AP Calculus BC FRQ Room

Absolute Value Function Limits

Examine the function $$f(x)=\frac{|x-2|}{x-2}$$.

Algebraic Manipulation with Radical Functions

Let $$f(x)= \frac{\sqrt{x+5}-3}{x-4}$$, defined for $$x\neq4$$. Answer the following:

Analyzing a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Analyzing Limits of a Combined Exponential‐Log Function

Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and

Analyzing Limits of a Composite Function

Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:

Application of the Squeeze Theorem with Trigonometric Functions

Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior

Caffeine Metabolism in the Human Body

A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu

Continuity and the Intermediate Value Theorem in Temperature Modeling

A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ

Continuity in a Parametric Function Context

A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an

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

Economic Model of Depreciating Car Value

A car purchased for $$30,000$$ dollars depreciates in value by $$15\%$$ each year. The value of the

Environmental Pollution Modeling

In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

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

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

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Graphical Analysis of Limits and Asymptotic Behavior

A graphical study titled 'Graph of Experimental Data' shows the measured concentration of a chemical

Graphical Analysis of Volume with a Jump Discontinuity

A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer

Horizontal and Vertical Asymptotes of a Rational Function

Let $$h(x)=\frac{2*x^2-3*x+1}{x^2-1}.$$ Answer the following:

Intermediate Value Theorem Application

Suppose $$f(x)$$ is a continuous function on the interval $$[1, 5]$$ with $$f(1) = -2$$ and $$f(5) =

Investigating Infinite Limits: Vertical and Horizontal Asymptotes

Given the function $$f(x)=\frac{2*x}{x-3}$$, answer the following questions: (a) Determine $$\lim_{x

Limits Involving Infinity and Vertical Asymptotes

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

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

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 Infinite Limits in Rational Functions

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

Piecewise Function Critical Analysis

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

Pollution Level Analysis and Removable Discontinuity

A function $$f(x)$$ represents the concentration of a pollutant (in mg/L) in a river as a function o

Saturation of Drug Concentration in Blood

A patient is given a drug with each dose containing 50 mg. However, due to metabolism, only 20% of t

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 $$x=0$$.

Temperature Change Analysis

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

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

Car Motion: Velocity and Acceleration

A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Derivatives of Inverse Functions

Let $$f(x)=\ln(x)$$ with inverse function $$f^{-1}(x)=e^x$$. Answer the following parts.

Differentiation in Exponential Growth Models

A population is modeled by $$P(t)=P_0e^{r*t}$$ with the initial population $$P_0=500$$ and growth ra

Error Bound Analysis for Cos(x) Approximations in Physical Experiments

In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Graphical Estimation of Tangent Slopes

Using the provided graph of a function g(t), analyze its rate of change at various points.

Growth Rate of a Bacterial Colony

The radius of a bacterial colony is modeled by $$r(t)= \sqrt{4*t+1}$$, where t (in hours) represents

Heat Transfer in a Rod: Modeling and Differentiation

The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me

Implicit Differentiation and Tangent Line Slope

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

Implicit Differentiation on a Circle

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

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

Linearization and Tangent Approximations

Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$

Maclaurin Series for e^x Approximation

Consider the function $$f(x)=e^x$$, which models many growth processes in nature. Use its Maclaurin

Optimization and Tangent Lines

A rectangular garden is to be constructed along a river with 100 meters of fencing available for thr

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Particle Motion in the Plane

A particle moves in the plane with its position given by $$x(t)=t^2-4*t+1$$ and $$y(t)=3*t-2.5$$, wh

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

Pollutant Levels in a Lake

A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre

Radioactive Decay and Derivative

A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is

Savings Account Growth: From Discrete Deposits to Continuous Derivatives

An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con

Secant and Tangent Slope Analysis

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

Taylor Expansion of a Polynomial Function Centered at x = 1

Given the polynomial function $$f(x)=3+2*x- x^2+4*x^3$$, analyze its Taylor series expansion centere

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

Vector Function and Motion Analysis

A particle moves according to the vector function $$\vec{r}(t)=\langle 2*\cos(t), 2*\sin(t)\rangle$$

Analyzing an Implicit Function with Mixed Variables

Consider the curve defined by $$x^3 + x*y + y^3 = 3$$. Analyze the derivative of the curve at a give

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)

Chain Rule with Trigonometric Composite Function

Examine the function $$ h(x)= \sin((2*x^2+1)^2) $$.

Coffee Cooling Dynamics using Inverse Function Differentiation

A cup of coffee cools according to the model $$T=100-a\,\ln(t+1)$$, where $$T$$ is the temperature i

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

Differentiation of an Inverse Function

Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a

Differentiation of an Inverse Trigonometric Form

Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.

