AP Calculus BC FRQ Room

Algebraic Method for Evaluating Limits

Consider the function $$h(x)=\frac{x^2-9}{x-3}.$$ Answer the following parts.

Analysis of a Jump Discontinuity

Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Approaching Vertical Asymptotes

Consider the function $$g(x)=\frac{3}{(x-2)^2}-\frac{1}{x-2}.$$ Answer the following:

Complex Rational Function and Continuity Analysis

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

Continuity Across Piecewise‐Defined Functions with Mixed Components

Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Continuity and Asymptotes of a Log‐Exponential Function

Examine the function $$f(x)= \ln(e^x + e^{-x})$$.

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 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 a Piecewise Function with Polynomial and Trigonometric Components

Consider the function $$f(x)= \begin{cases} x^2-1 & \text{if } x < \pi \\ \sin(x) & \text{if } x \ge

Continuity in Piecewise Functions with Parameters

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

Defining a Function with a Unique Limit Behavior

Construct a function $$f(x)$$ that meets the following conditions: - It is defined and continuous fo

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

Evaluating Limits Involving Radical Expressions

Consider the function $$h(x)= \frac{\sqrt{4x+1}-3}{x-2}$$.

Exploring the Squeeze Theorem

Define the function $$ f(x)= \begin{cases} x^2*\cos\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0

Factorable Discontinuity Analysis

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

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 Removable Discontinuity

A graph of a function f is provided (see stimulus). The graph shows that f has a hole at (2, 4) whil

Jump Discontinuity Analysis using Table Data

A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer

Limits and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits and Continuity in Particle Motion

A particle moves along a straight line with velocity given by $$v(t)=\frac{t^2-4}{t-2}$$ for t ≠ 2 s

Limits and Continuity of Radical Functions

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

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Limits Involving Absolute Value

Let $$h(x)=\frac{|x^2-9|}{x-3}.$$ Answer the following parts.

Limits Involving Exponential Functions

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

Limits Involving Radical Functions

Examine the function $$m(x)=\frac{\sqrt{x}-2}{x-4}$$.

Limits via Improper Integration Representation

Consider the function defined by the integral $$f(x)= \int_{1}^{x} \frac{1}{t^2} dt$$ for x > 1. Add

One-Sided Limits for a Piecewise Inflow

In a pipeline system, the inflow rate is modeled by the piecewise function $$R_{in}(t)= \begin{case

Oscillatory Behavior and Squeeze Theorem

Consider the function $$h(x)= x^2 \cos(1/x)$$ for $$x \neq 0$$ with $$h(0)=0$$.

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x\neq2$$. Answer the following:

Squeeze Theorem Application

Consider the function $$f(x)=\frac{\sin(3*x)}{x}$$ defined for x ≠ 0.

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 $$x=0$$.

Water Tank Flow Analysis

A water tank receives water from an inlet and drains water through an outlet. The inflow rate is giv

Applying the Quotient Rule

Let the function $$R(x)=\frac{x^2+1}{2*x-1}$$ represent a ratio used to gauge the rate of return on

Average vs Instantaneous Rate of Change in Temperature Data

The table below shows the temperature (in °C) recorded at several times during an experiment. Use t

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

Differentiation of a Trigonometric Function

Let $$f(x)=\sin(x)+x*\cos(x)$$. Differentiate the function using the sum and product rules.

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

Economic Model Rate Analysis

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

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Implicit Differentiation: Mixed Exponential and Polynomial Equation

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

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Logarithmic Differentiation Simplification

Consider the function $$h(x)=\ln\left( \frac{(x^2+1)^{3}*e^{2*x}}{\sqrt{x+2}} \right)$$.

