AP Calculus BC FRQ Room

Analyzing Continuity on a Closed Interval

Suppose a function $$f(x)$$ is continuous on the closed interval $$[0,5]$$ and differentiable on the

Comparing Methods for Limit Evaluation

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

Composite Functions: Limits and Continuity

Let $$f(x)=x^2-1$$, which is continuous for all $$x$$, and let $$g(x)=f(\sqrt{x+1})$$.

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 of an Integral Function

Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{

Continuity and Asymptotes of a Log‐Exponential Function

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

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

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:

Evaluating Limits Involving Absolute Value Functions

Consider the function $$f(x)= \frac{|x-4|}{x-4}$$.

Evaluating Limits Involving Exponential and Rational Functions

Consider the limits involving exponential and polynomial functions. (a) Evaluate $$\lim_{x\to\infty}

Identifying and Removing Discontinuities

The function $$f(x)=\frac{x^2-9}{x-3}$$ is undefined at x = 3.

Intermediate Value Theorem Application

Let $$f(x)=x^3-4*x+1$$, which is continuous on the real numbers. Answer the following:

Limits Involving Absolute Value Functions

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

Limits Involving Exponential Functions

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

Limits Involving Trigonometric Ratios

Consider the function $$f(x)= \frac{\sin(2*x)}{x}$$ for $$x \neq 0$$. A table of values near $$x=0$$

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

Pendulum Oscillations and Trigonometric Limits

A pendulum’s angular displacement from the vertical is given by $$\theta(t)= \frac{\sin(2*t)}{t}$$ f

Piecewise Functions and Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-1}{x-1}, & x \neq 1 \\ k, & x=1

Rational Function and Removable Discontinuity

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

Understanding Behavior Near a Vertical Asymptote

For the function $$f(x)=\frac{1}{(x-2)^2}$$, answer the following: (a) Determine $$\lim_{x\to2} f(x)

Advanced Analysis of a Composite Piecewise Function

Consider the function $$g(x)= \begin{cases} \frac{2*x^2-8}{x-2} & x \neq 2 \\ 5 & x=2 \end{cases}$$

Analyzing a Polynomial with Higher Order Terms

Consider the function $$f(x)=4*x^5 - 2*x^3 + x - 7$$. Answer the following:

Analyzing a Removable Discontinuity in a Rational Function

Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

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

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Car Motion and Critical Velocity

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

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction (in M) is recorded over time (in seconds) as

Complex Rational Differentiation

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

Derivative Estimation from a Graph

A graph of a function $$f(x)$$ is provided in the stimulus. Using the graph, answer the following pa

Derivative Using Limit Definition

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

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Differentiating a Piecewise-Defined Function

Consider the piecewise function $$f(x)=\begin{cases}x^2+2*x, & x \le 3 \\ 4*x-5, & x > 3 \end{cases}

Differentiating Composite Functions

Let $$f(x)=\sqrt{2*x^2+3*x+1}$$. (a) Differentiate $$f(x)$$ with respect to $$x$$ using the appropr

Exploration of the Definition of the Derivative as a Limit

Consider the function $$f(x)=\frac{1}{x}$$ for $$x\neq0$$. Answer the following:

Exponential Growth Derivative

In a model of bacterial growth, the population is described by $$f(t)=5*e^(0.2*t)+7$$, where \(t\) i

Finding and Interpreting Critical Points and Derivatives

Examine the function $$f(x)=x^3-9*x+6$$. Determine its derivative and analyze its critical points.

Finding the Derivative of a Logarithmic Function

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

Fuel Storage Tank

A fuel storage tank receives oil at a rate of $$F_{in}(t)=40\sqrt{t+1}$$ liters per hour and loses o

Implicit Differentiation of a Circle

Given the equation of a circle $$x^2 + y^2 = 25$$,

Implicit Differentiation: Mixed Exponential and Polynomial Equation

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

Motion Model with Logarithmic Differentiation

A particle moves along a track with its displacement given by $$s(t)=\ln(2*t+3)*e^{-t}$$, where $$t$

Plant Growth Rate Analysis

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

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

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:

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ (in m³) and radius $$r$$ (in m) satis