Differentiation of Composite Inverse Trigonometric Function involving a Rational Function

Differentiate the function $$f(x)= \arccos\left(\frac{3*x}{1+x^2}\right)$$ with respect to $$x$$ and

Enzyme Kinetics in a Biochemical Reaction

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

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

Higher-Order Derivatives via Implicit Differentiation

Consider the implicit relation $$x^2 + x*y + y^2 = 7$$.

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 in a Circle

Consider the circle defined by $$ x^2+y^2=49 $$.

Implicit Differentiation in a Cost-Production Model

In an economic model, the relationship between the production level $$x$$ (in units) and the average

Implicit Differentiation in Economic Equilibrium

In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand

Implicit Differentiation: Circle and Tangent Line

The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva

Inverse Function Derivative in an Exponential Model

Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p

Inverse Function Differentiation for a Quadratic Function

Let $$ f(x)= (x+1)^2 $$ with the domain $$ x\ge -1 $$. Consider its inverse function $$ f^{-1}(y) $$

Inverse Function Differentiation for a Trigonometric-Polynomial Function

Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with

Inverse Function Differentiation with Combined Logarithmic and Exponential Terms

Let $$f(x)=e^{x}+\ln(x)$$ for $$x>1$$ and let g be its inverse function. Answer the following.

Inverse Function Differentiation with Composite Trigonometric Functions

Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$

Multi-step Differentiation of a Composite Logarithmic Function

Consider the function $$F(x)= \sqrt{\ln\left(\frac{1+e^{2*x}}{1-e^{2*x}}\right)}$$, defined for valu

Rocket Fuel Consumption Analysis

A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)

Second Derivative of an Implicit Function

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

Tangent Line for a Parametric Curve

A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.

Trigonometric Composite Inverse Function Analysis

Consider the function $$f(x)=\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

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 Function Values Using Differentials

Let $$f(x)=\sqrt{x}$$. Use linearization near $$x=25$$ to approximate $$\sqrt{25.5}$$.

Differentiation and Concavity for a Non-Motion Problem: Water Filling a Tank

The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an

Estimating the Rate of Change from Reservoir Data

A reservoir's water level h (in meters) was recorded at different times, as shown in the table below

Exponential Function Inversion

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

Forensic Gas Leakage Analysis

A gas tank under investigation shows leakage at a rate of $$O(t)=3t$$ (liters per minute) while it i

GDP Growth Analysis

A country's GDP (in billions of dollars) is modeled by the function $$G(t)=200e^{0.04*t}$$, where t

Interpreting the Derivative in Straight Line Motion

A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Linearization Approximation

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

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

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 $

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Projectile Motion with Exponential Term

A projectile's height is given by $$h(t)=50t-5t^2+e^{-0.5t}$$, where h is measured in meters and t i

Related Rates in a Conical Water Tank

Water is being pumped into a conical tank at a rate of $$2\;m^3/min$$. The tank has a height of 6 m

Revenue and Marginal Analysis

A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit

River Flow Diversion

At a river junction, water flows in at a rate of $$I(t)=30+5t$$ (cubic feet per second) and exits at

Road Trip Distance Analysis

During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is

Series Integration for Work Calculation

A force along a displacement is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n+2}$$ (in Ne

Series Integration in Fluid Flow Modeling

The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$

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

Temperature Change of Coffee: Exponential Cooling

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

Analysis of an Exponential-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

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:

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Arc Length of a Parametric Circular Arc

A curve is defined parametrically by $$x(t) = 2*\cos(t)$$ and $$y(t) = 2*\sin(t)$$, where t varies f

Car Depreciation Analysis

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

Composite Functions and Derivatives

Let $$h(x)=f(g(x))$$ where $$f(u)=u^2+3$$ and $$g(x)=\sin(x)$$. Analyze the composite function on th

Concavity Analysis in a Revenue Model

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -

Concavity and Inflection Points

The function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$ models a certain process. Use the second derivative to

Concavity and Inflection Points of an Exponential Log Function

Consider the function $$f(x)= x\,e^{-x} + \ln(x)$$ for $$x > 0$$. Analyze the concavity of f.

Concavity and Points of Inflection

Consider the function $$f(x)=x^3 - 6*x^2 + 9*x + 2$$. Analyze the concavity of the function using th

Construction Payment Milestones

A construction project is structured around milestone payments. The first payment is $$10000$$ dolla

Differentiability and Optimization of a Piecewise Function

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

Error Estimation in Approximating $$e^x$$

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

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

Function Behavior Analysis

Consider the function \( f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1 \). Answer the following parts:

Garden Design Optimization

A gardener wants to design a rectangular garden adjacent to a river, so that fencing is required for

Implicit Differentiation and Tangent to an Ellipse

Consider the ellipse defined by the equation $$4*x^2 + 9*y^2 = 36$$. Answer the following parts:

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.

