AP Calculus BC FRQ Room

Algebraic Removal of Discontinuities in Rational Functions

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

Analysis of Rational Function Asymptotes and Removable Discontinuities

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

Analyzing Limits of a Combined Exponential‐Log Function

Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and

Applying the Squeeze Theorem

Let $$f(x)=x^2\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$. Use the Squeeze Theorem to evaluat

Approaching Vertical Asymptotes

Consider the function $$g(x)=\frac{3}{(x-2)^2}-\frac{1}{x-2}.$$ Answer the following:

Continuity Analysis in Road Ramp Modeling

A highway ramp is modeled by the function $$y(x)= \frac{(x-3)(x+2)}{x-3}$$ for $$x\neq3$$, where x i

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 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 of a Piecewise Log Function

Let $$g(x)= \begin{cases} \frac{\ln(1+x)-\sin(x)}{x} & x \neq 0 \\ a & x=0 \end{cases}.$$ Determine

Continuity of an Integral-Defined Function

Consider the function defined by the integral $$F(x)= \int_{0}^{x} \frac{t}{t^2+1} \; dt$$.

End Behavior of an Exponential‐Log Function

Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

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

L'Hôpital's Rule for Indeterminate Forms

Evaluate the limit $$h(x)=\frac{e^{2*x}-1}{\sin(3*x)}$$ as x approaches 0.

One-Sided Limits in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \sqrt{x+4}, & x < 5, \\ 3*x-7, & x \ge 5. \end{cases}$$ A

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

Real-World Temperature Sensor Analysis

A temperature sensor is modeled by the function $$T(t)=\frac{t^2-9}{t-3}$$ for t ≠ 3 (with t in minu

Trigonometric Limits

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

Water Flow Measurement Analysis

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

Water Tank Flow Analysis

A water tank receives water from an inlet and drains water through an outlet. The inflow rate is giv

Advanced Analysis of a Composite Piecewise Function

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

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Chain Rule in Biological Growth Models

A biologist models the growth of a bacterial population by the function $$P(t) = (5*t + 2)^4$$, wher

Derivative of a Function Involving an Absolute Value

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

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

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Evaluating the Derivative Using the Limit Definition

Consider the function $$f(x) = 3*x^2 - 2*x + 1$$. (a) Use the limit definition of the derivative:

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: Mixed Exponential and Polynomial Equation

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

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

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

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

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

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: Changing Shadow Length

A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts

River Flow Dynamics

A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$

Secant and Tangent Slope Analysis

Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Tangent Line to a Logarithmic Function

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

Temperature Change: Secant vs. Tangent Analysis

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

Urban Population Flow

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

Using 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

Analyzing an Implicit Function with Mixed Variables

Consider the curve defined by $$x^3 + x*y + y^3 = 3$$. Analyze the derivative of the curve at a give

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

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, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Chemical Mixing: Implicit Relationships and Composite Rates

In a chemical mix tank, the solute amount (in grams) and the concentration (in mg/L) are related by

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

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Differentiation Involving Inverse Trigonometric Functions

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

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 an Inverse Trigonometric Form

Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

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 in a Chemical Reaction

In a chemical process, the concentrations of two reactants, $$x$$ and $$y$$, satisfy the relation $$

Implicit Differentiation in a Hyperbola-like Equation

Consider the equation $$ x*y = 3*x - 4*y + 12 $$.

Implicit Differentiation on a Trigonometric Curve

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

Implicit Differentiation: Circle and Tangent Line

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

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 Analysis via Implicit Differentiation for a Transcendental Equation

Consider the equation $$e^{x*y}+x-y=0$$ defining y implicitly as a function of x near a point where

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 Derivatives in a Sensor Model

An instrument outputs a reading defined by $$f(x)= x^3 + 2$$, where $$x$$ represents the voltage inp

Inverse Function Differentiation Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

Inverse Function Differentiation for a Trigonometric-Polynomial Function

Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Inverse Function Differentiation in Economics

A product’s demand is modeled by a one-to-one differentiable function $$Q = f(P)$$, where $$P$$ is t

Inverse of a Radical Function with Domain Restrictions

Consider the function $$f(x)=\sqrt{1-x^2}$$. Analyze its invertibility.