Manufacturing Cost Function and Instantaneous Rate

The total cost (in dollars) to produce x units of a product is given by $$C(x)= 0.2x^3 - 3x^2 + 50x

Parametric Analysis of a Curve

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

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Product of Exponential and Trigonometric Functions

Let $$f(x)=e^(2*x)*\sin(x)$$. This function models oscillating growth. Answer the following:

Radioactive Decay with Logarithmic Correction

A radioactive substance decays following the model $$A(t)=A_0*e^{-k*t}+\ln(t+1)$$, where $$t$$ is th

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:

Tangent Line Approximation

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

Tangent Line Approximation vs. Taylor Series for ln(1+x)

An engineer studying the function $$f(x)=\ln(1+x)$$ is comparing the tangent line approximation with

Temperature Change with Provided Data

The temperature at different times after midnight is modeled by $$T(t)=5*\ln(t+1)+20$$, with $$t$$ i

Temperature Change: Secant vs. Tangent Analysis

A scientist recorded the temperature $$T$$ (in °C) at various times $$t$$ (in seconds) as shown in t

Using Taylor Series to Approximate the Derivative of sin(x²)

A physicist is analyzing the function $$f(x)=\sin(x^2)$$ and requires an approximation for its deriv

Velocity and Acceleration Analysis

A particle moving along a straight line has a velocity function given by $$v(t)=2*t^2 - 8*t + 3$$ (i

Water Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

Water Treatment Plant Simulator

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

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

Composite Function with a Radical in a Shadow Length Model

The length of a shadow cast by an object is modeled by the function $$s(t)= \sqrt{100+4*t^2}$$, wher

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Composite, Implicit, and Inverse: A Multi-Method Analysis

Let $$F(x)=\sqrt{\ln(5*x+9)}$$ for all x such that $$5*x+9>0$$, and let y = F(x) with g as the inver

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 Involving an Inverse Function and Logarithms

Let $$f(x)= x^2+ 3*x+ 2$$, and let y be defined by $$y= \ln(f^{-1}(e^{x}))$$, where $$f^{-1}$$ is th

Differentiation of a Log-Exponential Composition with Critical Points

Consider the function $$k(x)=x*\ln(e^{x}+3)$$. Answer the following parts.

Differentiation of an Inverse Trigonometric Form

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

Implicit Differentiation and Inverse Challenges

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

Implicit Differentiation for a Spiral Equation

Consider the curve given by the equation $$x^2 + y^2 = 4*x*y$$. Analyze its derivative using implici

Implicit Differentiation in a Hyperbola-like Equation

Consider the equation $$ x*y = 3*x - 4*y + 12 $$.

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 Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Implicit Differentiation with Logarithms and Products

Consider the equation $$ \ln(x+y) + x*y = \ln(4)+4 $$.

Indoor Air Quality Control

In a controlled laboratory environment, the rate of fresh air introduction is modeled by the composi

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

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 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 in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

Inverse Function Differentiation with a Logarithmic Function

Let $$ f(x)= \ln(x+3) $$. Consider its inverse function $$ f^{-1}(y) $$.

Inverse of a Radical Function with Domain Restrictions

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

Optimization with Composite Functions - Minimizing Fuel Consumption

A car's fuel consumption (in liters per 100 km) is modeled by $$F(v)= v^2 * e^{-0.1*v}$$, where $$v$

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 Growth Analysis Using Composite Functions

A population model is defined by $$P(t)= f(g(t))$$ where $$g(t)= e^{-t} + 3$$ and $$f(u)= 2*u^2$$. H

Power Series Representation and Differentiation of a Composite Function

Let $$f(x)= \sin(x^2)$$ and consider its Maclaurin series expansion.

Projectile Motion and Composite Exponential Functions

A projectile’s height at time $$t$$ (in seconds) is modeled by the function $$h(t)= e^{- (t-2)^2}$$.

Temperature Modeling and Composite Functions

A weather balloon ascends and the temperature at altitude x (in kilometers) is modeled by $$T(x) = \

Vector Function Trajectory Analysis

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

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 Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

Arc Length Calculation

Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.

Cubic Function with Parameter and Its Inverse

Examine the family of functions given by $$f(x)=x^3+k*x$$, where $$k$$ is a constant.

