AP Calculus BC FRQ Room

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 $$x \neq 2$$.

Composite Function and Continuity

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

Compound Interest and Loan Repayment

A simplified model for a loan repayment assumes that a borrower owes $$10,000$$ dollars and the rema

Continuity Analysis Involving Logarithmic and Polynomial Expressions

Consider the function $$f(x)= \begin{cases} \frac{\ln(x+2)}{x} & \text{if } x<0 \\ (x+1)^2 & \text{i

Continuity Analysis of a Piecewise Function

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

Continuity 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{

Economic Model of Depreciating Car Value

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

Evaluating Limits Involving Radical Expressions

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

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

Exponential Function Limits at Infinity

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

Graphical Analysis of a Continuous Polynomial Function

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

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

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 $$x=2$$.

Intermediate Value Theorem Application

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

Intermediate Value Theorem Application with a Cubic Function

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

Intermediate Value Theorem in Temperature Analysis

A city's temperature during a day is modeled by a continuous function $$T(t)$$, where t (in hours) l

Investigating a Function with a Removable Discontinuity

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

Investigating Limits at Infinity and Asymptotic Behavior

Given the rational function $$f(x)=\frac{5*x^2-3*x+2}{2*x^2+x-1}$$, answer the following: (a) Evalua

Limit and Continuity with Parameterized Functions

Let $$ f(x)= \frac{e^{3x} - 1 - 3x}{\ln(1+4x) - 4x}, $$ for $$x \neq 0$$ and define \(f(0)=L\) for c

Limit Evaluation Involving Radicals and Rationalization

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}$$.

Limit Evaluation Involving Trigonometric Functions

Let $$f(x)=\frac{\sin(4*x)}{\tan(2*x)}$$ for $$x\neq0$$, with f(0) defined separately. Answer the

Limits and Absolute Value Functions

Examine the function $$f(x)= \frac{|x-3|}{x-3}$$ defined for $$x \neq 3$$.

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 $$x\neq0$$.

Manufacturing Cost Sequence

A company's per-unit manufacturing cost decreases by $$50$$ dollars each year due to economies of sc

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 for a Piecewise Function

Consider the piecewise function $$f(x)= \begin{cases} 2*x+1 & \text{if } x< 3 \\ x^2-5*x+8 & \text{i

Radioactive Material Decay with Intermittent Additions

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

Squeeze Theorem with an Oscillatory Factor

Consider the function $$f(x)= x*\cos(\frac{1}{x})$$ for $$x \neq 0$$, with f(0) defined as 0. Use th

Telecommunications Signal Strength

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

Temperature Change Analysis

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

Water Filling a Leaky Tank

A water tank is initially empty. Every minute, 10 liters of water is added to the tank, but due to a

Zeno’s Maze Runner

A runner attempts to reach a wall 100 meters away by covering half of the remaining distance with ea

Analysis of Concavity and Second Derivative

Let $$f(x)=x^4-4*x^3+6*x^2$$. Analyze the concavity of the function and identify any inflection poin

Analyzing a Function with an Oscillatory Component

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

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

Circular Motion Analysis

An object moves along a circular path with angular position given by $$\theta(t)=2*t-0.1*t^2$$ (in r

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

Composite Function and Chain Rule Application

Consider the function $$h(x)=\sin(2*x^2+3)$$. Using the chain rule, answer the following parts:

Derivative from First Principles: Quadratic Function

Consider the function $$f(x)= 3*x^2 + 2*x - 5$$. Use the limit definition of the derivative to compu

Derivative of a Function Involving an Absolute Value

Consider the function $$f(x)=|x-3|+2$$. Answer the following:

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

Differentiability of an Absolute Value Function

Consider the function $$f(x) = |x|$$.

