AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

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

Absolute Value Limit Analysis II

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

Analyzing Continuity on a Closed Interval

Suppose a function $$f(x)$$ is continuous on the closed interval $$[0,5]$$ and differentiable on the

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Caffeine Metabolism in the Human Body

A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu

Computing a Limit Using Algebraic Manipulation

Evaluate the limit $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ using algebraic manipulation.

Continuity Analysis Involving Logarithmic and Polynomial Expressions

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

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Continuity and Asymptotes of a Log‐Exponential Function

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

Economic Growth and Continuity

The function $$E(t)$$ represents an economy's output index over time (in years). A table provides th

Examining Continuity with an Absolute Value Function

Consider the function defined by $$f(x)=\frac{|x-2|}{x-2}$$ for $$x \neq 2$$. (a) Evaluate $$\lim_{x

Fuel Efficiency and Speed Graph Analysis

A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph

Internet Data Packet Transmission and Error Rates

In a data transmission system, an error correction protocol improves the reliability of transmitted

Interplay of Polynomial Growth and Exponential Decay

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

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

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Limits Involving Radical Functions

Examine the function $$m(x)=\frac{\sqrt{x}-2}{x-4}$$.

Limits Involving Trigonometric Ratios

Consider the function $$f(x)= \frac{\sin(2*x)}{x}$$ for $$x \neq 0$$. A table of values near $$x=0$$

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

One-Sided Limits and Jump Discontinuities

Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en

Parameter Determination for Continuity

Let $$h(x)= \begin{cases} \frac{e^{2x} - 1 - a\,\ln(1+bx)}{x} & x \neq 0 \\ c & x = 0 \end{cases}.$$

Population Dynamics and Logistic Growth Model

A population of a species is modeled by the function $$P(t)=\frac{1000}{1+9e^{-0.3*t}},$$ where t is

Rational Function Limit and Continuity

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

Resistor Network Convergence

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

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Trigonometric Limits

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

Water Filling a Leaky Tank

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

Water Treatment Plant Discontinuity Analysis

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

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

Chemical Mixing Tank

In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t

Derivative via Quotient Rule: Fluid Flow Rate

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Differentiation of Implicitly Defined Functions

An ellipse is defined by the equation $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$. Use implicit differenti

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 $$x$$

Exponential Growth Derivative

In a model of bacterial growth, the population is described by $$f(t)=5*e^(0.2*t)+7$$, where \(t\) i

Higher Order Derivatives: Concavity and Inflection Points

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

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Limit Definition of the Derivative for a Trigonometric Function

Consider the function $$f(x)= \cos(x)$$.

Manufacturing Production Rates

A factory produces items at a rate given by $$P_{in}(t)=\frac{200}{1+e^{-0.3*(t-4)}}$$ items per hou

Polar Coordinates and Tangent Lines

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

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

River Flow Dynamics

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

Secant Line Approximation in an Experimental Context

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

Tangent Line Estimation from Experimental Graph Data

A function $$f(x)$$ is represented by the following graph of experimental data approximating $$f(x)=

Using the Limit Definition for a Non-Polynomial Function

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

Velocity and Acceleration Analysis

A particle moving along a straight line has a velocity function given by $$v(t)=2*t^2 - 8*t + 3$$ (i

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 Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Chain Rule Application: Differentiating a Nested Trigonometric Function

Consider the function $$f(x) = \sin(\cos(2*x))$$. Analyze its derivative using the chain rule.

Chain Rule in a Nested Composite Function

Consider the function $$f(x)= \sin\left(\ln((2*x+1)^3)\right)$$. Answer the following parts:

Chain Rule in the Context of Light Intensity Decay

The light intensity as a function of distance from the source is given by $$I(x) = 500 * e^{-0.2*\sq

Chain Rule with Exponential Function

Consider the function $$h(x)= e^{\sin(4*x)}$$ which models a process with exponential growth modulat

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

Composite Implicit Differentiation Involving Trigonometric and Polynomial Terms

Consider the relation $$\sin(x*y) + y^3 = x$$.

