AP Calculus BC FRQ Room

Algorithm Time Complexity

A recursive algorithm has an execution time that decreases with each iteration: the first iteration

Application of the Squeeze Theorem with Trigonometric Functions

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

Asymptotic Behavior and Horizontal Limits

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

Asymptotic Behavior in Rational Functions

Consider the rational function $$g(x)=\frac{2*x^3-5*x^2+1}{x^3-3*x+4}.$$ Answer the following parts

Continuity and the Intermediate Value Theorem in Temperature Modeling

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

Continuity for a Logarithmic Transform Function

Consider the function $$f(x)= \ln\Bigl(\frac{e^{3x}-1}{x}\Bigr)$$ for $$x \neq 0$$ and define $$f(0)

Determining Continuity via Series Expansion

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - x - 1}{x^2}$$ for $$x \neq 0$$ with $$f(0)=L$$.

Factorization and Limits

Consider the function $$f(x)=\frac{x^2-4 * x}{x-4}$$ defined for $$x \neq 4$$. Answer the following:

Finding a Parameter in a Limit Involving Logs and Exponentials

Consider the function $$ f(x)= \frac{\ln(1+kx) - (e^x - 1)}{x^2}, $$ for $$x \neq 0$$. Assume that $

Graphical Analysis of Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

Implicitly Defined Function and Differentiation

Consider the curve defined implicitly by the equation $$x*y + \sin(x) + y^2 = 10$$. Answer the follo

Infinite Limits and Vertical Asymptotes

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

Limit Evaluation Involving Radicals and Rationalization

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

Limits and Absolute Value Functions

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

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

Oscillatory Functions and Discontinuity

Consider the function $$f(x)= \begin{cases} \sin\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0 \end{

Physical Applications: Temperature Continuity

A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^

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 with Removable Discontinuity

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

Rational Functions and Limit at Infinity

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

Related Rates with an Expanding Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=100\

Removing a Removable Discontinuity in a Piecewise Function

Examine the function $$g(x)= \begin{cases} \frac{x^2-9}{x-3}, & x \neq 3 \\ m, & x=3 \end{cases}$$.

Resistor Network Convergence

A resistor network is constructed by adding resistors in a ladder configuration. The resistance adde

Seasonal Temperature Curve Analysis

A graph represents the average daily temperature (in $$^\circ C$$) as a function of the day of the y

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

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 Higher-Order Derivatives

Let $$f(x)=x*e^{-x}$$ model the concentration of a substance over time. Analyze both the first and s

Cooling Tank System

A laboratory cooling tank has heat entering at a rate of $$H_{in}(t)=200-10*t$$ Joules per minute an

Derivative via Quotient Rule: Fluid Flow Rate

A function describing the rate of fluid flow is given by $$f(x)= \frac{x^2+2}{3*x-1}$$.

Differentiability of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases} x^2, & x < 1 \\ 2*x, & x \ge 1 \end{cases}$$. A

Differentiation of Parametric Equations

A curve is defined by the following parametric equations: $$x(t)= t^2+1, \quad y(t)= 2*t^3-3*t+1.$$

Engineering Analysis of Log-Exponential Function

In an engineering system, the output voltage is given by $$V(x)=\ln(4*x+1)*e^{-0.5*x}$$, where $$x$$

Exploration of Derivative Notation and Higher Order Derivatives

Given the function $$f(x)=x^2*e^x$$, analyze its derivatives.

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}$

Implicit Differentiation for a Rational Equation

Consider the curve defined by $$\frac{x*y}{x+y} = 3$$.