Second Derivative and Concavity Analysis

Consider the function $$f(x)=x^3-6*x^2+12*x-5$$. Answer the following:

Tangent Line Approximation

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

Tangent Line to a Curve

Consider the function $$f(x)=\sqrt{x+4}$$ modeling a physical quantity. Analyze the behavior at $$x=

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Tracking a Car's Velocity

A car moves along a straight road according to the position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$,

Traffic Flow and Instantaneous Rate

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

Urban Population Flow

A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t

Using the Limit Definition for a Non-Polynomial Function

Consider the function $$f(x)= \sqrt{x+4}$$ for $$x\ge -4$$. Answer the following:

Vector Function and Motion Analysis

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

Warehouse Inventory Management

A warehouse receives shipments at a rate of $$I(t)=100e^{-0.05*t}$$ items per day and ships items ou

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 and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

Composite Function with Hyperbolic Sine

A cable's displacement over time is modeled by $$s(t)= \sinh(\ln(t+1))$$, where $$t$$ is in seconds.

Composite Population Growth Function

A population model is given by $$P(t)= e^{3\sqrt{t+1}}$$, where $$t$$ is measured in years. Analyze

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

Differentiation Involving Absolute Values and Composite Functions

Consider the function $$f(x)= \sqrt{|2*x - 3|}$$. Answer the following:

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 Nested Trigonometric Function

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

Implicit Differentiation and Concavity of a Logarithmic Curve

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

Implicit Differentiation in a Cost-Production Model

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

Implicit Differentiation in Trigonometric Equations

For the equation $$\cos(x*y) + x^2 - y^2 = 0$$, y is defined implicitly as a function of x.

Implicit Differentiation of a Circle

Consider the circle defined by $$x^2+y^2=25$$. Answer the following parts:

Implicit Differentiation on a Trigonometric Curve

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

Implicit Differentiation with an Exponential Function

Given the equation $$ e^{x*y}= x+y $$, use implicit differentiation.

Implicit Differentiation with Logarithmic Functions

Let $$x$$ and $$y$$ be related by the equation $$\ln(x*y) + x - y = 0$$.

Implicit Differentiation with Logarithms and Products

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

Implicit Equation with Logarithmic and Exponential Terms

The relation $$\ln(x+y)+e^{x-y}=3$$ defines y implicitly as a function of x. Answer the following pa

Inverse Analysis of an Exponential-Linear Function

Consider the function $$f(x)=e^{x}+x$$ defined for all real numbers. Analyze its inverse function.

Inverse Function Derivative in a Cubic Function

Let $$f(x)= x^3+ 2*x - 1$$, a one-to-one differentiable function. Its inverse function is denoted as

Inverse of a Piecewise Function

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

Investigating the Inverse of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$ with $$x \neq -3$$. Analyze its inverse.

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

Related Rates: Temperature Change in a Moving Object

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

Reservoir Levels and Evaporation Rates

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

Analysis of a Piecewise Function with Discontinuities

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

Analyzing a Motion Graph

A car's velocity over time is modeled by the piecewise function given in the graph. For $$0 \le t <

Applying L'Hospital's Rule to a Transcendental Limit

Evaluate the limit $$\lim_{x\to 0}\frac{e^{2*x}-1}{\sin(3*x)}$$.

Bacterial Population Growth

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

Chain Rule in Temperature Distribution along a Rod

A metal rod has a temperature distribution given by $$T(x)=25+15\sin\left(\frac{\pi*x}{8}\right)$$,

Chemical Concentration Rate Analysis

The concentration of a chemical in a reactor is given by $$C(t)=\frac{5*t}{t+2}$$ M (moles per liter

Chemical Reaction Temperature Change

In a laboratory experiment, the temperature T (in °C) of a reacting mixture is modeled by $$T(t)=20+

Chemistry: Rate of Change in a Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)= 50e^{-0.2*t}+5$$, wher

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.

Draining Conical Tank

Water is draining from a conical tank at a rate of $$5$$ m³/min. The tank has a height of $$10$$ m a

Draining Hemispherical Tank

A hemispherical tank of radius $$5$$ m is draining. The volume of water in the tank is given by $$V

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 Applications: Force and Motion

A force acting on a 4 kg object is given by $$F(t)= 12*t - 3$$ (Newtons), where $$t$$ is in seconds.