Differentiation in Exponential Growth Models

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

Differentiation of an Exponential Function

Let $$f(x)=e^{2*x}$$. Answer the following:

Hot Air Balloon Altitude Analysis

A hot air balloon’s altitude is modeled by the function $$h(t)=5*\sqrt{t+1}$$, where $$h$$ is in met

Implicit Differentiation of a Circle

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

Implicit Differentiation on a Circle

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

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 Polynomial Function

Consider the function $$f(x)=2*x^3 - 5*x^2 + 3*x - 7$$ which represents the position (in meters) of

Instantaneous vs. Average Rate of Change

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

Interpreting Graphical Slope Data

A laboratory experiment measures the velocity (in m/s) of a moving object over time. A graph of the

Maclaurin Polynomial for √(1+x)

A scientist approximates the function $$f(x)=\sqrt{1+x}$$ for small values of x using its Maclaurin

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

Pollutant Levels in a Lake

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

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

Quotient Rule in a Chemical Concentration Model

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

Revenue Change Analysis via the Product Rule

A company’s revenue (in thousands of dollars) is modeled by $$R(x) = (2*x + 3)*(x^2 - x + 4)$$, wher

Secant Line Estimation for a Radical Function

Consider the function $$f(x)= \sqrt{x}$$.

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

Using the Limit Definition for a Non-Polynomial Function

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

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

Chain Rule and Quotient Rule for a Rational Composite Function

Let $$f(x)= \frac{(3*x^2 + 2)^4}{(1+x)^{1/2}}$$. Answer the following:

Chain Rule in Oscillatory Motion

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

Composite and Implicit Differentiation with Trigonometric Functions

Consider the equation $$\sin(x*y)+x^2=y^2$$ which relates x and y. Answer the following parts:

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

Design Optimization for a Cylindrical Can

A manufacturer wants to design a cylindrical can that holds a fixed volume of $$V = 1000$$ cm³. The

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 Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \arctan(\sqrt{x})$$.

Differentiation of an Inverse Trigonometric Composite Function

Let $$y = \arcsin(\sqrt{x})$$. Answer the following:

Engine Air-Fuel Mixture

In an engine, the fuel injection rate is modeled by the composite function $$F(t)=w(z(t))$$, where $

Implicit Differentiation and Inverse Functions in a Trigonometric Equation

Consider the equation $$x*y + \sin(x+y)= 1$$ which defines y implicitly as a function of x.

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 Involving Logarithms

Consider the equation $$\ln(x+y)=x*y$$ which relates x and y. Answer the following parts:

Implicit Differentiation of a Circle

Consider the circle described by $$x^2 + y^2 = 25$$. A table of select points on the circle is given

Implicit Differentiation of an Implicit Curve

Consider the curve defined by $$x*y + x^2 - y^2 = 5$$. Answer the following parts.

Implicit Differentiation with Logarithmic Equation

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

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 of a Shifted Logarithmic Function

Analyze the function $$f(x)=\ln(x-1)+2$$ defined for $$x>1$$ and its inverse.

Projectile Motion and Composite Exponential Functions

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

Reservoir Level: Inverse Function Application

A reservoir's water level $$h$$ (in feet) is related to time $$t$$ (in minutes) through an invertibl

Trigonometric Composite Inverse Function Analysis

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

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Application of L’Hospital’s Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 2x + 1}{3x^3 + 7}$$ using L’Hospital’s Rule.

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

Differentials and Function Approximation

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

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 Rates: Marginal Profit Analysis

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

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.

