AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

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

Algebraic Manipulation with Radical Functions

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

Calculating Tangent Line from Data

The table below gives a function $$f(x)$$ representing the distance (in meters) of a moving object f

Complex Rational Function and Continuity Analysis

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

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 Analysis from Table Data

The water level (in meters) in a reservoir is recorded at various times as shown in the table below.

Continuity in Composition of Functions

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

Continuity Involving a Radical Expression

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

Endpoint Behavior of a Continuous Function

Let $$m(x)=\sqrt{x+4}$$ be defined on the interval $$[-4,5]$$. Answer the following:

Epsilon-Delta Proof for a Polynomial Function

Let $$f(x)=x^2+3*x+2$$. Answer the following:

Evaluating a Limit with Algebraic Manipulation

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

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Exploring the Squeeze Theorem

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

Graph Analysis of a Discontinuous Function

Examine a function $$f(x)=\frac{x^2-4}{x-2}$$. A graph of the function is provided in the attached s

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

Identifying and Removing a Discontinuity

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, which is undefined at $$x=2$$.

Interplay of Polynomial Growth and Exponential Decay

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

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

One-Sided Infinite Limits in Rational Functions

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

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

One-Sided Limits and Jump Discontinuity Analysis

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

Radioactive Material Decay with Intermittent Additions

A laboratory maintains a radioactive material by adding 10 units of the substance at the beginning o

Rational Function Limit and Continuity

Consider the function $$f(x)=\frac{x^2+5*x+6}{x+3}$$ defined for $$x\neq -3$$. Notice that the funct

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Removable Discontinuity in a Trigonometric Function

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

Trigonometric Rate Function Analysis

A pump’s output is modified by a trigonometric factor. The outflow rate is recorded as $$R(t)=\frac{

Analysis of a Piecewise Function's Differentiability

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

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Bacterial Culture Growth: Discrete to Continuous Analysis

In a controlled laboratory, a bacterial culture doubles every hour. The discrete model after n hours

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

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 Series Representing a Function

Consider the function defined by the infinite series $$S(x)= \sum_{n=0}^\infty \frac{(-1)^n * x^{2*

Differentiating Composite Functions using the Chain Rule

Consider the function $$S(x)=\sin(3*x^2+2)$$ which might model the stress on a structure as a functi

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 Population Growth in Ecology

A certain species in a reserve is observed to grow according to the function $$P(t)=1000*e^{0.05*t}$

Higher Order Derivatives: Concavity and Inflection Points

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

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation: Mixed Exponential and Polynomial Equation

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

Instantaneous Rate of Change and Series Approximation for √(1+x)

A company models its cost using the function $$C(x)=\sqrt{1+x}$$. To understand small changes in cos

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)

Optimization and Tangent Lines

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

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

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

Projectile Motion Analysis

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

Projectile Trajectory: Rate of Change Analysis

The height of a projectile is given by $$h(t)= -4.9t^2 + 20t + 1.5$$ in meters, where t is in second

Reconstructing Position from a Velocity Graph

A velocity versus time graph for a moving object is provided in the stimulus. Use the graph to answe

Related Rates: Two Moving Vehicles

A car is traveling east at 60 km/h and a truck is traveling north at 80 km/h. Let $$x$$ and $$y$$ be

Second Derivative Test and Stability

Consider the function $$f(x)=x^4-8*x^2+16$$.

Tangent and Normal Lines

Consider the function $$g(x)=\sqrt{x}$$ defined for $$x>0$$. 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

Temperature Change: Secant vs. Tangent Analysis

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

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 with Nested Radicals

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

Calculating an Inverse Trigonometric Derivative in a Physics Context

A pendulum's angle is modeled by $$\theta = \arcsin(0.5*t)$$, where $$t$$ is time in seconds and $$\

Chain Rule in Oscillatory Motion

A mass-spring system has its displacement modeled by $$ s(t)= e^{-0.5*t}\cos(3*t) $$.