Estimating Rate of Change from Table Data

The following table shows the velocity (in m/s) of a car at various times recorded during an experim

Forensic Gas Leakage Analysis

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

L'Hôpital's Rule Application

Evaluate the limit: $$\lim_{t \to \infty} \frac{5*t^3 - 4*t^2 + 7}{7*t^3 + 2*t - 6}$$ using L'Hôpita

Linearization Approximation Problem

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

Linearization of Trigonometric Implicit Function

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

Logarithmic Differentiation and Asymptotic Behavior

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

Maximizing a Rectangular Enclosure Area

A farmer has 100 m of fencing to enclose a rectangular area. Answer the following:

Motion with Non-Uniform Acceleration

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

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Pollutant Scrubber Efficiency

A factory emits pollutants at a rate given by $$I(t)=100e^{-0.3t}$$ (kg per hour), and a scrubber re

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

Radical Function Inversion

Let $$f(x)=\sqrt{2*x+5}$$ represent a measurement function. Analyze its inverse.

Series Expansion in Vibration Analysis

A vibrating system has its displacement modeled by $$y(t)= \sum_{n=0}^{\infty} \frac{(-1)^n (2t)^{2*

Analysis of a Logarithmic Function

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

Application of Rolle's Theorem

Let f be a function that is continuous on $$[2,5]$$ and differentiable on $$(2,5)$$, with $$f(2) = f

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

Derivative Sign Chart and Function Behavior

Given the function $$ f(x)=\frac{x}{x^2+1},$$ answer the following parts:

Differentiability of a Piecewise Function

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

Discounted Cash Flow Analysis

A project is expected to return cash flows that decrease by 10% each year from an initial cash flow

Economic Optimization: Maximizing Profit

The profit function for a product is given by $$P(x) = -2*x^3 + 27*x^2 - 108*x + 150$$, where \(x\)

Elasticity Analysis of a Demand Function

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

Fractal Tree Branch Lengths

A fractal tree is constructed as follows: The trunk has a length of 10 meters. At each generation, e

Investment with Increasing Contributions and Interest

An investor begins with an account balance of $$5000$$ dollars which earns an annual interest rate o

Loan Amortization with Increasing Payments

A loan of $$20000$$ dollars is to be repaid in equal installments over 10 years. However, the repaym

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

MVT Application: Rate of Temperature Change

The temperature in a room is modeled by $$T(t)= -2*t^2+12*t+5$$, where $$t$$ is in hours. Analyze th

Optimization: Maximizing Rectangular Area with a Fixed Perimeter

A farmer has 300 meters of fencing to enclose a rectangular field that borders a straight river (no

Population Growth Modeling

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

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

Retirement Savings with Diminishing Deposits

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

Road Trip Analysis

A car's speed (in mph) during a road trip is recorded at various times. Use the table provided to an

Rolle's Theorem on a Cubic Function

Consider the cubic function $$f(x)= x^3-3*x^2+2*x$$ defined on the interval $$[0,2]$$. Verify that t

Salt Tank Mixing Problem

In a mixing tank, salt is added at a constant rate of $$A(t)=10$$ grams/min while the salt solution

Series Manipulation and Transformation in an Economic Forecast Model

A forecast model is given by the series $$F(x)=\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^2} * x^n$$. Ans

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Square Root Function Inverse Analysis

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

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 an Integral Function: $$F(x)=\int_0^x \sin(t^2)\,dt$$

Because the antiderivative of $$\sin(t^2)$$ cannot be expressed in closed form, use its power series

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Accumulation Function from a Rate Function

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

Antiderivatives and the Constant of Integration

The function $$f(x)=3*x^{2}$$ has an antiderivative $$F(x)$$.