Differentiation of an Inverse Trigonometric Form

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

Implicit Differentiation in a Cost-Production Model

In an economic model, the relationship between the production level $$x$$ (in units) and the average

Implicit Differentiation in Geometric Optics

A parabolic mirror used in a geometric optics experiment is described by the implicit equation $$x^2

Implicit Differentiation in Mixed Function Equation

Consider the relation $$x^2*y+\sin(y)=5*x$$. Analyze this relation using implicit differentiation.

Implicit Differentiation Involving Logarithms

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

Implicit Differentiation of a Product Equation

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

Implicit Differentiation: Second Derivative of Exponential-Trigonometric Equation

Consider the equation $$e^{x*y} + \sin(y) - x^2 = 0$$ where $$y$$ is defined implicitly as a functio

Implicit Differentiation: Second Derivatives of a Circle

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

Inverse Function Derivative for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Inverse Function Derivative in an Exponential Model

Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p

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

Logarithmic and Exponential Composite Function with Transformation

Let $$g(x)=\ln((3*x+1)^2)-e^{x}$$. Answer the following questions.

Multi-step Differentiation of a Composite Logarithmic Function

Consider the function $$F(x)= \sqrt{\ln\left(\frac{1+e^{2*x}}{1-e^{2*x}}\right)}$$, defined for valu

Nested Composite Function Differentiation

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

Power Series Representation and Differentiation of a Composite Function

Let $$f(x)= \sin(x^2)$$ and consider its Maclaurin series expansion.

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Second Derivative via Chain Rule

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

Taylor Polynomial and Error Bound for a Trigonometric Function

Let $$f(x) = \cos(2*x)$$. Develop a second-degree Taylor polynomial centered at 0, and analyze the a

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 Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

Analyzing Motion on a Curved Path

A particle moves along a path defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$t \in [0,2\pi]$

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

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 Differentials

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

Bacterial Population Growth Analysis

A laboratory culture of bacteria has an initial population of $$P_0=1000$$ and grows according to th

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

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 $

Conical Tank Filling

A conical water tank has a height of $$10$$ m and a top radius of $$4$$ m. The water in the tank for

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

Graphical Analysis of an Inverse Function

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

Inflating Spherical Balloon

A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1

Interpreting the Derivative in Straight Line Motion

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

Inverse of a Trigonometric Function

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

Inversion in a Light Intensity Decay Model

A laboratory experiment records the decay of light intensity over time, modeled by $$f(t)=80*e^{-0.2

Linearization and Differentials: Approximating Function Values

Consider the function $$f(x)= x^4$$. Use linearization to estimate the value of the function for a s

Linearization in Inverse Function Approximation

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

Linearization to Estimate Change in Electrical Resistance

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

Maximizing Efficiency: Derivative Analysis in a Production Process

The efficiency of a production process is modeled by $$E(x)=50+10*\ln(x)-0.5*x$$, where $$x$$ repres

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

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.

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

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

Particle Motion with Changing Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=8-3*t$$ (in m/s²). Given

Population Growth Analysis

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

Quadratic Function Inversion with Domain Restriction

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

Rational Function Particle Motion Analysis

A particle moves along a straight line with its position given by $$s(t)=\frac{t^2+1}{t-1}$$, where

Related Rates: Circular Oil Spill

An oil spill on a lake forms a circular patch whose area is given by $$A= \pi*r^2$$, where $$r$$ is

Series Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Series Approximation for a Displacement Function

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

Series Approximation in an Exponential Population Model

A population is modeled by $$P(t)= 1000 \times \sum_{n=0}^{\infty} \frac{(0.05t)^n}{n!}$$, which is

Temperature Change of Coffee: Exponential Cooling

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

Water Tank Filling: Related Rates

A cylindrical tank has a fixed radius of 2 m. The volume of water in the tank is given by $$V=\pi*r^