Data Table Inversion

A function $$f(x)$$ is represented by the following data table. Use the data to analyze the inverse

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Economics: Cost Function and Marginal Analysis

A company's cost function is given by $$C(x)= 0.5*x^3 - 4*x^2 + 10*x + 100$$, where $$x$$ represents

Ellipse Tangent Line Analysis

Consider the ellipse given by the implicit equation $$9*x^2 + 4*y^2 = 36$$. Answer the following par

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Implicit Differentiation: Tangent to a Circle

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

Industrial Mixer Flow Rates

In an industrial mixer, an ingredient is added at a rate of $$I(t)=7t$$ (kg per minute) and is consu

Inflating Balloon: Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of 10 in³/s.

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

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

L'Hôpital’s Rule in Chemical Reaction Rates

In a chemical reaction, the ratio of certain concentrations is modeled by $$R(x)=\frac{3*x^2-2*x+1}{

Linearization Approximation Problem

Given the function $$f(x)=\sqrt{x+4}$$, use linearization to approximate the value of $$f(5.1)$$.

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

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 Function Series Analysis

The function $$L(x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n}$$ represents $$\ln(x)$$ centere

Maclaurin Series for ln(1+x)

Consider the function $$f(x)= \ln(1+x)$$. Its Maclaurin series may be used to approximate values of

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Piecewise Velocity and Acceleration Analysis

A particle moves along a straight line with its velocity given by $$ v(t)= \begin{cases} t^2-4*t+3,

Population Growth: Rate of Change Analysis

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

Related Rates: Inflating Spherical Balloon

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

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 Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Series Integration in Fluid Flow Modeling

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

Series Representation of a CDF

A cumulative distribution function (CDF) is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^

Spherical Balloon Inflation

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

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

Application in Motion: Approximate Velocity using Taylor Series

A particle’s position is given by $$s(t)=e^{-t}+t^2$$. Using Taylor series approximations near $$t=0

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

Average and Instantaneous Velocity Analysis

A bird’s position is given by $$s(t)=2*t^2-t+1$$ (in meters) for $$t\in[0,3]$$ seconds.

Average vs. Instantaneous Profit Rate

A company’s profit is modeled by the function $$P(t)= 0.2*t^3 - 3*t^2 + 10*t$$, where $$t$$ is the t

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

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Concavity and Inflection Points in a Trigonometric Function

Consider the function $$f(x)=\sin(x)-\frac{1}{2}*x$$ on the interval [0, 2π]. Answer the following p

Convergence and Series Approximation of a Simple Function

Consider the function defined by the power series $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n}{n+1} * x^n$

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

Examine the function $$f(x)= x^2\,e^{-x} + \ln(x)$$ defined for $$x > 0$$. Apply methods to find its

Graph Analysis of Experimental Data

A set of experimental measurements was recorded over time. Analyze the following data regarding the

Graph Interpretation of a Function's Second Derivative

Using the provided graph of the second derivative, analyze the concavity of the original function $$

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

Logistic Growth in Biology

The logistic growth of a species is modeled by $$P(t) = \frac{1}{1 + e^{-0.5*(t-4)}}$$, where t is i

Mean Value Theorem in Motion

A car travels along a straight road and its position is modeled by $$s(x) = x^2$$ (in kilometers), w

Modeling Population Growth: Rate of Change

A population is modeled by the function $$ P(t)=100e^{0.05t}-20t, \quad 0 \le t \le 10,$$ where $$t

Rational Function Discontinuities

Consider the rational function $$ R(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}.$$ Answer the following parts:

Relative Extrema Using the First Derivative Test

Consider the function $$ f(x)=e^{-x^2}.$$ Answer the following parts:

Retirement Savings with Diminishing Deposits

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

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

River Sediment Transport

Sediment enters a river from a landslide at a rate of $$S_{in}(t)=4*\exp(0.2*t)$$ tonnes/day and is

Series Approximation in Engineering: Oscillation Amplitude

An engineer models the oscillation amplitude by $$A(t)=\sin(0.2*t)\,e^{-0.05*t}$$. Derive the Maclau

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Volume of a Solid of Revolution Using the Washer Method

Find the volume of the solid obtained by revolving the region bounded by $$y=\sqrt{x}$$, $$y=\frac{x

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

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

Analyzing and Integrating a Function with a Removable Discontinuity

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

Antiderivative with an Initial Condition

Given the function $$f(x)=6*x$$, find a function $$F(x)$$ such that $$F'(x)=f(x)$$ and $$F(2)=5$$.

