 = y$$. Answer the following parts:

Implicit Differentiation with Trigonometric Components

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

Inverse Derivative via Chain Rule for a Logarithmic-Exponential Function

Let $$f(x)=\ln(1+e^x)$$. Analyze its inverse derivative.

Inverse Differentiation of a Trigonometric Function

Consider the function $$f(x)=\arctan(2*x)$$ defined for all real numbers. Analyze its inverse functi

Inverse Function Differentiation for a Cubic Function

Let $$f(x)= x^3 + x$$ be an invertible function with inverse $$g(x)$$. Use the inverse function deri

Inverse Function Differentiation in a Science Experiment

In an experiment, the relationship between an input value $$x$$ and the output is given by $$f(x)= \

Inverse Function Differentiation with a Logarithmic Function

Let the function $$f(x)=\ln(2+x^2)$$ be differentiable and one-to-one, and let its inverse be $$g(y)

Inverse of a Radical Function with Domain Restrictions

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

Inverse Trigonometric Functions: Analysis and Application

Consider the function $$f(x) = \arctan(3*x)$$. Analyze its rate of change and the equation of the ta

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Maximizing the Garden Area

A rectangular garden is to be built alongside a river, so that no fence is needed along the river. T

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

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

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

Analyzing a Production Cost Function

A company's cost function for producing goods is given by $$C(x)=x^3-12x^2+40x+100$$, where x repres

Analyzing Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

Analyzing Rate of Approach in a Pursuit Problem

Two cars are traveling on perpendicular roads. Car A is moving east at 60 km/h and is 3 km from the

Approximating Changes with Differentials

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

Bacterial Population Growth

The population of a bacterial culture is modeled by $$P(t)=1000e^{0.3*t}$$, where $$P(t)$$ is the nu

Boat Crossing a River: Relative Motion

A boat must cross a 100 m wide river. The boat's speed relative to the water is 5 m/s (directly acro

Complex Limit Analysis with L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2*x} - 1 - 2*x}{x^2}$$ using L'Hôpital's Rule. Answer t

Cooling Coffee Temperature

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

Cubic Curve Linearization

Consider the curve defined implicitly by $$x^3 + y^3 - 3*x*y = 0$$.

Differentials and Function Approximation

Consider the function $$f(x)=x^{1/3}$$. At $$x=8$$, answer the following parts.

Drug Concentration in the Blood

A patient's drug concentration is modeled by $$C(t)=20e^{-0.5t}+5$$, where $$t$$ is measured in hour

Engineering Linearization for Error Approximation

An engineer is working with the function $$f(x)= \sqrt{x}$$ where \(x\) is a measured quantity. To s

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Expanding Circular Pool

A circular pool is being designed such that water flows in uniformly, expanding its surface area. Th

Exponential Function Inversion

Consider the function $$f(x)=e^{2*x}+3$$ which models the growth of a certain variable. Analyze the

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume is given by $$V= \frac{4}{3}*\pi*r^3$$, w

Limit Evaluation Using L'Hôpital's Rule

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

Linearization Approximation

Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans

Maximizing Revenue in a Business Model

A company’s revenue as a function of price is given by $$R(p)= p*(100 - 2*p)$$, where $$p$$ is the p

Optimization of a Rectangular Enclosure

A rectangular enclosure is to be built adjacent to a river. Only three sides of the enclosure requir

Parametric Curve Motion

A particle’s trajectory is given by the parametric equations $$x(t)=t^2-1$$ and $$y(t)=2*t+3$$ for $

Particle Motion Analysis Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Rate of Change in Logarithmic Brightness

The brightness of a star, measured on a logarithmic scale, is given by $$B(t)=\ln(100+t^2)$$, where

Sediment Transport in a Riverbank

In a riverbank environment, sediment is deposited at a rate of $$D(t)=20-0.5t$$ (kg/min) while simul

Series Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

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

Series Solution of a Drug Concentration Model

The drug concentration in the bloodstream is modeled by $$C(t)= \sum_{n=0}^{\infty} \frac{(-t)^n}{n!