Arc Length of $$y=x^{3/2}$$ on $$[0,4]$$

The curve defined by $$y=x^{3/2}$$ is given for $$x\in[0,4]$$. The arc length of a curve is determin

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 with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

Distance from Acceleration Data

A car's acceleration is recorded in the table below. Given that the initial velocity is $$v(0)= 10$$

Drug Concentration in a Bloodstream

A patient receives an intravenous drug infusion at a rate $$R_{in}(t)=3e^{-0.1*t}$$ mg/min for $$0 \

Evaluating an Integral Involving an Exponential Function

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

Implicit Differentiation Involving an Integral

Consider the relationship $$y^2 + \int_{1}^{x} \cos(t)\, dt = 4$$. Answer the following parts.

Improper Integral Evaluation

Evaluate the integral $$\int_{1}^{\infty} \frac{1}{t^2}\, dt$$ by answering the following parts.

Inverse Functions in Economic Models

Consider the function $$f(x) = 3*x^2 + 2$$ defined for $$x \ge 0$$, representing a demand model. Ans

Investigating Partition Sizes

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

Limit of a Riemann Sum as a Definite Integral

Consider the limit of the Riemann sum given by $$\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{6}{

Net Displacement vs. Total Distance Traveled

A particle moving along a straight line has a velocity function given by $$v(t)= t^2 - 4*t + 3$$ (in

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

Parametric Integral and Its Derivative

Let $$I(a)= \int_{0}^{a} \frac{t}{1+t^2}dt$$ where a > 0. This integral is considered as a function

Radioactive Decay: Accumulated Decay

A radioactive substance decays according to $$m(t)=50 * e^(-0.1*t)$$ (in grams), with time t in hour

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

Recovering Accumulated Change

A company’s revenue rate changes according to $$R'(t)=8*t-12$$ (in dollars per day). If the revenue

Recovering Position from Velocity

A particle moves along a straight line with a velocity given by $$v(t)=6*t-2$$ (in m/s) for $$t\in [

Riemann and Trapezoidal Sums with Inverse Functions

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

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Water Accumulation in a Reservoir

A reservoir receives water at an inflow rate modeled by $$r(t)=\frac{20}{t+1}$$ (in cubic meters per

Work Done by a Variable Force

A variable force acting along a track is given by $$F(x)=6*\sqrt{x}$$ (in Newtons). Compute the work

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,

Chemical Reactor Mixing

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

Cooling of an Object Using Newton's Law of Cooling

An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto

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

Direction Fields and Isoclines

Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying

Epidemic Spread Modeling

An epidemic in a closed population of $$N=10000$$ individuals is modeled by the logistic equation $$

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

FRQ 2: Separable Differential Equation with Initial Condition

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

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

FRQ 14: Dynamics of a Car Braking

A car braking is modeled by the differential equation $$\frac{dv}{dt} = -k*v$$, where the initial ve

Interpreting Slope Fields for a Differential Equation

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

Loan Balance with Continuous Interest and Payments

A loan has a balance $$L(t)$$ (in dollars) that accrues interest continuously at a rate of $$5\%$$ p

Logistic Growth: Time to Half-Capacity

Consider a logistic population model governed by the differential equation $$\frac{dP}{dt}=kP\left(1

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Mixing Problem with Constant Flow Rate

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

Modeling Free Fall with Air Resistance

An object falls under gravity while experiencing air resistance proportional to its velocity. The mo

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie

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

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

Projectile Motion with Drag

A projectile is launched horizontally with an initial velocity $$v_0$$. Due to air resistance, the h

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

Radioactive Decay with Constant Production

A radioactive substance decays at a rate proportional to its current amount but is also produced at

Second-Order Differential Equation: Oscillations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}= -9*y$$ with initial conditions

Separation of Variables with Trigonometric Functions

Solve the differential equation $$\frac{dy}{dx} = \frac{\sin(x)}{1+y^2}$$ by using separation of var

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

Area Between a Function and Its Tangent Line

Let $$f(x)=x^3-x$$. At the point $$x=1$$, find the tangent line to the curve and determine the area

Area Between Curves: Enclosed Region

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

Area Between Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

Area Calculation: Region Under a Parabolic Curve

Let $$f(x)=4-x^2$$. Consider the region bounded by the curve $$f(x)$$ and the x-axis.