Inverse of a Shifted Logarithmic Function

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

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Parameter Dependent Composite Function

The temperature in a metal rod is modeled by $$T(x)= e^{a*x}$$, where the parameter $$a$$ changes wi

Physics Lab: Logarithmic Chain Rule in a Kinetics Experiment

In a kinetics experiment, the reactant concentration is modeled by $$C(t)=\ln(3*e^{2*t}+4)$$, where

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Second Derivative via Implicit Differentiation

Given the relation $$x^2 + x*y + y^2 = 7$$, answer the following:

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

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.

Chain Rule in Temperature Distribution along a Rod

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

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

City Population Migration

A city's population is influenced by immigration at a rate of $$I(t)=100e^{-0.2t}$$ (people per year

Comparison of Series Convergence and Error Analysis

Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

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

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Differentiation of a Product Involving Exponentials and Logarithms

Consider the function $$f(t)=e^{-t}\ln(t+2)$$, defined for t > -2. Answer the following:

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

Drug Concentration in the Blood

A patient's drug concentration is modeled by $$C(t)=20e^{-0.5t}+5$$, where $$t$$ is measured in hour

Ellipse Tangent Line Analysis

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

Exponential Relation

Consider the equation $$e^{x*y} = x + y$$.

Filling a Conical Tank

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

Implicit Differentiation in a Tank Filling Problem

A tank's volume and liquid depth are related by $$V=10y^3$$, where y (in meters) is the depth. Water

Ladder Sliding Down a Wall

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

Maclaurin Series for ln(1+x)

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

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Motion Model Inversion

Suppose that the position of a particle moving along a line is given by $$f(t)=t^3+t$$. Analyze the

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

Optimization in Design: Maximizing Inscribed Rectangle Area

A rectangle is inscribed in a semicircle of radius $$R$$ (with the rectangle's base along the diamet

Optimization with Material Costs

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

Parametric Motion Analysis

A particle moves such that its position is described by the parametric equations $$x(t)= t^2 - 4*t$$

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Population Growth Rate

The population of a bacteria culture is given by $$P(t)= 500e^{0.03*t}$$, where $$t$$ is in hours. A

Rate of Change in Logarithmic Brightness

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

Revenue Concavity Analysis

A company's revenue over time is modeled by $$R(t)=100\ln(t+1)-2t$$. Answer the following:

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 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

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

Varying Acceleration and Particle Motion

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

Analyzing a Function with Implicit Logarithmic Differentiation

Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating

Application of Rolle's Theorem

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

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:

Arc Length Approximation

Let $$f(x) = \sqrt{x}$$ be defined on the interval [1,9].

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Combining Series and Integration to Analyze a Population Model

A population's growth rate is approximated by the series $$P'(t)=\sum_{n=0}^\infty \frac{t^n}{(n+1)!

Concavity & Inflection Points for a Rational Polynomial Function

Examine the function $$f(x)= \frac{x}{x^2+1}$$ to determine its concavity and identify any inflectio

Concavity and Inflection Points

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Convergence and Series Approximation of a Simple Function

Consider the function defined by the power series $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n}{n+1} * x^n$

Economic Equilibrium and Implicit Differentiation

An economic equilibrium is modeled by the implicit equation $$e^{p}*q + p^2 = 100$$, where \( p \) r

Error Approximation using Linearization

Consider the function $$f(x) = \sqrt{4*x + 1}$$.

Exponential Decay in Velocity

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

Finding and Interpreting Inflection Points in a Complex Function

Analyze the function $$f(x)= e^{-x}\,\ln(x)$$ for $$x > 0$$. Investigate the points of inflection an

Fractal Tree Branch Lengths

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

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Mean Value Theorem with a Trigonometric Function

Let $$f(x)=\sin(x)$$ be defined on the interval $$[0,\pi]$$. Answer the following parts:

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

Optimal Timing via the Mean Value Theorem

A particle’s position is given by $$s(t)=t^2e^{-t}+3$$ for $$t\in[0,3]$$.