Composite Differentiation in Polynomial Functions

Consider the function $$f(x)= (2*x^3 - x + 1)^4$$. Use the chain rule to differentiate f(x).

Continuity and Differentiability of a Piecewise Function

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

Differentiation of a Logarithmic-Square Root Composite Function

Let $$f(x)= \ln(\sqrt{1+ 3*x^2})$$. Differentiate the function with respect to $$x$$, simplify your

Implicit Differentiation in a Logarithmic Equation

Given the equation $$\ln(x*y) + x - y = 0$$, answer the following:

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 Exponential Functions

Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.

Implicit Differentiation Involving Inverse Trigonometric Functions

Consider the equation $$\theta = \arctan\left(\frac{y}{x}\right)$$, where $$y$$ is a differentiable

Implicit Differentiation of a Product Equation

Consider the equation $$ x*y + x + y = 10 $$.

Implicit Differentiation: Circle and Tangent Line

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

Infinite Series in a Financial Deposit Model

An investor makes monthly deposits that follow a geometric sequence, with the deposit in the nth mon

Inverse Analysis of a Composite Exponential-Trigonometric Function

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

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse Function Differentiation in a Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Inverse Function Differentiation in a Sensor

A sensor produces a reading described by the function $$f(t)= \ln(t+1) + t^2$$, where $$t$$ is in se

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Inverse Trigonometric Differentiation

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

Modeling with Composite Functions: Pollution Concentration

The pollutant concentration in a lake is modeled by $$C(t) = \sqrt{100 - 2*e^{-0.1*t}}$$, where $$t$

Parametric Curve Analysis with Composite Functions

A curve is defined by the parametric equations $$x(t)=\ln(1+t^2)$$ and $$y(t)=\sqrt{t+4}$$, where t

Projectile Motion and Composite Exponential Functions

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

Revenue Model and Inverse Analysis

A company's revenue (in millions) is modeled by $$R(x)=\sqrt{x+4}+3$$, where x represents production

Rocket Fuel Consumption Analysis

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

Vector Function Trajectory Analysis

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

Analyzing Rate of Approach in a Pursuit Problem

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

Approximating Changes with Differentials

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

Approximating Function Values Using Differentials

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

Arc Length Calculation

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

Car Motion with Changing Acceleration

A car's velocity is given by $$v(t) = 3*t^2 - 4*t + 2$$, where $$t$$ is in seconds. Answer the follo

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

Comparing Rates: Temperature Change and Coffee Cooling

The temperature of a freshly brewed coffee is modeled by $$T(t)=95*e^{-0.05*t}+25$$ (in °F), where $

Compound Interest Rate Change

An investment grows according to $$A(t)=5000e^{0.07t}$$, where t is measured in years. Answer the fo

Cooling Coffee Temperature Change

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

Differentiating a Product: f(x)=x sin(x)

Let \(f(x)=x\sin(x)\). Analyze the behavior of \(f(x)\) near \(x=0\).

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

Economic Marginal Cost Analysis

A manufacturer’s total cost for producing $$x$$ units is given by $$C(x)= 0.01*x^3 - 0.5*x^2 + 10*x

Economic Optimization: Profit Maximization

A company's profit (in thousands of dollars) is modeled by $$P(x) = -2x^2 + 40x - 150$$, where $$x$$

Ellipse Tangent Line Analysis

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

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

Filling a Conical Tank

A conical water tank has its radius related to its height by $$r=\frac{h}{2}$$, and its volume is gi

Graph Interpretation of Experimental Data

A laboratory experiment measured the concentration of a chemical reactant over time. The following g

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Graphical Interpretation of Slope and Instantaneous Rate

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

Hyperbolic Motion

A particle moves along a path given by the hyperbola $$x*y = 16$$. The particle's position changes w

Implicit Differentiation in Astronomy

The trajectory of a comet is given by the ellipse $$x^2 + 4*y^2 = 16$$, where \(x\) and \(y\) (in as

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 on an Ellipse

Consider the ellipse defined by $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, which represents a track. A runn

