AP Calculus BC FRQ Room

Algorithm Time Complexity

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

Calculating Tangent Line from Data

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

Comparing Methods for Limit Evaluation

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

Continuity and Asymptotes of a Log‐Exponential Function

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

Continuity Conditions for a Piecewise-Defined Function

Consider the function defined by $$ f(x)= \begin{cases} 2*x+1, & x < 3 \\ ax^2+ b, & x \ge 3 \end{c

Continuity of a Trigonometric Function Near Zero

Consider the function defined by $$ f(x)= \begin{cases} \frac{\sin(5*x)}{x}, & x \neq 0 \\ L, & x =

Continuity of Log‐Exponential Function

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

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 $$f(0)=L$$.

End Behavior Analysis of a Rational Function

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

Evaluating a Limit Involving a Radical and Trigonometric Component

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

Evaluating Limits Involving Exponential and Rational Functions

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

Factorable Discontinuity Analysis

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

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

Investigating Limits and Areas Under Curves

Consider the region bounded by the curve $$y=\frac{1}{x}$$, the vertical line $$x=1$$, and the verti

Jump Discontinuity Analysis using Table Data

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

Limits and Continuity of Radical Functions

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

Limits Involving Exponential Functions

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

Limits Involving Infinity and Vertical Asymptotes

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

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

Piecewise Function Continuity and Differentiability

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

Rational Function and Removable Discontinuity

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

Rational Function Limit and Continuity

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

Rational Function with Removable Discontinuity

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

Trigonometric Function and the Squeeze Theorem

Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Water Treatment Plant Discontinuity Analysis

A water treatment plant monitors the inflow to a reservoir. Due to sensor calibration, the inflow ra

Analyzing Car Speed from a Distance-Time Table

A car's position (in meters) is recorded at various times (in seconds) as shown in the table. Use th

Average and Instantaneous Growth Rates in a Bacterial Culture

A bacterial population is modeled by the function $$P(t)= e^{0.3*t} + 10$$, where $$t$$ is measured

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

Chemical Reaction Rate Control

During a chemical reaction in a reactor, reactants enter at a rate of $$R_{in}(t)=\frac{10*t}{t+2}$$

Complex Rational Differentiation

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

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 the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Differentiating Composite Functions

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

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

Differentiation of a Trigonometric Function

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

Error Bound Analysis for Cos(x) Approximations in Physical Experiments

In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze

Implicit Differentiation for a Rational Equation

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

Implicit Differentiation on a Circle

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

Implicit Differentiation: Cost Allocation Model

A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1

Market Price Rate Analysis

The market price of a product (in dollars) has been recorded over several days. Use the table below

Optimization Using Derivatives

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

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

Related Rates: Draining Conical Tank

Water is draining from an inverted conical tank with a height of 6 m and a top radius of 3 m. The vo

Satellite Orbit Altitude Modeling

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

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

Tangent Line Approximation for a Parabolic Arch

Engineers design a parabolic arch described by $$y(x)= -0.5*x^2 + 4*x$$.

Tangent Line Estimation in Transportation Modeling

A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote

Taylor Series Expansion of ln(x) About x = 2

For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

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

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

Velocity Function from a Cubic Position Function

An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a

Biological Growth Model Differentiation

In a biological model, the concentration of a chemical is modeled by $$C(t)=e^{-0.5*t}+\ln(2*t+3)$$.

Chain Rule Combined with Inverse Trigonometric Differentiation

Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:

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 Differentiation with Nested Logarithmic Functions

Consider the function $$F(x)= \sqrt{\ln(3*x^2+1)}$$.

Composite Function with Implicitly Defined Inner Function

Let the function $$h(x)$$ be defined implicitly by the equation $$h(x) - \ln(h(x)) = x$$, and consid

Differentiation Involving Absolute Values and Composite Functions

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

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

Differentiation of a Nested Trigonometric Function

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

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Drug Concentration in the Bloodstream

A drug is infused into a patient's bloodstream at a rate given by the composite function $$R(t)=k(m(

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 a Radical Equation

The relationship between $$x$$ and $$y$$ is given by $$\sqrt{x} + \sqrt{y} = 6$$.