Mean Value Theorem in River Flow

A river cross‐section’s depth (in meters) is modeled by the function $$f(x) = x^3 - 4*x^2 + 3*x + 5$

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 Rectangular Garden

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

Optimizing Material for a Container

An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.

Parameter Estimation in a Log-Linear Model

In a scientific experiment, the data is modeled by $$P(t)= A\,\ln(t+1) + B\,e^{-t}$$. Given that $$P

Radiocarbon Dating in Artifacts

An archaeological artifact contains a radioactive isotope with an initial concentration of 100 units

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ increases at a constant rate of $$\fr

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

Revenue Optimization in Business

A company’s price-demand function is given by $$P(x)= 50 - 0.5*x$$, where $$x$$ is the number of uni

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

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.

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Analyzing a Cumulative Distribution Function (CDF)

A chemical reaction has a time-to-completion modeled by the cumulative distribution function $$F(t)=

Approximating Water Volume Using Riemann Sums

A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a

Arc Length of an Architectural Arch

An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len

Area Estimation Using Riemann Sums for $$f(x)=x^2$$

Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef

Continuous Antiderivative for a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,

Cyclist's Distance Accumulation Function

A cyclist’s total distance traveled is modeled by $$D(t)= \int_{0}^{t} (5+\sin(u))\, du + 2$$ kilom

Evaluating an Integral Involving an Exponential Function

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

Evaluating an Integral Using U-Substitution

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

Finding Area Between Two Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.

Fundamental Theorem and Total Accumulated Growth

A bacteria culture grows according to the logistic model $$\frac{dN}{dt}=N\left(1-\frac{N}{10000}\r

Improper Integral Convergence

Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.

Integration by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Integration of a Piecewise Function for Total Area

Consider the piecewise function $$f(x)$$ defined by: $$f(x)=3-x$$ for $$0 \le x \le 3$$, and $$f(x)=

Integration Using U-Substitution

Evaluate the definite integral $$\int_{0}^{2} (3*x+1)^{4} dx$$ using u-substitution. Answer the foll

Logistic Growth and Population Integration

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

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu

Particle Displacement and Total Distance

A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le

Population Model Inversion and Accumulation

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

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Total Cost from a Marginal Cost Function

A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre

Trapezoidal Approximation for a Curved Function

Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:

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

Volume of a Solid by the Shell Method

Consider the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and the vertical line $$x=4$$.

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 \(\

Work on a Nonlinear Spring

A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di

Analysis of a Piecewise Function with Potential Discontinuities

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

Bacterial Growth with Predation

A bacterial culture grows according to the differential equation $$\frac{dB}{dt}= r*B - P$$, where $

Compound Interest and Investment Growth

An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the

Cooling and Mixing Combined Problem

A container holds 2 L of water initially at 80°C. Cold water at 20°C flows into the container at a r

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

Drug Concentration in the Bloodstream

A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif

Estimating Total Change from a Rate Table

A car's velocity (in m/s) is recorded at various times according to the table below:

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

Exact Differential Equation

Examine the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y)\,dy = 0 $$. Determine if the

Exact Differential Equations

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2+2*x*y)\,dy = 0 $$. Answer the followi

FRQ 12: Bacterial Growth with Limiting Resources

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where

FRQ 16: Harvesting in a Predator-Prey Model

A prey population $$P(t)$$ in a marine ecosystem is modeled by the differential equation $$\frac{dP}

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

Logistic Population Growth Model

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

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

A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti

Newton's Law of Cooling

A hot liquid cools in a room maintained at a constant temperature $$T_{room}$$. The temperature $$T(

RC Circuit: Voltage Decay

In an RC circuit, the voltage across a capacitor satisfies $$\frac{dV}{dt} = -\frac{1}{R*C} * V$$. G

Relative Motion with Acceleration

A car starts from rest and its velocity $$v(t)$$ (in m/s) satisfies the differential equation $$\fra

Separable Differential Equation with Absolute Values

Consider the differential equation $$\frac{dy}{dx} = \frac{|x|}{y}$$ with the condition that $$y>0$$

Separable Differential Equation with Initial Condition

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

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

Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}= x$$. A slope field for this differential equatio

Temperature Change with Variable Ambient Temperature

A heated object is cooling in an environment where the ambient temperature changes over time. For $$

Variable Carrying Capacity in Population Dynamics

In a modified logistic model, the carrying capacity of a population is time-dependent and given by $

Viral Spread on Social Media

Let $$V(t)$$ denote the number of viral posts on a social media platform. Posts go viral at a consta

Approximating Functions using Taylor Series

Consider the function $$f(x)= \ln(1+2*x)$$. Use Taylor series methods to approximate and analyze thi

Area Between Curves from Experimental Data

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

Area Enclosed by a Cardioid in Polar Coordinates

Consider the polar curve given by $$r(\theta)=1+\cos(\theta)$$.