Firework Trajectory Analysis

A firework is launched and its height (in meters) is modeled by the function $$h(t)=-4.9t^2+30t+5$$,

GDP Growth Analysis

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

Graphical Analysis of an Inverse Function

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

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

Implicit Differentiation on an Ellipse

An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo

Inflating Balloon: Related Rates

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

L’Hôpital’s Rule for an Exponential Ratio

Analyze the limit of the function $$f(t)=\frac{e^{2*t}-1}{t}$$ as $$t\to 0$$. Answer the following:

Linearization Approximation

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

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

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

Maximizing Revenue in a Business Model

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

Optimal Dimensions of a Cylinder with Fixed Volume

A closed right circular cylinder must have a volume of $$200\pi$$ cubic centimeters. The surface are

Parametric Curve Motion

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

Particle Motion Analysis Using Cubic Position Function

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

Pollutant Scrubber Efficiency

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

Population Growth Analysis

A certain bacterial population in a lab grows according to the model $$P(t)=100\cdot e^{0.03*t}$$, w

Population Growth Differential

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

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

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

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Series Expansion in Vibration Analysis

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

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

Surface Area of a Solid of Revolution

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

Urban Traffic Flow Analysis

An urban highway ramp experiences an inflow of cars at a rate of $$I(t)=40+2t$$ (cars per minute) an

Air Pollution Control in an Enclosed Space

In an enclosed environment, contaminated air enters at a rate of $$I(t)=15-\frac{t}{2}$$ m³/min and

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

Analysis of an Exponential Function

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

Asymptotic Behavior and Limits of a Logarithmic Model

Examine the function $$f(x)= \ln(1+e^{-x})$$ and its long-term behavior.

Concavity and Inflection Points

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

Concavity and Points of Inflection

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

Differentiability and Critical Points of a Piecewise Function

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

Fractal Tree Branch Lengths

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

Instantaneous vs. Average Rates in a Real-World Model

A company’s monthly revenue is modeled by $$ R(t)=0.5t^3-4t^2+12t+100, \quad 0 \le t \le 6,$$ where

Inverse Analysis for a Function with Multiple Transformations

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

Inverse Analysis for a Logarithmic Function

Let $$f(x)= \ln(2*x+5)$$ for $$x > -2.5$$. Answer the following parts.

Light Reflection Between Mirrors

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

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

Maclaurin Series for $$\arctan(x)$$

Let $$f(x)=\arctan(x)$$. Develop its Maclaurin series expansion, determine the corresponding 5th deg

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

Minimizing Production Cost

A company’s production cost is modeled by the function $$C(x)=0.5*x^2 - 20*x + 300$$, where $$x$$ re

Optimization in Particle Motion

A particle moves along a line with position given by $$ s(t)=t^3-6t^2+9t+4, \quad t\ge0.$$ Answer t

Optimization with a Combined Logarithmic and Exponential Function

A company's revenue is modeled by $$R(x)= x\,e^{-0.05x} + 100\,\ln(x)$$, where x (in hundreds) repre

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

Taylor Polynomial for $$\cos(x)$$ Centered at $$x=\pi/4$$

Consider the function $$f(x)=\cos(x)$$. You will generate the second degree Taylor polynomial for f(

Taylor Series for $$\ln\left(\frac{1+x}{1-x}\right)$$

Let $$f(x)=\ln\left(\frac{1+x}{1-x}\right)$$. Derive its Taylor series expansion about $$x=0$$, dete

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

Temperature Variations

The daily temperature of a city (in °C) is recorded at various times during the day. Use the tempera

Volume Using Cylindrical Shells

The region bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is revolved about the y-axis to form a solid.

Accumulated Earnings Over Time

A freelancer’s earning rate is modeled by $$E'(t)=15+4*\sin\left(\frac{\pi*t}{12}\right)$$ dollars p

Analyzing a Cumulative Distribution Function (CDF)

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

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

Biomedical Modeling: Drug Concentration Dynamics

A drug concentration in the bloodstream is modeled by $$f(t)= 5\left(1 - e^{-0.3*t}\right)$$ for $$t

Car Acceleration, Velocity, and Distance

In a physics experiment, the acceleration of a car is modeled by the function $$a(t)=4*t-1$$ (in m/s

Charging a Battery

An electric battery is charged with a variable current given by $$I(t)=4+2*\sin\left(\frac{\pi*t}{6}

Composite Functions and Inverses

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

Comprehensive Integration of a Polynomial Function

Consider the function $$f(x)=(x-3)(x+2)^2$$ on the interval $$[1,5]$$. This problem involves multipl

Consumer Spending Accumulation

Consumers' marginal spending over a 10-hour day is modeled by $$S(t)= 100*e^{-0.2*t}$$ dollars per h

Continuous Antiderivative for a Piecewise Function

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

Cost Function Accumulation

A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

Distance Traveled by a Particle

A particle has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t\in [0,5]$$ seconds.