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation with Inverse Functions

Suppose a differentiable function $$f$$ satisfies the equation $$f(x) + f^(-1)(x) = 2*x$$ for all x

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 Versus Average Rates: A Comparative Study

Examine the function $$f(x)=\ln(x)$$. Analyze its average and instantaneous rates of change over a g

Manufacturing Cost Function and Instantaneous Rate

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

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$

Optimization Problem via Derivatives

A manufacturer’s cost in dollars for producing $$x$$ units is modeled by the function $$C(x)= x^3 -

Particle Motion on a Straight Line: Average and Instantaneous Rates

A particle moving along a straight line has its position given by $$s(t)=t^3 - 6*t^2 + 9*t + 4$$ for

Population Growth Rates

A city’s population (in thousands) was recorded over several years. Use the data provided to analyze

Product of Exponential and Trigonometric Functions

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

Product Rule Application in Kinematics

A particle’s distance along a path is given by $$s(t)= t*e^(2*t)$$, where $$t$$ is in seconds. Answe

Radioactive Decay and Derivative

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

Related Rates in a Conical Tank

Water is draining from a conical tank. The tank has a total height of 10 m and its radius is always

Satellite Orbit Altitude Modeling

A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}

Secant to Tangent Convergence

Consider the natural logarithm function $$f(x)=\ln(x)$$ for \(x>0\). Answer the following:

Second Derivative of a Composite Function

Consider the function $$f(x)=\cos(3*x^2)$$. Answer the following:

Tangent Line Approximation for a Combined Function

Consider the function $$f(x)= \sin(x) + x^2$$. Use the concept of the tangent line to approximate ne

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

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

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

Chain Rule and Inverse Trigonometric Differentiation

Consider the function $$f(x)= 3*\arccos\left(\frac{x}{4}\right) + \sqrt{1-\frac{x^2}{16}}$$. Answer

Chain Rule in Economic Utility Functions

A consumer's utility function is given by $$U(x,y)=\sqrt{x+y^2}$$, where x and y represent quantitie

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

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:

Composite and Inverse Differentiation in an Electrical Circuit

In an electrical circuit, the current is modeled by $$ I(t)= \sqrt{20*t+5} $$ and the voltage is giv

Composite Functions in Biological Growth

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

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 an Arctan Composite Function

For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$

Differentiation of an Inverse Exponential Function

Let $$f(x)=e^{2*x}-7$$, and let g denote its inverse function. Answer the following parts.

Exponential Composite Function Differentiation

Consider the function $$f(x)= e^{3*x^2+2*x}$$.

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Implicit Differentiation in Economic Equilibrium

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

Implicit Differentiation of an Implicit Curve

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

Implicit Differentiation: Second Derivatives of a Circle

Given the circle $$x^2+y^2=10$$, answer the following parts:

Inverse Differentiation of a Trigonometric Function

Consider the function $$f(x)=\arctan(2*x)$$ defined for all real numbers. Analyze its inverse functi

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

Inverse Function Differentiation in a Science Experiment

In an experiment, the relationship between an input value $$x$$ and the output is given by $$f(x)= \

Inverse Function Differentiation in a Sensor

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

Inverse Function in Logistic Population Growth

A population model is given by $$P(t)=\frac{100}{1+4e^{-0.5*t}}$$ for t \ge 0. Analyze the inverse f

Maximizing the Garden Area

A rectangular garden is to be built alongside a river, so that no fence is needed along the river. T

Multilayer Composite Differentiation in a Climate Model

A climate model gives the temperature $$T(t)$$ (in °C) as a function of time $$t$$ (in years) by $$T

Tangent Line for a Parametric Curve

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

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 $

Analyzing a Motion Graph

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

Analyzing Temperature Change of Coffee

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

Applying L'Hospital's Rule to a Transcendental Limit

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

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

Business Profit Optimization

A firm's profit is modeled by $$P(x)= -4*x^2 + 240*x - 1000$$, where $$x$$ (in hundreds) represents

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+

Circular Motion and Angular Rate

A point moves along a circle of radius 5 meters. Its angular position is given by $$\theta(t)=2*t^2-

Continuity in a Piecewise-Defined Function

Let $$g(x)= \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 2*x + k & \text{if } x \ge 1 \end{cases}$$.

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

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.