Infrared Sensor Distance Analysis

An infrared sensor measures the distance to a moving target using the function $$d(t)=50*e^{-0.2*t}+

Integration of Flow Rates Using the Trapezoidal Rule

A tank is being filled with water, and the flow rate Q (in L/min) is recorded at several time interv

L'Hôpital's Rule in Context

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

Linearization for Approximating Function Values

Let $$f(x)= \sqrt{x}$$. Use linearization at $$x=10$$ to approximate $$\sqrt{10.1}$$. Answer the fol

Linearization in Inverse Function Approximation

Let $$f(x)=x^5+2*x+1$$ be a one-to-one function. Although its inverse cannot be found explicitly, li

Linearization to Estimate Change in Electrical Resistance

The resistance of a wire is modeled by $$R(T)=R_0(1+\alpha*T)$$, where $$R_0=100$$ ohms and $$\alpha

Logarithmic Differentiation and Asymptotic Behavior

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

Maclaurin Series for ln(1+x)

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

Optimization with Material Costs

A company plans to design an open-top rectangular box with a square base that must have a volume of

Particle Motion Along a Line with Polynomial Velocity

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

Population Growth Differential

Consider an implicit relationship between a population $$N$$ and time $$t$$ given by $$\ln(N) + t =

Quadratic Function Inversion with Domain Restriction

Let $$f(x)=x^2+4$$. Since quadratic functions are not one-to-one over all real numbers, consider an

Rate of Change in Logarithmic Brightness

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

Related Rates in Conical Tank Draining

Water is draining from a conical tank. The volume of water is given by $$V=\frac{1}{3}\pi*r^2*h$$, a

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 Approximation for a Displacement Function

A displacement function is modeled by $$s(t)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1} t^n}{n}$$, which

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 Representation of a CDF

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

Series-Based Analysis of Experimental Data

An experiment models a measurement function as $$g(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x/4)^n}{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

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Analysis of a Decay Model with Constant Input

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

Analysis of a Rational Function and the Mean Value Theorem

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$. Answer the following parts.

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 of the Mean Value Theorem in Motion

A car's position on a straight road is given by the function $$s(t)=t^3-6*t^2+9*t+5$$, where t is in

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

Area and Volume of Region Bounded by Exponential and Linear Functions

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+2$$. The region enclosed by these curves will be

Area Enclosed by a Polar Curve

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

Bank Account Growth and Instantaneous Rate

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

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$

Determining the Meeting Point of Two Functions

Consider the functions $$f(x)= e^x$$ and $$g(x)= 3 + \ln(x)$$ representing two different processes.

Discounted Cash Flow Analysis

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

Exponential Decay in Velocity

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

Extreme Value Theorem: Finding Global Extrema

Consider the function $$f(x)= x^3-6*x^2+9*x+2$$ on the closed interval $$[0,4]$$. Use the Extreme Va

Fuel Consumption in a Generator

A generator operates with fuel being supplied at a constant rate of $$S(t)=5$$ liters/hour and consu

Graph Interpretation of a Function's First Derivative

A graph of the derivative function is provided below. Use it to determine the behavior of the origin

Interpreting a Velocity-Time Graph

A particle’s velocity over the interval $$[0,6]$$ seconds is depicted in the graph provided.

Linear Particle Motion Analysis

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

Mean Value Theorem Application

Let \( f(x) = x^3 - 3*x^2 + 2 \) be defined on the closed interval \([0,3]\). Answer the following p

Modeling Exponential Population Growth

A population is modeled by the function $$P(t)=500*e^{0.2*t}$$, where \(t\) is measured in years.