Application of the Mean Value Theorem in Motion

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

Arc Length of a Parametric Circular Arc

A curve is defined parametrically by $$x(t) = 2*\cos(t)$$ and $$y(t) = 2*\sin(t)$$, where t varies f

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

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

Concavity and Inflection Points

Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:

Concavity in an Economic Model

Consider the function $$f(x)= x^3 - 3*x^2 + 2$$, which represents a simplified economic trend over t

Convergence and Differentiation of a Series with Polynomial Coefficients

The function $$P(x)=\sum_{n=0}^\infty \frac{n^2 * (x-1)^n}{3^n}$$ is used to model stress in a mater

Differentiability and Optimization of a Piecewise Function

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

Drug Dosage Infusion

A patient receives an intravenous drug infusion at a rate given by $$D(t)=4*\exp(-0.2*t)$$ mg/min. A

Epidemic Infection Model

In a community experiencing an epidemic, the rate of new infections is modeled by $$I(t)=\frac{200}{

Exponential Decay in Velocity

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

Ink Drop Diffusion and Intensity Loss

When a drop of ink is placed in water, it spreads out in concentric rings. The intensity of the ink

Instantaneous vs. Average Rates in a Real-World Model

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

Inverse Analysis for a Function with Multiple Transformations

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

Light Reflection Between Mirrors

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

Loan Amortization with Increasing Payments

A loan of $$20000$$ dollars is to be repaid in equal installments over 10 years. However, the repaym

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 in Motion

A car travels along a straight highway with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t + 5$$

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

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

Real-World Modeling: Radioactive Decay with Logarithmic Adjustment

A radioactive substance decays according to $$N(t)= N_0\,e^{-0.03t}$$. In an experiment, the recorde

Region Area and Volume: Polynomial and Linear Function

A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo

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

River Pollution Analysis

A river receives a pollutant through industrial discharge at a rate of $$P_{in}(t)=10*\exp(-0.1*t)$$

Tangent Line and Linearization

Consider the function $$ f(x)=\sqrt{x+5}.$$ Answer the following parts:

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

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

Taylor Series in Economics: Cost Function

An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro

Transformation Between Function and Inverse

Consider the function $$f(x)= (x-2)^3 + 5$$ defined for all real numbers. Answer the following parts

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Volume of a Solid of Revolution Using the Washer Method

Find the volume of the solid obtained by revolving the region bounded by $$y=\sqrt{x}$$, $$y=\frac{x

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Antiderivatives and the Constant of Integration

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

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

Arc Length of $$y=x^{3/2}$$ on $$[0,4]$$

The curve defined by $$y=x^{3/2}$$ is given for $$x\in[0,4]$$. The arc length of a curve is determin

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{2}{3}*x^{3/2}$$ for $$x$$ in the interval $$[1,9]$$. Answ

Area Estimation with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

Area Under a Piecewise Function

A function is defined piecewise as follows: $$f(x)=\begin{cases} x & 0 \le x \le 2,\\ 6-x & 2 < x \

Bacterial Growth Accumulation

The instantaneous growth rate of a bacterial culture is modeled by $$r(t)= 0.3*t$$ million cells per

Calculating Work Using Integration

A variable force is given by $$F(x)=5*x^2-2*x$$ (in Newtons) and is applied along the direction of m

Comprehensive Integration of a Polynomial Function

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

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

Error Bound Analysis for the Trapezoidal Rule

For the function $$f(x)=\ln(x)$$ on the interval $$[1,2]$$, the error bound for the trapezoidal rule

Evaluation of an Improper Integral

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

Integration by Parts: Logarithmic Function

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

Integration of a Piecewise Function for Total Area

Consider the piecewise function $$f(x)$$ defined by: $$f(x)=3-x$$ for $$0 \le x \le 3$$, and $$f(x)=

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

Integration via Substitution and Numerical Methods

Evaluate the integral $$\int_0^2 \frac{2*x}{\sqrt{1+x^2}}\,dx$$.