Differentials and Function Approximation

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

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

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

Economics: Cost Function and Marginal Analysis

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

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Expanding Circular Pool

A circular pool is being designed such that water flows in uniformly, expanding its surface area. Th

Expanding Rectangle: Related Rates

A rectangle has a length $$l$$ and width $$w$$ that are changing with time. At a certain moment, the

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

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 an Ellipse

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

Implicit Differentiation: Tangent to a Circle

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

Inflating Spherical Balloon

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

Linearization Approximation Problem

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

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Linearization of a Power Function

Let $$f(x)=x^4$$. Use linearization at $$x=4$$ with $$\Delta x=-0.02$$ to approximate $$(3.98)^4$$.

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Parametric Motion in the Plane

A particle moves in the plane according to the parametric equations $$x(t)=t^2-2*t$$ and $$y(t)=3*t-

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 =

Related Rates: Expanding Circular Ripple

A circular ripple in a pond expands such that its area, given by $$A=\pi r^2$$, is increasing at a c

Related Rates: Inflating Spherical Balloon

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

Revenue Concavity Analysis

A company’s revenue from sales is modeled by the function $$R(x)= 300*x - 2*x^2$$, where \(x\) repre

River Flow Diversion

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

Series Analysis in Acoustics

The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^

Series Differentiation and Approximation of Arctan

Consider the function $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^{2*n+1}}{2*n+1}$$, which represents

Shadow Length Rate

A 6-foot lamp post casts a shadow from a 5-foot-tall person walking away from it. Let $$x$$ represen

Solids of Revolution: Washer vs Shell Methods

Consider the region enclosed by $$y = \sin(x)$$ and $$y = \cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$

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 Quartic Function as a Perfect Power

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

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.

Analysis of an Absolute Value Function

Consider the function $$f(x)=|x^2-4|$$. Answer the following parts:

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:

Application of the 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

Derivative Sign Chart and Function Behavior

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

Determining the Meeting Point of Two Functions

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

Energy Consumption Rate Model

A household's energy consumption rate (in kW) is modeled by $$E(t) = 2*t^2 - 8*t + 10$$, where t is

Evaluating an Improper Integral using Series Expansion

The function $$I(x)=\sum_{n=0}^\infty (-1)^n * \frac{(2*x)^{n}}{n!}$$ converges to a known function.

Implicit Differentiation and Inverse Function Analysis

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, where y is a function of x near the point wh

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 Logarithmic Function

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

Lake Ecosystem Nutrient Dynamics

In a lake, nutrients (phosphorus) enter at a rate given by $$N_{in}(t)=5*\sin(t)+10$$ mg/min and are

Logistic Growth Model Analysis

Consider the logistic growth model given by $$P(t)=\frac{100}{1+9e^{-0.5*t}}$$. Answer the following

Manufacturing Optimization in Production

A company’s profit (in thousands of dollars) from producing x (in thousands of units) is given by $$

Mean Value Theorem Application for Mixed Log-Exponential Function

Let $$h(x)= \ln(x) + e^{-x}$$ be defined on the interval [1,3]. Analyze the function using the Mean

Motion with a Piecewise-Defined Velocity Function

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

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

Population Growth Model Analysis

A population of organisms is modeled by the function $$P(t)= -2*t^2+20*t+50$$, where $$t$$ is measur

Rate of Change in Biological Growth

A bacteria population is modeled by \( P(t)=100*e^{0.03*t} \), where \( t \) is the time in hours. A

Related Rates: Draining Conical Tank

Water is draining from a conical tank with a height of \(10\,m\) and a top diameter of \(8\,m\). Wat

Related Rates: Expanding Balloon

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

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

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

Taylor Series in Differential Equations: $$y'(x)=y(x)\cos(x)$$

Consider the initial value problem $$y'(x)= y(x)\cos(x)$$ with $$y(0)=1$$. Assume a power series sol

Water Tank Dynamics

A water tank receives water from a pipe at a rate of $$R(t)=3*t+5$$ liters/min and loses water throu

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Accumulated Change Prediction

A population grows continuously at a rate proportional to its size. Specifically, the growth rate is

Analyzing a Cumulative Distribution Function (CDF)

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

Antiderivative with Initial Condition

Find the general antiderivative of the function $$f(x)=5*x^3-2*x+6$$ and determine the particular an

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

Applying the Fundamental Theorem of Calculus

Consider the function $$f(x)=2*x$$. Use the Fundamental Theorem of Calculus to evaluate the definite

Area Between Curves

Consider the curves given by $$f(x)=x^2$$ and $$g(x)=2*x$$. A graph of these curves is provided. Det

Area Between Inverse Functions

Consider the functions $$f(x)=\sqrt{x}$$ and $$g(x)=x-2$$.