Area Under a Parametric Curve

Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u

Average Cost Function in Production

A factory’s cost function for producing $$x$$ units is modeled by $$C(x)=0.5*x^2+3*x+100$$, where $$

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 Computation

Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=

Average Value of a Piecewise Function

Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x

Average Value of a Piecewise Function

Consider the piecewise function defined on $$[0,4]$$ by $$ f(x)= \begin{cases} x^2 & \text{for } 0

Average Velocity of a Car

A car's velocity is given by $$v(t)=20-4*\ln(t+1)$$ (in m/min) for $$t$$ in minutes on the interval

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

Center of Mass of a Lamina

A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a

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

Cyclist Average Speed Calculation

A cyclist’s velocity is given by $$v(t) = t^2 - 4*t + 6$$ (in m/s) for $$t$$ in the interval $$[0,4]

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Inflow vs Outflow: Water Reservoir Capacity

A reservoir receives water with an inflow rate given by $$I(t)=20+5\sin(t)$$ (liters/min) and discha

Moment of Inertia of a Thin Plate

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

Optimization and Integration: Maximum Volume

A company designs open-top cylindrical containers to hold $$500$$ liters of liquid. (Recall that $$1

Optimizing the Thickness of a Cooling Plate

The local heat conduction efficiency at a point on a cooling plate is modeled by the function $$A(x)

Polar Coordinates: Area of a Region

A region in the plane is described in polar coordinates by the equation $$r= 2+ \cos(θ)$$. Determine

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

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 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 by the Shell Method

Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is revolved about t

Volume of a Solid via the Disc Method

The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th

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

Volume of an Irregular Tank

A water tank has a varying cross-sectional profile described by $$y(x)=\sqrt{25 - (x-5)^2}$$, for $$

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Work Done by a Variable Force

A variable force given by $$F(x)= 2*x + 3$$ (in Newtons) is applied to an object as it moves along a

Work Done by a Variable Force

A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as

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 of a Parametric Curve with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Arc Length of a Polar Curve

Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Area Enclosed by a Polar Curve: Lemniscate

The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.

Comparing Parametric, Polar, and Cartesian Representations

An object moves along a curve described by the parametric equations $$x(t)= \frac{t}{1+t^2}$$ and $$

Conversion Between Polar and Cartesian Coordinates

Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.

Conversion to Cartesian and Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.

Converting and Analyzing a Polar Equation

Examine the polar equation $$r=2+3\cos(\theta)$$.

Exploring Polar Curves: Spirals and Loops

Consider the polar curve $$r=θ$$ for $$0 \le θ \le 4\pi$$, which forms a spiral. Analyze the spiral

Integration of Speed in a Parametric Motion

For the parametric curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$,

Intersection of Polar and Parametric Curves

Consider the polar curve given by $$r = 2\cos(θ)$$ and the parametric curve defined by $$x(t)= 1+t^2

Lissajous Figures and Their Properties

A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)

Motion Analysis via a Vector-Valued Function

An object's position is described by the vector function $$\mathbf{r}(t)= \langle e^{-t}, \; \ln(1+t

Motion in a Damped Force Field

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t)

Oscillatory Behavior in Vector-Valued Functions

Examine the vector-valued function $$\mathbf{r}(t)=\langle \cos(2*t), \sin(3*t), \cos(t)\sin(2*t) \r

Oscillatory Motion in a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in

Parametric Curves and Intersection Points

Two curves are defined by $$C_1: x(t)=t^2,\, y(t)=2*t+1$$ and $$C_2: x(s)=4-s^2,\, y(s)=3*s$$. Find

Parametric Intersection and Enclosed Area

Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given

Parametric Motion and Change of Direction

A particle moves along a path defined by the parametric equations $$x(t)=t^3-3t$$ and $$y(t)=2t^2$$

Parametric Representation of an Ellipse

An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Projectile Motion via Parametric Equations

A projectile is launched with initial speed $$v_0 = 20\,m/s$$ at an angle of $$45^\circ$$. Its motio

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

Vector-Valued Function and Particle Motion

Consider the vector-valued function $$\vec{r}(t)= \langle e^t, \sin(t), \cos(t) \rangle$$ representi

Vector-Valued Functions and Curvature

Let the vector-valued function be $$\vec{r}(t)= \langle t, t^2, t^3 \rangle$$.

Vector-Valued Kinematics

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

Wind Vector Analysis in Navigation

A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)