Antiderivatives and the Fundamental Theorem

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

Area Between a Curve and Its Tangent

For the function $$f(x)=x^3-3*x^2+2*x$$, analyze the area between the curve and its tangent line at

Area Between the Curves f(x)=x² and g(x)=2x+3

Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete

Average Temperature from a Continuous Function

Along a metal rod, the temperature is modeled by $$f(t)= t^3 - 3*t^2 + 2*t$$ (in $$^\circ C$$) for

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Bacterial Growth with Logarithmic Integration

A bacterial culture grows at a rate given by $$P'(t)=100/(t+2)$$ (in bacteria per hour). Given that

Calculating Work Using Integration

A variable force is given by $$F(x)=5*x^2-2*x$$ (in Newtons) and is applied along the direction of m

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

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

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

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

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

Probability Density Function and Expected Value

Let the probability density function (pdf) be defined by $$f(x)=k*x*e^{-x}$$ for $$x\ge0$$.

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

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

Riemann and Trapezoidal Sums with Inverse Functions

Consider the function $$f(x)= 3*\sin(x) + 4$$ defined on the interval \( x \in [0, \frac{\pi}{2}] \)

Temperature Change in a Material

A laser heats a material such that its temperature changes at a rate given by $$\frac{dT}{dt} = 2*\s

Volume by Disk Method of a Rotated Region

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

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Chemical Reaction Rate

A chemical reaction causes the concentration $$A(t)$$ of a reactant to decrease according to the rat

Chemical Reactor Mixing

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

Combined Differential Equations and Function Analysis

Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y

Cooling Model Using Newton's Law

Newton's law of cooling states that the temperature T of an object changes at a rate proportional to

FRQ 1: Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}=x$$. Answer the following parts.

FRQ 5: Mixing Problem in a Tank

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Implicit Differentiation in a Differential Equation Context

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

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

Logistic Differential Equation Analysis

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

Modeling Cooling and Heating: Temperature Differential Equation

Suppose the temperature of an object changes according to the differential equation $$\frac{dT}{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

Newton's Law of Cooling

A cup of coffee at an initial temperature of $$90^\circ C$$ is placed in a room maintained at a cons

Newton's Law of Cooling

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

Non-linear Differential Equation using Separation of Variables

Consider the differential equation $$\frac{dy}{dx}= \frac{x*y}{x^2+1}$$. Answer the following questi

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, where $

RC Circuit Differential Equation

In an RC circuit, the capacitor charges according to the differential equation $$\frac{dQ}{dt}=\frac

Separable DE with Trigonometric Component

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

Separable Differential Equation and Slope Field Analysis

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

Separable Differential Equation with Initial Condition

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

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Slope Field and Solution Curve Sketching

Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq

Water Pollution with Seasonal Variation

A river receives a pollutant with a time-varying influx modeled by $$I(t)=20+5\cos(0.5*t)$$ kg/day,

Arc Length of a Logarithmic Curve

Consider the curve defined by $$y = \ln(\sec(t))$$ for $$t$$ in the interval $$[0,\pi/4]$$. Determin

Area Between Curves: Parabolic and Linear Functions

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu

Area Between Exponential Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=e^{-2*x}$$ for $$x\ge0$$. Answer the following:

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

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 and Instantaneous Acceleration

For a particle, the acceleration is given by $$a(t)=4*\sin(t)-t$$ (in m/s²) for $$t\in[0,\pi]$$. Giv

Average Power Consumption

A household's power consumption is modeled by the function $$P(t)=3+2*\sin\left(\frac{\pi}{12}*t\rig

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Value and Critical Points of a Trigonometric Function

Consider the function $$f(x)=\sin(2*x)+\cos(2*x)$$ on the interval $$\left[0,\frac{\pi}{2}\right]$$.