Optimization in Production Costs

In an economic context, consider the cost function $$C(x)=0.5*x^3-6*x^2+25*x+100$$, where C(x) repre

Parameter Estimation in a Log-Linear Model

In a scientific experiment, the data is modeled by $$P(t)= A\,\ln(t+1) + B\,e^{-t}$$. Given that $$P

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Population Growth Modeling

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

Projectile Motion Analysis

A projectile is launched vertically with its height given by $$s(t) = -16*t^2 + 64*t + 80$$ (in feet

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: Changing Shadow Length

A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp

Relative Extrema and Critical Points of a Cubic Polynomial

Consider the function $$f(x)=x^3 - 6*x^2 + 9*x + 2$$. Use the analytical techniques of differentiati

River Sediment Transport

Sediment enters a river from a landslide at a rate of $$S_{in}(t)=4*\exp(0.2*t)$$ tonnes/day and is

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 $$\ln(x)$$ about $$x=1$$

For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,

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

Accumulated Displacement from Acceleration

A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini

Accumulation Function and the Fundamental Theorem of Calculus

Let $$F(x) = \int_{2}^{x} \sqrt{1 + t^3}\, dt$$. Answer the following parts regarding this accumulat

Accumulation Function from a Rate Function

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

Analyzing an Invertible Cubic Function

Consider the function $$f(x) = x^3 + 2*x + 1$$ defined for all $$x$$. Answer the following questions

Antiderivatives and the Constant of Integration

Consider the rate function $$ r(t)= 2*t + 3 $$ where t represents time in seconds.

Comparing Integration Approximations: Simpson's Rule and Trapezoidal Rule

A student approximates the definite integral $$\int_{0}^{4} (x^2+1)\,dx$$ using both the trapezoidal

Composite Functions and Inverses

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

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 $$x\ge2$$,

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

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Differentiation and Integration of a Power Series

Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.

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.

Finding Area Between Two Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.

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 Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \tan^{-1}(x)$$. Answer the following questions regarding its inverse a

Integration of a Trigonometric Function by Two Methods

Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me

Integration Using U-Substitution

Evaluate the integral $$\int (3*x+2)^5\,dx$$ using u-substitution.

Integration Using U-Substitution

Evaluate the definite integral $$\int_{0}^{2} (3*x+1)^{4} dx$$ using u-substitution. Answer the foll

Integration via Partial Fractions

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

Integration via U-Substitution for a Composite Function

Evaluate the integral of a composite function and its definite form. In particular, consider the fun

Interpreting the Constant of Integration in Cooling

An object cools according to the differential equation $$\frac{dT}{dt}=-k*(T-20)$$ where $$T(t)$$

Inverse Functions in Economic Models

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

Particle Displacement and Total Distance

A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Recovering Accumulated Change

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

Riemann and Trapezoidal Sums with Inverse Functions

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

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

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

Vehicle Distance Estimation from Velocity Data

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

Volume of a Solid by the Shell Method

Consider the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and the vertical line $$x=4$$.

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 on a Nonlinear Spring

A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di

Analysis of a Piecewise Function with Potential Discontinuities

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

Chemical Reaction Rate

A chemical reaction causes the concentration $$A(t)$$ of a reactant to decrease according to the rat

Chemical Reaction Rate

In a chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to the first-or

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Chemical Reaction Rate Modeling

In a chemical reaction, the concentration $$C(t)$$ (in moles per liter) of a reactant decreases acco

Compound Interest and Investment Growth

An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the

Cooling Coffee Data Analysis

A cup of coffee cools down in a room according to Newton's Law of Cooling. The temperature $$T(t)$$

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

Exact Differential Equation

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

FRQ 18: Enzyme Reaction Rates

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

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 $$x$$

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

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 with Time-Dependent Inflow

A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa

Modeling Ambient Temperature Change

The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT

Modeling Disease Spread with Differential Equations

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

Modeling Free Fall with Air Resistance

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

Modeling Medication Concentration in the Bloodstream

A patient receives an intravenous drug at a constant rate $$R$$ (mg/min) and the drug is eliminated

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

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

A fish population in a lake is modeled by the logistic equation with harvesting: $$\frac{dP}{dt}=r\,

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, where $

Related Rates: Conical Tank Overflow

A conical tank has a height of $$10\,m$$ and a base radius of $$4\,m$$. Water is draining from the t

RL Circuit Analysis

An RL circuit is described by the differential equation $$L\frac{di}{dt} + R*i = V$$, where $$L=0.5\

Salt Tank Mixing Problem

A tank contains $$100$$ L of water with $$10$$ kg of salt. Brine containing $$0.5$$ kg of salt per l

Simplified Predator-Prey Model

A simplified model for a predator population is given by the differential equation $$\frac{dP}{dt} =

Temperature Change and Differential Equations

A hot liquid cools in a room at $$20^\circ C$$ according to the differential equation $$\frac{dT}{dt