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

Average Value and Accumulated Change

For the function $$f(x)= x^2+1$$ defined on the interval [0, 4], find the average value of the funct

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

Convergence of an Improper Integral

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{x^{p}}\,dx$$, where $$p$$ is a positive

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

Definite Integral Involving an Inverse Function

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

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

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

Evaluation of an Improper Integral

Consider the integral $$\int_{1}^{\infty} \frac{1}{x^{2}} dx$$. Answer the following:

Graphical Transformations and Inverse Functions

Consider the linear function $$f(x)= \frac{1}{2}*x + 5$$ defined for all real $$x$$. Answer the foll

Implicit Differentiation Involving an Integral

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

Integration via Partial Fractions

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

Interpreting Area Under a Curve from a Graph

A graph displays the function $$f(x)=0.5*x+1$$ over the interval $$[0,6]$$.

Investigating Partition Sizes

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

Midpoint Riemann Sum Estimation

The function $$f(x)$$ is sampled at the following (possibly non-uniform) x-values provided in the ta

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

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

Particle Motion with Variable Acceleration

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

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Scooter Motion with Variable Acceleration

A scooter's acceleration is given by $$a(t)= 2*t - 1$$ (m/s²) for $$t \in [0,5]$$, with an initial v

Volume by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

Water Pollution Accumulation

In a river, a pollutant is introduced at a rate $$P_{in}(t)=8-0.5*t$$ (kg/min) and is simultaneously

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

Area and Volume from a Differential Equation-derived Family

Consider the family of curves that are solutions to the differential equation $$\frac{dy}{dx} = 2*x$

Autonomous ODE: Equilibrium and Stability

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(2-y)*(y+1)$$. Answer the following

Chain Reaction in a Nuclear Reactor

A simplified model for a nuclear chain reaction is given by the logistic differential equation $$\fr

Cooling and Mixing Combined Problem

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

Differential Equation in a Gravitational Context

Consider the differential equation $$\frac{dv}{dt}= -G\,\frac{M}{(R+t)^2}$$, which models a simplifi

Epidemic Spread Modeling

In a simplified epidemic model, the number of infected individuals $$I(t)$$ is modeled by the logist

Exact Differential Equation

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

Exponential Growth with Variable Rate

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

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 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 7: Projectile Motion with Air Resistance

A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

FRQ 13: Cooling of a Planetary Atmosphere

A planetary atmosphere cools according to Newton's Law of Cooling: $$\frac{dT}{dt}=-k(T-T_{eq})$$, w

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Investment Growth Model

An investment account grows continuously at a rate proportional to its current balance. The balance

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: Time to Half-Capacity

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

Logistic Model in Population Dynamics

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

Maclaurin Series Solution for a Differential Equation

Given the differential equation $$\frac{dy}{dx} = y * \cos(x)$$ with initial condition $$y(0)=1$$, f

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

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Projectile Motion with Air Resistance

A projectile is fired vertically upward with an initial velocity of $$50\,m/s$$. The projectile expe

Radioactive Decay Differential Equation

A radioactive substance decays according to the differential equation $$\frac{dM}{dt} = -k*M$$, wher

Saltwater Mixing Problem

A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of

Separable DE with Exponential Function

Solve the differential equation $$\frac{dy}{dx}=y\cdot\ln(y)$$ for y > 0 given the initial condition