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 Population in a Logistic Model

A population is modeled by a logistic function $$P(t)=\frac{500}{1+2e^{-0.3*t}}$$, where $$t$$ is me

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

Average Value of a Population Growth Rate

The instantaneous growth rate of a bacterial population is modeled by the function $$r(t)=0.5*\cos(0

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

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

Drug Concentration Profile Analysis

The functions $$f(t)=5*t+10$$ and $$g(t)=2*t^2+3$$ (where t is in hours and concentration in mg/L) r

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

Fluid Flow Rate and Total Volume

A river has a flow rate given by $$Q(t)=50+10*\cos(t)$$ (in cubic meters per second) for $$t\in[0,\p

Logarithmic and Exponential Equations in Integration

Let $$f(x)=\ln(x+2)$$. Consider the expression $$\frac{1}{6}\int_0^6 [f(x)]^2dx=k$$, where it is giv

Moment of Inertia of a Thin Plate

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

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

Projectile Maximum Height

A ball is thrown upward with an acceleration of $$a(t)=-9.8$$ m/s², an initial velocity of $$v(0)=20

Shadow Length Related Rates

A 1.8-meter tall man is walking away from a 5-meter tall lamp post at a constant speed of $$1.5$$ 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

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 of a Solid with the Washer Method

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

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

Acceleration Analysis in a Vector-Valued Function

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

Analysis of Particle Motion Using Parametric Equations

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

Analysis of Vector Trajectories

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

Arc Length Calculation of a Cycloid

Consider a cycloid described by the parametric equations $$x(t)=r*(t-\sin(t))$$ and $$y(t)=r*(1-\cos

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 Polar Curve

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$ for \(0 \le \theta \le \pi\).

Arc Length of a Polar Curve

Consider the polar curve given by $$r(\theta)=1+\frac{\theta}{\pi}$$ for $$0 \le \theta \le \pi$$. A

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

Area Between Polar Curves

In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t

Area between Two Polar Curves

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

Area Enclosed by a Polar Curve

Consider the polar curve given by $$r = 2*\sin(\theta)$$.

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

Designing a Race Track with Parametric Equations

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

Designing a Roller Coaster: Parametric Equations

The path of a roller coaster is modeled by the equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ f

Error Analysis in Taylor Approximations

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

Limit Evaluation of a Trigonometric Piecewise Function

Define the function $$f(θ)= \begin{cases} \frac{1-\cos(θ)}{θ}, & θ \neq 0 \\ 0, & θ=0 \end{cases}$$

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Modeling Circular Motion with Vector-Valued Functions

An object moves along a circle of radius $$3$$ with its position given by $$\mathbf{r}(t)=\langle 3\

Motion Along a Parametric Curve

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

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)

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

Parametric Intersection of Curves

Consider the curves $$C_1: x(t)=\cos(t),\, y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$ and $$C_2: x(s)=1

Parametric Motion with Damping

A particle's motion is modeled by the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t

Particle Motion in Circular Motion

A particle moves along a circular path given by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

Polar Coordinates: Analysis of $$r = 2+\cos(\theta)$$

The polar curve $$r= 2+\cos(\theta)$$ is given. Analyze various aspects of this curve.

Polar to Cartesian Conversion

Consider the polar curve defined by $$r = 4*\cos(\theta)$$.

Polar to Cartesian Conversion and Tangent Slope

Consider the polar curve $$r=2*(1+\cos(\theta))$$. Answer the following parts.

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

Spiral Path Analysis

A spiral is defined by the vector-valued function $$r(t) = \langle e^{-t}*\cos(t), e^{-t}*\sin(t) \r

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Time of Nearest Approach on a Parametric Path

An object travels along a path defined by $$x(t)=5*t-t^2$$ and $$y(t)=t^3-6*t$$ for $$t\ge0$$. Answe

Vector-Valued Function of Particle Trajectory

A particle in space follows the vector function $$\mathbf{r}(t)=\langle t, t^2, \sqrt{t} \rangle$$ f

Vector-Valued Functions and Curvature

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

Vector-Valued Functions: Velocity and Acceleration

A particle's position is given by the vector-valued function $$\vec{r}(t) = \langle \sin(t), \ln(t+1

Vector-Valued Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$