Temperature Control in a Chemical Reaction Vessel

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

Temperature Regulation in Biological Systems

In a biological system, the temperature \(T(t)\) (in °C) of an organism is modeled by the differenti

Tumor Growth Under Chemotherapy

A tumor's size $$S(t)$$ (in cm³) grows at a rate proportional to its size, at $$0.08*S(t)$$, but che

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

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Area Between a Parabola and a Line

Consider the region bounded by the curves $$y=5*x - x^2$$ and $$y=x$$ where they intersect. Answer t

Area Between a Rational Function and Its Asymptote

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

Area Between Nonlinear Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=\frac{x}{3}$$. Determine the area between these tw

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 Value of a Polynomial Function

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

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Car Braking and Stopping Distance

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

Center of Mass of a Nonuniform Rod

A thin rod extends from $$x=0$$ to $$x=3$$ and has a linear density given by $$\delta(x)=1+x$$ (in k

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

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

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 Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Implicit Differentiation with Trigonometric Function

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

Inverse Function Analysis

Consider the function $$f(x)=3*x^3+2$$ defined for all real numbers.

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

Pumping Water from a Conical Tank

An inverted right circular conical tank has a height of $$10$$ meters and a top radius of $$4$$ mete

Surface Area of a Rotated Curve

Consider the curve $$y=x^3$$ on the interval $$[0,2]$$. This curve is rotated about the x-axis, form

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

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

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

Work Done by a Variable Force

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=5*x$$ (in Newtons), where $$x$$ is

Analysis of a Polar Curve: The Limaçon

Consider the polar curve $$r(θ)= 2+\cos(θ)$$ for $$0 \le θ \le 2\pi$$. Answer the following:

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 and Surface Area of Revolution from a Parametric Curve

Consider the curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\ln(\sec(t)+\tan(t))$$ for $$0 \le t < \fr

Arc Length of a Cycloid

Consider the cycloid defined by the parametric equations $$x(t)= t - \sin(t)$$ and $$y(t)= 1 - \cos(

Arc Length of a Parametric Curve

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

Arc Length of a Vector-Valued Curve

A vector-valued function is given by $$\mathbf{r}(t)=\langle e^t,\, \sin(t),\, \cos(t) \rangle$$ for

Area Between Two Polar Curves

Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:

Area of a Region in Polar Coordinates with an Internal Boundary

Consider a region bounded by the outer polar curve $$R(\theta)=5$$ and the inner polar curve $$r(\th

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

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

Conversion between Polar and Cartesian Coordinates

The polar equation $$r = 2 + 2\cos(\theta)$$ describes a limaçon. Analyze this curve by converting i

Curvature of a Space Curve

Consider the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t), t \rangle$$ for $$t \i

Displacement from a Vector-Valued Velocity Function

A particle's velocity is given by $$\vec{v}(t)=\langle \cos(t), \sin(t), t \rangle$$ for $$t \in [0,

Dynamics in Polar Coordinates

A particle moves such that its polar coordinates are given by $$ r(\theta)=1+\theta $$, where $$ \th

Exponential and Logarithmic Dynamics in a Polar Equation

Consider the polar curve defined by $$r=e^{\theta}$$. Answer the following:

Integrating a Vector-Valued Function

A particle has a velocity given by $$\vec{v}(t)= \langle e^t, \cos(t) \rangle$$. Its initial positio

Kinematics in Polar Coordinates

A particle’s position in polar coordinates is given by $$r(t)= \frac{5*t}{1+t}$$ and $$\theta(t)= \f

Length of a Polar Spiral

For the polar spiral defined by $$r=\theta$$ for $$0 \le \theta \le 2\pi$$, answer the following:

Maclaurin Series for Trigonometric Functions

Let $$f(x)=\sin(x)$$.

Motion Along a Helix

A particle moves along a helix defined by $$\mathbf{r}(t)=\langle \cos(t), \sin(t), t \rangle$$.

Motion Along an Elliptical Path

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

Oscillatory Behavior in Vector-Valued Functions

Examine the vector-valued function $$\mathbf{r}(t)=\langle \cos(2*t), \sin(3*t), \cos(t)\sin(2*t) \r

Polar Differentiation and Tangent Lines

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$.

Polar to Cartesian Conversion

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

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Tangents and Normals of a Parametric Curve

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