Surface Area of a Solid of Revolution

Consider the curve $$y = \ln(x)$$ for $$x \in [1, e]$$. Find the surface area of the solid formed by

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

Vector Function: Particle Motion in the Plane

A particle moves in the plane with a position vector given by $$\mathbf{r}(t)=\langle t, t^2 \rangle

Vehicle Motion on a Curved Path

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

Aircraft Climb Analysis

An aircraft's vertical motion is modeled by a vertical velocity function given by $$v(t)=20-2*t$$ (i

Analysis of a Piecewise Function's Differentiability and Extrema

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

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Application of Rolle's Theorem to a Trigonometric Function

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

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Car Depreciation Analysis

A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de

Composite Function and Inverse Analysis

Let $$f(x)= e^(x) - x$$ defined for all real numbers, and consider its behavior.

Energy Consumption Rate Model

A household's energy consumption rate (in kW) is modeled by $$E(t) = 2*t^2 - 8*t + 10$$, where t is

Error Approximation using Linearization

Consider the function $$f(x) = \sqrt{4*x + 1}$$.

Error Estimation in Approximating $$e^x$$

For the function $$f(x)=e^x$$, use the Maclaurin series to approximate $$e^{0.3}$$. Then, determine

Expanding Oil Spill - Related Rates

A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius

Implicit Differentiation and Inverse Function Analysis

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, where y is a function of x near the point wh

Inverse Analysis for a Function with Multiple Transformations

Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.

Inverse Derivative Analysis of a Quartic Polynomial

Consider the function $$f(x)= x^4 - 4*x^2 + 2$$ defined for $$x \ge 0$$. Answer the following.

Inverse Function Derivative for a Piecewise Function

Suppose f is defined piecewise by $$f(x)= x^2$$ for $$x \ge 0$$ and $$f(x)= -x$$ for $$x < 0$$. Cons

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Investment Portfolio Dividends

A company pays annual dividends that form an arithmetic sequence. The dividend in the first year is

Light Reflection Between Mirrors

A beam of light is directed between two parallel mirrors. With each reflection, 70% of the light’s i

Linear Particle Motion Analysis

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

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Projectile Trajectory: Parametric Analysis

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

Roller Coaster Height Analysis

A roller coaster's height (in meters) as a function of time (in seconds) is given by $$h(t) = -0.5*t

Series Convergence and Integration in a Physical Model

A physical process is modeled by the power series $$g(x)=\sum_{n=1}^\infty \frac{2^n}{n!} * (x-3)^n$

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

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

Taylor Series for $$e^{\sin(x)}$$

Let $$f(x)=e^{\sin(x)}$$. First, obtain the Maclaurin series for $$\sin(x)$$ up to the $$x^3$$ term,

Taylor Series in Economics: Cost Function

An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro

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.

Water Tank Dynamics

A water tank receives water from a pipe at a rate of $$R(t)=3*t+5$$ liters/min and loses water throu

Accumulated Displacement from a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

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

Antiderivatives and the Constant of Integration

Consider the rate function $$ r(t)= 2*t + 3 $$ where t represents time in seconds.

Antiderivatives and the Fundamental Theorem of Calculus

Given the function $$f(x)= 2*x+3$$, use the Fundamental Theorem of Calculus to evaluate the definite

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 the Curves: Linear and Quadratic Functions

Consider the curves $$y = 2*t$$ and $$y = t^2$$. Answer the following parts to find the area of th

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

Area Under a Piecewise Function

A function is defined piecewise as follows: $$f(x)=\begin{cases} x & 0 \le x \le 2,\\ 6-x & 2 < x \

Bacteria Population Accumulation

A bacteria culture grows at a rate given by $$r(t)=0.5*t^2-t+3$$ (in thousand bacteria per hour) for

Center of Mass of a Rod with Variable Density

A thin rod of length 10 m has a linear density given by $$\rho(x)= 2 + 0.3*x$$ (in kg/m), where x is

Comparing Integration Approximations: Simpson's Rule and Trapezoidal Rule

A student approximates the definite integral $$\int_{0}^{4} (x^2+1)\,dx$$ using both the trapezoidal

Composite Functions and Inverses

Consider \(f(x)= x^2+1\) for \(x \ge 0\). Answer the following questions regarding \(f\) and its inv

Consumer Surplus in an Economic Model

For a particular product, the demand function is given by $$D(p)=100 - 5p$$ and the supply function

Cost Accumulation via Integration

A manufacturing process has a marginal cost function given by $$MC(x)= 4 + 3*x$$ dollars per item, w

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Estimating Area Under a Curve Using Riemann Sums

Consider the function $$f(x)$$ whose values on the interval $$[0,10]$$ are given in the table below.