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 [

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

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

Error Analysis in Taylor Polynomial Approximations

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

Finding the Centroid of a Planar Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ between the vertical lines $$x=0$$ a

Force on a Submerged Plate

A vertical rectangular plate is submerged in water. The plate is 3 m wide and extends from a depth o

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. Answer the following:

Particle Motion with Velocity Reversal

A particle moves along a straight line with an acceleration given by $$a(t)=12-6*t$$ (in m/s²) for $

Profit-Cost Area Analysis

A company’s profit (in thousands of dollars) is modeled by $$P(x) = -x^2 + 10*x$$ and its cost by $$

Projectile Motion with Constant Acceleration

A ball is thrown upward and moves under the constant acceleration due to gravity $$a(t)=-9.8$$ (in m

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

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

Total Change in Temperature Over Time (Improper Integral)

An object cools according to the function $$\Delta T(t) = e^{-0.5*t}$$, where $$t\ge 0$$ is time in

Total Distance Traveled with Changing Velocity

A runner’s velocity is given by $$v(t)=3*(t-1)*(t-4)$$ m/s for $$0 \le t \le 5$$ seconds. Note that

Volume by Cross-Section: Rotated Region

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$, with the intersection points form

Volume of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Volume of a Solid by the Washer Method

The region bounded by $$y=x^2$$ and $$y=4$$ is rotated about the x-axis, forming a solid with a hole

Volume Using Washer Method

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat

Volume with Square Cross Sections

The region in the $$xy$$-plane is bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. A solid is formed

Work Done by a Variable Force

A force acting on an object is given by the function $$F(x)=3*x^2$$ (in Newtons). The object moves a

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

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Arc Length of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=3*\cos(t)$$ and $$y(t)=3*\sin(t)$$ for

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 \

Area Between Polar Curves

Consider the polar curves $$ r_1=2+\cos(\theta) $$ and $$ r_2=1+\cos(\theta) $$. Although the curves

Area between Two Polar Curves

Given two polar curves: $$r_1 = 1+\cos(\theta)$$ and $$r_2 = 2\cos(\theta)$$, consider the region wh

Average Position from a Vector-Valued Function

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

Computing the Area Between Two Polar Curves

Consider the polar curves given by $$R(\theta)=3+2*\cos(\theta)$$ (outer curve) and $$r(\theta)=1+\c

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 \

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,

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

Implicit Differentiation with Implicitly Defined Function

Consider the equation $$x^2+xy+y^2=7$$, which defines $$y$$ implicitly as a function of $$x$$.

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

Intersection Points of Polar Curves

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

Motion in the Plane: Logarithmic and Radical Components

A particle’s position in the plane is given by the vector-valued function $$\mathbf{r}(t)=\langle \l

Parametric Particle with Acceleration and Jerk

A particle's motion is given by the parametric equations $$x(t)=t^4-6*t^2$$ and $$y(t)=2*t^3-9*t$$ f

Parametric Representation of Circular Motion

An object moves along a circle of radius $$5$$, with its position given by $$x(t)=5\cos(t)$$ and $$y

Parametric Slope and Arc Length

Consider the parametric curve defined by $$x(t)= t-\ln(t)$$ and $$y(t)= t\cdot\ln(t)$$ for $$t > 1$$

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

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

Related Rates: Expanding Balloon

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

Self-Intersection in a Parametric Curve

Consider the parametric curve defined by $$ x(t)=t^2-t $$ and $$ y(t)=t^3-3*t $$. Investigate whethe

Spiral Motion with a Damped Vector Function

An object moves according to the spiral vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t),\; e^{