Implicit Differentiation Involving Exponential Functions

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

Implicit Differentiation with Exponential and Trigonometric Components

Consider the relation $$ (x^2 + y^2) * e^{y} = x $$. Answer the following:

Implicit Differentiation with Logarithmic Functions

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

Implicit Differentiation: Circle and Tangent Line

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

Implicit Differentiation: Conic Section Analysis

Consider the conic section defined by $$x^2 + 3*x*y + y^2 = 5$$. Answer the following:

Inverse Functions in Economic Modeling

Let the cost function be given by $$f(x)= 4*x + \sqrt{x}$$, where x represents the number of items p

Logarithmic and Composite Differentiation

Let $$g(x)= \ln(\sqrt{x^2+1})$$.

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

Rainwater Harvesting System

A rainwater harvesting system collects water in a reservoir. The inflow rate is given by the composi

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

Second Derivative via Chain Rule

Let $$h(x)=(e^{2*x}+1)^4$$. Answer the following parts.

Tangent Line to an Ellipse

Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan

Temperature Control: Heating Element Dynamics

A room's temperature is controlled by a heater whose output is given by the composite function $$H(t

Analysis of a Piecewise Function with Discontinuities

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

Analyzing Motion on an Inclined Plane

A sled moves along an inclined plane with displacement given by $$s(t)=5*t^2-0.5*t^3$$, where $$s$$

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

Business Profit Optimization

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

Chemical Reaction Rate Model

A chemical reaction has its reactant concentration modeled by $$C(t)= 0.5*t^2 - 3*t + 4$$, where \(C

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 $

Cubic Curve Linearization

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

Data Table Inversion

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

Expanding Rectangle: Related Rates

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

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

Ladder Sliding Down a Wall

A 10-meter ladder leans against a vertical wall and begins to slide. The bottom slides away from the

Linearization of Implicit Equation

Consider the implicit equation $$x^2 + y^2 - 2*x*y = 1$$, which defines $$y$$ as a function of $$x$$

Linearization of Trigonometric Implicit Function

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

Marginal Cost and Revenue Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$C(x)$$ is measured in dollars

Motion along a Curved Path

A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo

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

Radical Function Inversion

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

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

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.

Related Rates: Pool Water Level

Water is being drained from a circular pool. The surface area of the pool is given by $$A=\pi*r^2$$.

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

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 Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

Series Convergence and Approximation for f(x) Centered at x = 2

Consider the function $$f(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^{2*n}}{n+1}$$. Answer the follo

Series Integration in Fluid Flow Modeling

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

Series Representation of a CDF

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

Shadow Length Rate

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

Spherical Balloon Inflation

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

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

Absolute Extrema via Candidate's Test

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

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

Amusement Park Ride Braking Distance

An amusement park ride uses a sequence of friction pads to stop a roller coaster. The first pad diss

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:

Analyzing Convergence of a Modified Alternating Series

Consider the series $$S(x)=\sum_{n=1}^\infty (-1)^n * \frac{(x+2)^n}{n}$$. Answer the following.

Application in Motion: Approximate Velocity using Taylor Series

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

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Bouncing Ball with Energy Loss

A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of the height fr

Composite Functions and Derivatives

Let $$h(x)=f(g(x))$$ where $$f(u)=u^2+3$$ and $$g(x)=\sin(x)$$. Analyze the composite function on th

Concavity and Inflection Points Analysis

Consider the function \( f(x)=\ln(x) - x \) where \( x > 0 \). Answer the following parts:

Concavity of an Integral Function

Let $$F(x)= \int_0^x (t^2-4*t+3)\,dt$$. Analyze the concavity of $$F(x)$$.

Derivatives and Inverses

Consider the function $$f(x)=\ln(x)+x$$ for x > 0, and let g(x) denote its inverse function. Answer

Differentiability of a Piecewise Function

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

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Elasticity Analysis of a Demand Function

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

Fuel Consumption in a Generator

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

Increasing/Decreasing Intervals for a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$, defined for $$x > -2$$ (with $$x \neq -2$$).

Interpreting a Velocity-Time Graph

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

Inverse Analysis of a Cubic Polynomial

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

Inverse Derivative Analysis of a Quartic Polynomial

Consider the function $$f(x)= x^4 - 4*x^2 + 2$$ defined for $$x \ge 0$$. Answer the following.