Motion with a Piecewise-Defined Velocity Function

A particle travels along a line with a piecewise velocity function defined by $$ v(t)=\begin{cases}

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

Rate of Change in a Chemical Reaction

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

Skier's Speed Analysis

A skier's speed (in m/s) on a slope was recorded at various time intervals. Use the data provided to

Stock Price Analysis

The daily closing price of a stock (in dollars) is recorded at various days. Use the stock price dat

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

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

Accumulated Change via U-Substitution

Evaluate the definite integral representing the accumulated amount of a substance given by $$\int_{1

Antiderivatives and the Fundamental Theorem of Calculus

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

Area Between Two Curves

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

Area 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

Consumer Surplus in an Economic Model

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

Cyclist's Displacement from Variable Acceleration

A cyclist's acceleration is given by $$a(t)= 7 - 3*t$$ (in m/s²). At time $$t=0$$, the cyclist has a

Distance vs. Displacement from a Velocity Function

A runner's velocity is modeled by $$v(t)=5-0.5*t$$ (in m/s) for $$0\le t\le10$$. The runner may chan

Estimating Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$. The following table provides the values of

Estimating Chemical Production via Riemann Sums

In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th

Improper Integral and the p-Test

Determine whether the improper integral $$\int_1^{\infty} \frac{1}{x^2}\,dx$$ converges, and if it c

Improper Integral Evaluation

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

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)=

Logistic Growth and Population Integration

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

Modeling Water Inflow Using Integration

Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$

Optimizing the Inflow Rate Strategy

A municipality is redesigning its water distribution system. The water inflow rate is modeled by $$F

Power Series Approximation of an Integral Function

The function $$f(x)=e^{-x^2}$$ does not have an elementary antiderivative. Its definite integral can

Rewriting Functions for Integration

Consider the function $$f(x)=\frac{1}{\sqrt{x}} - \frac{1}{x+1}$$. Rewrite this function in a form s

Riemann Sums and Inverse Analysis from Tabular Data

Let $$f(x)= 2*x + 1$$ be defined on the interval $$[0,5]$$. Answer the following questions about $$f

Solving for Unknowns using Logarithmic Properties in Integration

Consider the definite integral $$\int_(a)^(b) \frac{3}{x} dx$$ which is given to equal 6, where a is

U-Substitution Integration

Evaluate the definite integral $$\int_1^5 (2*x-3)^4 dx$$ using the method of u-substitution.

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

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

Autonomous Differential Equations and Stability Analysis

An autonomous differential equation has the form $$\frac{dy}{dt} = f(y)$$ with critical points at $$

Bacteria Culture with Regular Removal

A bacterial culture has a population $$B(t)$$ that grows at a continuous rate of $$12\%$$ per hour,

Depreciation Model of a Vehicle

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

Differential Equations in Economic Modeling

An economist models the rate of change of a commodity price $$P(t)$$ with the differential equation

Exponential Growth with Variable Rate

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

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 6: Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda * N$$

Investment Growth with Nonlinear Dynamics

The rate of change of an investment amount $$I$$ is modeled by the nonlinear differential equation $

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

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 Differential Equations

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

Newton's Law of Cooling

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

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

Projectile Motion with Air Resistance

A projectile is launched with an initial speed $$v_0$$ at an angle $$\theta$$ relative to the horizo

Radioactive Decay

A radioactive substance decays according to the law $$\frac{dN}{dt} = -k*N$$. The half-life of the s

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Relative Motion with Acceleration

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

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(

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

Temperature Control in a Chemical Reaction Vessel

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

Variable Carrying Capacity in Population Dynamics

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

Area Between a Rational Function and Its Asymptote

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

Area Between Curves: Parabolic & Linear Regions

Consider the curves $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Answer the following questions regarding the re

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 Curves: Supply and Demand Analysis

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

Area Between 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

Consider the curves $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. These curves enclose a region in the plane.