Integration via Substitution and Numerical Methods

Evaluate the integral $$\int_0^2 \frac{2*x}{\sqrt{1+x^2}}\,dx$$.

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:

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Partial Fractions Integration

Evaluate the integral $$\int_1^3 \frac{4*x-2}{(x-1)(x+2)} dx$$ by decomposing the integrand into p

Population Increase from a Discontinuous Growth Rate

A sudden migration event alters the population growth rate. The growth rate (in individuals per year

Population Model Inversion and Accumulation

Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Trigonometric Integral via U-Substitution

Evaluate the integral $$\int_{0}^{\pi/2} \sin(2*x)\,dx$$ using an appropriate substitution.

Water Pollution Accumulation

In a river, a pollutant is introduced at a rate $$P_{in}(t)=8-0.5*t$$ (kg/min) and is simultaneously

Work Done by a Variable Force

A variable force given by $$F(x)= 3*x^2$$ (in Newtons) acts on an object as it moves along a straigh

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Car Engine Temperature Dynamics

The temperature $$T(t)$$ (in °C) of a car engine is modeled by the differential equation $$\frac{dT}

Chain Reaction in a Nuclear Reactor

A simplified model for a nuclear chain reaction is given by the logistic differential equation $$\fr

Chemical Reactor Mixing

In a chemical reactor, the concentration $$C(t)$$ (in M) of a chemical is governed by the equation $

City Population with Migration

The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe

Differential Equation Involving Logarithms

Consider the differential equation $$\frac{dy}{dx} = (y-1)*\ln|y-1|$$ with the initial condition $$y

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

Flow Rate in River Pollution Modeling

A river system is modeled to study pollutant concentration $$C(t)$$ (in mg/L). Polluted water with c

FRQ 3: Population Growth and Logistic Model

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt} = k*P\le

Implicit Differentiation in a Differential Equation Context

Suppose the function $$y(x)$$ satisfies the implicit equation $$x\,e^{y}+y^2=7$$. Differentiate both

Implicit Solution of a Separable Differential Equation

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

Integrating Factor Application

Solve the first order linear differential equation $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ with the initi

Integrating Factor Method

Solve the differential equation $$\frac{dy}{dx} + \frac{2}{x} y = \frac{\sin(x)}{x}$$ for $$x>0$$.

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

Logistic Growth Model

A population P grows according to the logistic differential equation $$\frac{dP}{dt} = rP\left(1-\fr

Logistic Population Model

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

Mixing Problem with Constant Rates

A tank contains $$200\,L$$ of a well-mixed saline solution with $$5\,kg$$ of salt initially. Brine w

Mixing Problem with Differential Equations

A tank initially contains $$S(0)=S_0$$ grams of salt dissolved in a volume $$V$$ liters of water. Br

Modeling Ambient Temperature Change

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

Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Piecewise Differential Equation with Discontinuities

Consider the following piecewise differential equation defined for a function $$y(x)$$: For $$x < 2

Radioactive Decay with Constant Source

A radioactive material is produced at a constant rate S while simultaneously decaying. This process

Series Solution for a Second-Order Differential Equation

Consider the differential equation $$y'' - y = 0$$ with the initial conditions $$y(0)=1$$ and $$y'(0

Sketching a Solution Curve from a Slope Field

A slope field for the differential equation $$\frac{dy}{dt}=y(1-y)$$ is provided. Use the slope fiel

Solution Curve from Slope Field

A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio

Solution Curve Sketching Using Slope Fields

Given the differential equation $$\frac{dy}{dx} = x - y$$, a slope field is provided. Use the field

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Accumulated Rainfall

The rate of rainfall over a 12-hour storm is modeled by $$r(t)=4*\sin\left(\frac{\pi}{12}*t\right) +

Analyzing the Inverse of an Exponential Function

Let $$f(x)=\ln(2*x+1)$$, defined for $$x\ge0$$.