Estimating Area Under a Curve from Tabular Data

A function $$f(t)$$ is sampled at discrete time points as given in the table below. Using these data

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Evaluating a Piecewise Function with a Removable Discontinuity

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

Flow of Traffic on a Bridge

Cars cross a bridge at a rate modeled by $$R(t)=300+50*\cos\left(\frac{\pi*t}{6}\right)$$ vehicles p

Integrated Growth in Economic Modeling

A company experiences revenue growth at an instantaneous rate given by $$r(t)=0.5*t+2$$ (in millions

Integration via Partial Fractions

Evaluate the integral $$\int_{0}^{1} \frac{2*x+3}{(x+1)(x+2)} dx$$. Answer the following:

Logarithmic Functions in Ecosystem Models

Let \(f(t)= \ln(t+2)\) for \(t \ge 0\) model an ecosystem measurement. Answer the following question

Optimizing the Inflow Rate Strategy

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

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

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+

Tank Filling Problem

Water flows into a tank at a rate given by $$R(t)=8e^{-0.5*t}+2$$ (in liters per minute) for $$t\geq

Total Cost from a Marginal Cost Function

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

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

Vehicle Motion and Inverse Time Function

A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Bacteria Growth with Antibiotic Treatment

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

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Cooling Cup of Coffee

A cup of coffee at an initial temperature of $$95^\circ C$$ is placed in a room. For the first 5 min

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

Direction Fields and Isoclines

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

Direction Fields and Stability Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=y(1-y)$$. Answer the following parts.

Euler's Method Approximation

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

Flow Rate in River Pollution Modeling

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

FRQ 3: Population Growth and Logistic Model

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

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

Integrating Factor Application

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

Investment Growth with Nonlinear Dynamics

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

Logistic Growth in Population Dynamics

The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l

Logistic Growth Model

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

Medicine Infusion and Elimination Model

A patient receives an intravenous infusion of a drug such that the infusion rate is $$R(t)=e^{0.2*t}

Mixing Problem in a Saltwater Tank

A tank initially contains $$100$$ liters of water with a salt concentration of $$2\,g/l$$. Brine wit

Mixing Problem with Constant Rates

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

Modeling Disease Spread with Differential Equations

In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin

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

Nonlinear Differential Equation with Implicit Solution

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

Particle Motion with Damping

A particle moving along a straight line is subject to damping and its motion is modeled by the secon

Population Dynamics in Ecology

Consider the differential equation that models the growth of a fish population in a lake: $$\frac{dP

RC Circuit Differential Equation

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

Separable Differential Equation with Parameter Identification

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -a*C$$, where $$C(t)$$

Series Convergence and Error Analysis

Consider the power series representation $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Slope Field and Solution Curve Sketching

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

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Approximating Functions using Taylor Series

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

Arc Length in Polar Coordinates

Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

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

Average Chemical Concentration Analysis

In a chemical reaction, the concentration of a reactant (in M) is recorded over time as given in the

Average Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Center of Mass of a Lamina with Constant Density

A thin lamina occupies the region in the first quadrant bounded by $$y=x^2$$ and $$y=4$$. The densit

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Cyclist's Journey: Displacement versus Total Distance

A cyclist's velocity is given by $$v(t)=\sin(t)$$ (in m/s) for $$t\in[0,2\pi]$$. Answer the followin

Determining the Length of a Curve

Find the arc length of the curve given by $$y=\sqrt{4*x}$$ for $$x\in[0,9]$$.