Average Daily Temperature

The temperature during a day is modeled by $$T(t)=10+5*\sin((\pi/12)*t)$$ (in °C), where $$t$$ is th

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 Polynomial Function

Consider the function $$f(x)=2*x^2 - 4*x + 3$$ defined on the closed interval $$[0,4]$$. Answer the

Balloon Inflation Related Rates

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

Car Braking and Stopping Distance

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

Car Motion Analysis

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

Cost Function from Marginal Cost

A manufacturing process has a marginal cost function given by $$MC(q)=3*\sqrt{q}$$, where $$q$$ (in

Displacement and Distance from a Variable Velocity Function

A particle moves along a straight line with velocity function $$v(t)= \sin(t) - 0.5$$ for $$t \in [0

Electric Current and Charge

An electric current in a circuit is defined by $$I(t)=4*\cos\left(\frac{\pi}{10}*t\right)$$ amperes,

Optimization and Integration: Maximum Volume

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

Particle Motion: Position, Velocity, and Acceleration

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

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Surface Area of a Rotated Parabolic Curve

The curve $$y = x^2$$ is rotated about the x-axis for $$x$$ in the interval $$[0,3]$$ to form a surf

Volume of a Region via Washer Method

The region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$ is rotated about the x-

Volume of a Solid by the Disc Method

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

Volume of a Solid 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

Volume of a Solid: ln(x) Region Rotated

Consider the region in the $$xy$$-plane bounded by $$y=\ln(x)$$, $$y=0$$, $$x=1$$, and $$x=e$$. This

Volume Using the Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ with $$x\ge0$$. This region is rotated about th

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

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=5+3*x$$ (in newtons), where $$x$$ is the displacement

Arc Length of a Parabolic Curve

The parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ models a portion of a parabolic path for

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)= 3*\sin(t)$$ and $$y(t)= 3*\cos(t)$$ for $$0 \le t \le \frac{\p

Arc Length of a Parametric Curve

The curve defined by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$t \in [0,4]$$ is given.

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 Two Polar Curves

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

Converting Polar to Cartesian and Computing Slope

The polar curve is given by the equation $$r=4\cos(\theta)$$. Answer the following:

Determining Curvature from a Vector-Valued Function

Consider the curve defined by $$\mathbf{r}(t)=\langle t, t^2, t^3 \rangle$$ for $$t \ge 0$$. Analyze

Finding the Slope of a Tangent to a Parametric Curve

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

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 Analysis with the Line y = x

Given the parametric equations $$x(t)=\ln(t+2)$$ and $$y(t)=t^2-1$$ for $$t \ge 0$$, answer the foll

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

Motion Along an Elliptical Path

Consider a particle moving along the curve defined by $$ x(t)=2*\cos(t) $$ and $$ y(t)=3*\sin(t) $$

Motion of a Particle in the Plane

A particle moves in the plane with parametric equations $$x(t)=t^2-4*t$$ and $$y(t)=2*t^3-6*t^2$$ fo

Parametric Curve: Intersection with a Line

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

Parametric Oscillations and Envelopes

Consider the family of curves defined by the parametric equations $$x(t)=t$$ and $$y(t)=e^{-t}\sin(k

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 with Logarithmic Component

A particle moves along a path given by $$x(t)= \frac{t}{t+1}$$ and $$y(t)= \ln(t+1)$$, where $$t \ge

Polar Curve Sketching and Area Estimation

A polar curve is described by sample data given in the table below.

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 using Parametric Equations

A projectile is launched with an initial speed of $$v_0 = 20\,\text{m/s}$$ at an angle of $$30^\circ

Roller Coaster Design: Parametric Path

A roller coaster is modeled by the parametric equations $$x(t)=t-\cos(t)$$ and $$y(t)=t-\sin(t)$$ fo

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 Analysis through Polar Conversion

Consider the polar curve defined by $$r(θ)= 4\sin(θ)$$. Answer the following:

Tangent Line to a 3D Vector-Valued Curve

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

Taylor/Maclaurin Series: Approximation and Error Analysis

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

Vector-Valued Function Analysis

Let the vector-valued function be given by $$\vec{r}(t)=<e^{t},\, \sin(t),\, \cos(t)>$$ for $$0\leq

Vector-Valued Functions: Tangent and Normal Components

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

Vector-Valued Motion: Acceleration and Maximum Speed

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

Work Done by a Force along a Vector Path

A force field is given by $$\mathbf{F}(t)=\langle2*t,\;3\sin(t)\rangle$$ and an object moves along a