Maclaurin Approximation for $$\ln(1+2*x)$$

Consider the function $$f(x)=\ln(1+2*x)$$. In this problem, you will generate the Maclaurin series f

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

Mean Value Theorem with Trigonometric Function

Consider the function $$f(x)= \sin(x)$$ on the interval $$[0,\pi]$$.

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

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

Rate of Reaction: Concentration Change

In a chemical reaction, the concentration (in mM) of a reactant is modeled by $$C(t) = 50*e^{-0.3*t}

Rolle's Theorem: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

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 $$\arcsin(x)$$

Derive the Maclaurin series for $$f(x)=\arcsin(x)$$ up to the $$x^5$$ term, determine the radius of

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

Travel Distance from Speed Data

A traveler’s speed (in km/h) is recorded at various times during a trip. Use the data to approximate

Vector Analysis of Particle Motion

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

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

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

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

Area Under a Piecewise-Defined Curve with a Jump Discontinuity

Consider the function $$ g(x)= \begin{cases} 2x+1 & \text{if } 0 \le x < 2, \\ 3x-2 & \text{if } 2 \

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

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Continuity and Integration of a Sinc-like Function

Consider the function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\ 1 & \text{i

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 $$x\ge2$$,

Definite Integral Involving an Inverse Function

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

Definite Integration of a Polynomial Function

For the function $$f(x)=5*x^{3}$$ defined on the interval $$[1,2]$$, determine the antiderivative an

Displacement vs. Total Distance Traveled

A particle moves along a straight line with the velocity function given by $$v(t)=t^2 - 4*t + 3$$. O

Drug Concentration in a Bloodstream

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

Energy Consumption in a Household

A household's power usage is modeled by $$P(t)= 3\sin((\pi/12)*t)+3$$ kW for $$t \in [0,24]$$ hours.

Evaluating an Integral Involving an Exponential Function

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

Implicit Differentiation Involving an Integral

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

Improper Integral Convergence

Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.

Integration by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Midpoint Riemann Sum Estimation

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

Net Change in Drug Concentration

The rate of change of a drug's concentration in the bloodstream is given by $$R(t)=8*e^{-0.5*t}$$ (i

Net Displacement vs. Total Distance Traveled

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

Particle Motion with Variable Acceleration and Displacement Analysis

A particle moves along a straight line with acceleration given by $$a(t)=4-2*t$$ (in m/s²). At time

Rainfall Accumulation Over Time

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

Revenue Accumulation and Constant of Integration

A company's revenue is modeled by $$R(t) = \int_{0}^{t} 3*u^2\, du + C$$ dollars, where t (in years)

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

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

Variable Interest Rate and Continuous Growth

An investment grows continuously with a variable interest rate given by $$r(t)=0.05+0.01*t$$. The in

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Volume of a Solid: Cross-Sectional Area

A solid has cross-sectional area perpendicular to the x-axis given by $$A(x)= (4-x)^2$$ for $$0 \le

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 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 \(\

Analysis of a Piecewise Function with Potential Discontinuities

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

Economic Growth Model

An economy's output \(Y(t)\) is modeled by the differential equation $$\frac{dY}{dt}= a\,Y - b\,Y^2$

Epidemic Spread Modeling

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

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 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

FRQ 17: Slope Field Analysis and Particular Solution

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

FRQ 18: Enzyme Reaction Rates

A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\

Implicit Differentiation in a Differential Equation Context

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

Implicit Solution of a Separable Differential Equation

Solve the differential equation $$\frac{dy}{dx}=\frac{y+1}{x}$$ with the initial condition $$y(1)=2$

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 in a Water Tank

A tank initially contains $$100$$ liters of saltwater with $$5$$ kg of salt dissolved in it. Pure wa

Modeling Currency Exchange Rates with Differential Equations

Suppose the exchange rate $$E(t)$$ (domestic currency per foreign unit) evolves according to the dif

Non-linear Differential Equation Modeling Spill Rate

Water leaks from a reservoir at a rate proportional to the square root of the volume. This is modele

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=6-4*t$$. At time t = 0,

Piecewise Differential Equation with Discontinuities

Consider the following piecewise differential equation defined for a function $$y(x)$$: For $$x < 2

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Projectile Motion with Air Resistance

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

Reservoir Contaminant Dilution

A reservoir has a constant volume of 10,000 L and contains a pollutant with amount $$Q(t)$$ (in kg)