Draining a Conical Tank Related Rates

Water is draining from a conical tank that has a height of $$8$$ meters and a top radius of $$3$$ me

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

Inflow Rate to a Reservoir

The inflow rate of water into a reservoir is given by $$R(t)=\frac{100*t}{5+t}$$ (in cubic meters pe

Inflow vs Outflow: Water Reservoir Capacity

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

Integral Approximation Using Taylor Series

Approximate the integral $$\int_{0}^{0.2} \sin(2*x)\,dx$$ by using the Taylor series expansion of $$

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

Movement Under Variable Acceleration

A car accelerates along a straight road with acceleration given by $$a(t)=2*t - 3$$ (in m/s²) and ha

Particle Acceleration and Turning Points

A particle moves along a straight line with velocity $$v(t)= t^3 - 6*t^2 + 9*t + 2$$ (in m/s) and ac

Particle Position and Distance Traveled

A particle moves along a line with velocity $$v(t)=t^3-6*t^2+9*t$$ (m/s) for $$t\in[0,5]$$. Given th

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 $$

Surface Area of a Solid of Revolution

Consider the curve $$y=\sqrt{x}$$ on the interval $$[0,9]$$. When this curve is rotated about the x-

Volume of a Solid Using the Shell Method

The region in the first quadrant bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is rotated about the y-axi

Volume of a Solid via the Disc Method

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

Volume of an Irregular Tank

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

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 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

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

Arc Length and Speed from Parametric Equations

Consider the curve defined by $$x(t)=e^t$$ and $$y(t)=e^{-t}$$ for $$-1 \le t \le 1$$. Analyze the a

Arc Length of a Decaying Spiral

Consider the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t}\sin(t)$$ for $$t \ge 0$

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t) = t^2$$ and $$y(t) = t^3$$ for $$0 \le t \le 2$$.

Area Between Polar Curves

Consider the polar curves defined by $$r_1= 4$$ and $$r_2= 2+2\cos(\theta)$$. Find the area of the r

Circular Motion Analysis

A particle moves in a circle according to the vector-valued function $$\vec{r}(t)=<3\cos(t),\, 3\sin

Conversion from Polar to Cartesian Coordinates

The polar equation $$r(\theta)=4*\cos(\theta)$$ represents a curve.

Conversion of Polar to Parametric Form

A particle’s motion is given in polar form by the equations $$r = 4$$ and $$\theta = \sqrt{t}$$ wher

Curvature of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t^2$$ for \(t>0\).

Differentiation and Integration of a Vector-Valued Function

Let $$\mathbf{r}(t)=\langle e^{-t}, \sin(t), \cos(t) \rangle$$ for $$t \in [0,\pi]$$.

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Helical Motion with Damping

A particle moves along a helical path with damping, described by the vector function $$\vec{r}(t)= \

Intersection and Area Between Polar Curves

Two polar curves are given by $$r_1(\theta)=2\sin(\theta)$$ and $$r_2(\theta)=1+\cos(\theta)$$.

Optimization of Walkway Slope with Fixed Arc Length

A walkway is designed with its shape given by the parametric equations $$x(t)= t$$ and $$y(t)= c*t*(

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

Parametric Curve with a Loop and Tangent Analysis

Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An

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 Representation of an Ellipse

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

Particle Trajectory in Parametric Motion

A particle moves along a curve with parametric equations $$x(t)= t^2 - 4*t$$ and $$y(t)= t^3 - 3*t$$

Polar Coordinates and Dynamics

A point moves along a spiral defined by the polar equation $$r=3\theta$$, where $$\theta$$ is given

Real-World Data Analysis from Tabular Measurements

A vehicle's distance (in meters) along a straight road is recorded at various times (in seconds) as

Vector-Valued Function Analysis

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

Work Done by a Force along a Path

A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra

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