Solving a Linear Differential Equation using an Integrating Factor

Consider the linear differential equation $$\frac{dy}{dx} + \frac{2}{x} * y = \frac{\sin(x)}{x}$$ wi

Temperature Control in a Chemical Reaction Vessel

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

Accumulation of Rainwater in a Reservoir

During a storm lasting 6 hours, rain falls on a reservoir at a rate given by $$R(t)=3+2\sin(t)$$ (cm

Area Between Curves in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

Average and Instantaneous Acceleration

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

Average Fuel Consumption and Optimization

A vehicle's fuel consumption rate is modeled by the function $$f(x)=2*x^2-8*x+10$$, where $$x$$ repr

Average Temperature Analysis

A research team models the ambient temperature in a region over a 24‐hour period with the function $

Average Temperature Analysis

A meteorological station recorded the temperature in a region as a function of time given by $$T(t)

Average Temperature Over a Day

The temperature in a city over a 24-hour period is modeled by $$T(t)=10+5\sin\left(\frac{\pi}{12}*t\

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Center of Mass of a Rod

A thin rod of length 10 m has a linear density given by $$\rho(x)=3+0.4*x$$ (in kg/m) where $$x$$ is

Determining the Arc Length of a Curve

Consider the curve defined by $$y=\frac{1}{2}*e^{x/2}$$ over the interval $$[0,2]$$.

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

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

Electrical Charge Distribution

A wire of length 3 meters carries a charge distribution given by $$\rho(x)=\frac{5}{1+x^2}$$ (in cou

Fluid Pressure on a Submerged Plate

A vertical rectangular plate with a width of 3 ft and a height of 10 ft is submerged in water so tha

Implicit Differentiation with Trigonometric Function

Consider the equation $$\cos(x * y) + x = y$$. Answer the following:

Motion Analysis of a Car

A car has an acceleration given by $$a(t)=2-0.5*t$$ for $$0\le t\le8$$ seconds. The initial velocity

Oval Path Implicit Differentiation

A particle moves along a path described by the equation $$x^2 + 2 * x * y + y^2 = 10$$. Answer the f

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (in m

Population Change via Rate Integration

A small town’s population changes at a rate given by $$P'(t)=100*e^{-0.3*t}$$ (persons per year) wit

Position and Velocity from Tabulated Data

A particle’s velocity (in m/s) is measured at discrete time intervals as shown in the table. Use the

Sand Pile Dynamics

Sand is being added to a pile at a rate given by $$A(t)=8-0.5*t$$ (kg/min) for $$0\le t\le12$$ minut

Solid of Revolution via Disc Method

Consider the region bounded by the curve $$y = x^2$$ and the x-axis for $$0 \le x \le 3$$. This regi

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 Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of an Irregular Tank

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

Volume with Equilateral Triangle Cross Sections

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

Water Tank Dynamics: Inflow and Outflow

A water tank receives water through an inflow at a rate given by $$I(t)=10+2*t$$ (liters per minute)

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Work Done by a Variable Force

A variable force is applied along a frictionless surface and is given by $$F(x)=6-0.5*x$$ (in Newton

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

Analyzing the Concavity of a Parametric Curve

A curve is defined by $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$.

Arc Length of a Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

Area Between Two Polar Curves

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

Average Position from a Vector-Valued Function

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

Concavity and Inflection in Parametric Curves

A curve is defined by the parametric functions $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$ for \(-2\le t\le2\)

Conversion Between Polar and Cartesian Coordinates

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

Implicit Differentiation with Implicitly Defined Function

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

Intersection of Polar and Parametric Curves

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

Intersections in Polar Coordinates

Two polar curves are given by $$r = 3 - 2*\sin(\theta)$$ and $$r = 1 + \cos(\theta)$$.

Optimization in Garden Design using Polar Coordinates

A garden is to be designed in the shape of a circular sector with radius $$r$$ and central angle $$\

Parametric Particle with Acceleration and Jerk

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

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 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

Spiral Motion in Polar Coordinates

A particle moves in polar coordinates with \(r(\theta)=4-\theta\) and the angle is related to time b