Area Between Curves from Experimental Data

In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres

Area Between Curves: Supply and Demand Analysis

In an economic model, the supply and demand functions for a product (in hundreds of units) are given

Area between Parabola and Tangent

Consider the parabola defined by $$y^2 = 4 * x$$. Let $$P = (1, 2)$$ be a point on the parabola. Ans

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

Average Value of a Temperature Function

A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Car Motion Analysis

A car's acceleration is given by $$a(t) = 4 - 2 * t$$ (in m/s²) for $$0 \le t \le 4$$ seconds. The c

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

Error Analysis in Taylor Polynomial Approximations

Let $$h(x)= \cos(3*x)$$. Analyze the error involved when approximating $$h(x)$$ by its third-degree

Fluid Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

Power Series Representation for ln(x) about x=4

The function $$f(x)=\ln(x)$$ is to be expanded as a power series centered at $$x=4$$. Find this seri

Probability Density Function and Expected Value

A continuous random variable $$X$$ has a probability density function defined by $$f(x)=k*x$$ for $$

River Cross Section Area

The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$

River Crossing: Average Depth and Flow Calculation

The depth of a river along a 100-meter cross-section is modeled by $$d(x)=2+\cos\left(\frac{\pi}{50}

Surface Area of a Solid of Revolution

Consider the curve $$y= \sqrt{x}$$ over the interval $$0 \le x \le 4$$. When this curve is rotated a

Volume by Cross-Sectional Area (Square Cross-Sections)

A solid has a base in the xy-plane bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4

Volume by Shell Method: Rotated Parabolic Region

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

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 of Revolution Using the Disc Method

Let R be the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is rotated about th

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

Work Done in Lifting a Cable

A cable of length 10 m with a uniform mass density of 5 kg/m hangs vertically from a winch. The winc

Analysis of a Cycloid

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Analyzing Oscillatory Motion in Parametric Form

The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$

Arc Length and Surface Area of Revolution from a Parametric Curve

Consider the curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\ln(\sec(t)+\tan(t))$$ for $$0 \le t < \fr

Arc Length of a Polar Curve

Consider the polar curve given by $$r = 2 + 2*\sin(\theta)$$ for $$0 \le \theta \le \pi$$.

Arc Length of a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \ln(t+1), \sqrt{t}, e^t \rangle$$ defined

Arc Length of an Elliptical Curve

The parametric equations $$x(t)= 4\cos(t)$$ and $$y(t)= 3\sin(t)$$, for $$0 \le t \le \frac{\pi}{2}$

Circular Motion in Vector-Valued Form

A particle moves along a circle of radius 5 with its position given by $$ r(t)=\langle 5*\cos(t),\;

Conversion and Differentiation of a Polar Curve

Consider the polar curve defined by $$ r=2+\sin(\theta) $$. Study its conversion to Cartesian coordi

Displacement from a Vector-Valued Velocity Function

A particle's velocity is given by $$\vec{v}(t)=\langle \cos(t), \sin(t), t \rangle$$ for $$t \in [0,

Drone Altitude Measurement from Experimental Data

A drone’s altitude (in meters) is recorded at various times (in seconds) as shown in the table below

Exponential Decay in Vector-Valued Functions

A particle moves in the plane with its position given by the vector-valued function $$\vec{r}(t)=\la

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Exponential-Logarithmic Particle Motion

A particle moves in the plane with its position given by the parametric equations $$x(t)=e^{t}+\ln(t

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

Motion Along a Helix

A particle moves along a helix defined by $$\mathbf{r}(t)=\langle \cos(t), \sin(t), t \rangle$$.

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

Parametric Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^2-

Parametric Tangent Line and Curve Analysis

For the curve defined by the parametric equations $$x(t)=t^{2}$$ and $$y(t)=t^{3}-3t$$, answer the f

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Projectile Motion in Parametric Form

A projectile is launched with an initial speed of $$20\,m/s$$ at an angle of $$30^\circ$$ above the

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume increases at a constant rate of $$30\,cm^3/

Vector-Valued Kinematics

A particle follows a path in space described by the vector-valued function $$r(t) = \langle \cos(t),

Weather Data Analysis from Temperature Table

A meteorologist records the temperature (in $$^\circ C$$) at a weather station at various times (in