Second-Order Differential Equation: Oscillations

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

Separable DE with Exponential Function

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

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx}= \frac{4*x}{y}$$ with the initial condition $$y(0

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

Slope Field and Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$

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

Solution and Analysis of a Linear Differential Equation with Equilibrium

Consider the differential equation $$\frac{dy}{dx} = 3*y - 2$$, with the initial condition $$y(0)=1$

Solution Curve from Slope Field

A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Temperature Change with Variable Ambient Temperature

A heated object is cooling in an environment where the ambient temperature changes over time. For $$

Temperature Control in a Chemical Reaction Vessel

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

Vibration of a Suspension Bridge

A suspension bridge’s vertical displacement is modeled by the differential equation $$\frac{d^2y}{dt

Analyzing a Reservoir's Volume Over Time

Water flows into a reservoir at a variable rate given by $$R(t)=50e^{-0.1*t}$$ m³/hour and simultane

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Area Between Exponential Curves

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

Average Population Density on a Road

A town's population density along a road is modeled by the function $$P(x)=50*e^{-0.1*x}$$ (persons

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 Population Growth Rate

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

Averaging Chemical Concentration in a Reactor

In a chemical reactor, the concentration of a substance is given by $$C(t)=100*e^{-0.5*t}+20$$ (mg/L

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

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

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Cost Function from Marginal Cost

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

Determining the Length of a Curve

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

Drone Motion Analysis

A drone’s vertical acceleration is modeled by $$a(t) = 6 - 2*t$$ (in m/s²) for time $$t$$ in seconds

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Flow Rate into a Tank

Water flows into a tank at a rate given by $$Q(t)=\frac{100}{1+t^2}$$ liters per hour on the interva

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Particle Motion with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

Polar Coordinates: Area of a Region

A region in the plane is described in polar coordinates by the equation $$r= 2+ \cos(θ)$$. Determine

Pollution Concentration in a Lake

A lake has a pollution concentration modeled by $$C(x) = 16 - x^2$$ (in mg/L), where $$x$$ (in meter

Total Charge in an Electrical Circuit

In an electrical circuit, the current is given by $$I(t)=5*\cos(0.5*t)$$ (in amperes), where \(t\) i

Volume by Revolution: Washer Method

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$. When this region is rotated about

Volume by the Shell Method: Rotating a Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the line $$y=0$$, and the vertical line $$x

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

Work Done on a Non-linear Spring

A non-linear spring exerts a force given by $$F(x) = 3 * x^2 + 2 * x$$ (in Newtons), where $$x$$ (in

Acceleration in Polar Coordinates

An object moves in the plane with its position expressed in polar coordinates by $$r(t)= 4+\sin(t)$$

Analysis of a Vector-Valued Function

Consider the vector-valued function $$\mathbf{r}(t)= \langle t^2+1,\; t^3-3*t \rangle$$, where $$t$$

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

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 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 with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

Area Between Polar Curves

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

Area Between Polar Curves: Annulus with a Hole

Two polar curves are given by \(R(\theta)=3\) and \(r(\theta)=2+\cos(\theta)\) for \(0\le\theta\le2\

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 and Differentiation of a Polar Curve

Consider the polar curve defined by $$ r=2+\sin(\theta) $$. Study its conversion to Cartesian coordi

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

Enclosed Area of a Parametric Curve

A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t

Intersection of Parametric Curves

Consider two particles moving along different paths: Particle A: $$x_A(t)= t^2, \quad y_A(t)= 2t +

Intersection Points of Polar Curves

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

Kinematics in the Plane: Parametric Motion

A particle moves in the plane with its position given by the parametric equations $$ x(t)=t^2-2*t $$

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

Optimization on a Parametric Curve

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

Parametric Egg Curve Analysis

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

Parametric Equations and Tangent Slopes

Consider the parametric equations $$x(t)= t^3 - 3*t$$ and $$y(t)= t^2$$, for $$t \in [-2, 2]$$. Anal

Particle Motion in Circular Motion

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

Particle Trajectory in Parametric Motion

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

Relative Motion of Two Objects

Two objects A and B move in the plane with positions given by the vector functions $$\vec{r}_A(t)= \

Taylor/Maclaurin Series: Approximation and Error Analysis

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

Time of Nearest Approach on a Parametric Path

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

Vector-Valued Function Analysis

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