Investigating Partition Sizes

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

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

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

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

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

Population Model Inversion and Accumulation

Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Total Cost from a Marginal Cost Function

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

U-Substitution in Accumulation Functions

In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe

Bacteria Growth with Antibiotic Treatment

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

Braking of a Car

A car decelerates according to the differential equation $$\frac{dv}{dt} = -k*v$$, where k is a posi

Chemical Reaction in a Closed System

The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential

Combined Differential Equations and Function Analysis

Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y

Differential Equations in Economic Modeling

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

Direction Field Analysis for Differential Equation

Consider the differential equation $$\frac{dy}{dx} = y\,(1-y)$$. A direction field for this equation

Economic Model: Differential Equation for Cost Function

A company’s marginal cost is represented by $$MC(x)=\frac{dC}{dx}=3+2*x$$, where $$x$$ is the number

Exact Differential Equation

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

Exact Differential Equations

Consider the differential equation $$ (2*x + y) + (x + 3*y)\,\frac{dy}{dx} = 0$$.

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

FRQ 2: Separable Differential Equation with Initial Condition

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

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 $$x$$

Logistic Growth Model in Population Dynamics

A population is modeled by the logistic differential equation $$\frac{dy}{dt} = 0.5*y\left(1-\frac{y

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

Modeling Cooling in a Variable Environment

Suppose the cooling of a heated object is modeled by the differential equation $$\frac{dT}{dt} = -k*

Modeling Disease Spread with Differential Equations

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

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

A cup of coffee at an initial temperature of $$90^\circ C$$ is placed in a room maintained at a cons

Newton's Law of Cooling

An object with an initial temperature of $$90\,^{\circ}C$$ is placed in a room maintained at $$20\,^

Nonlinear Differential Equation with Implicit Solution

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

Piecewise Differential Equation with Discontinuities

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

Projectile Motion with Air Resistance

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

Radioactive Decay with Constant Production

A radioactive substance decays at a rate proportional to its current amount but is also produced at

Radioactive Decay with External Source

A radioactive substance decays while receiving a constant external activation. Its behavior is model

Slope Field Analysis and Solution Curve Sketching for $$\frac{dy}{dx}= x - y$$

Consider the differential equation $$\frac{dy}{dx} = x - y$$ with initial condition $$y(0)=1$$. You

Solution Curve from Slope Field

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

Solution Curve Sketching Using Slope Fields

Given the differential equation $$\frac{dy}{dx} = x - y$$, a slope field is provided. Use the field

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Accumulated Change in a Population Model

A population of insects grows at a rate given by $$P'(t)=10e^{-0.2*t}$$, where $$t$$ is in days and

Accumulated Interest in a Savings Account

An investor’s savings account experiences continuous deposits and withdrawals. The deposit rate is g

Accumulated Rainfall

The rainfall intensity in a region is given by $$R(t)=0.2*t^2+1$$ (in cm/hour), where $$t$$ is measu

Advanced Parameter-Dependent Integration Problem

Consider the function $$g(x)=e^{-a*x}$$, where $$a>0$$ and $$x$$ lies within $$[0,b]$$. The average

Arc Length of a Cable

A cable hanging in a particular configuration follows the curve $$y=\ln(x+1)$$ for $$x\in[0,4]$$. De

Area Between a Parabola and a Line

Let $$f(x)= x^2$$ and $$g(x)= 2*x + 3$$. Determine the area of the region bounded by these two curve

Average Car Speed Analysis from Discrete Data

A car's speed (in km/h) is recorded at equal time intervals over a 1-hour journey. Analyze the car's

Average Temperature Over a Day

A function modeling the temperature (in °F) throughout a day is given by $$T(t)= 3*\sin\left(\frac{\

Average Value of a Trigonometric Function

A function representing sound intensity is given by $$I(t)= 4*\cos(2*t) + 10$$ over the time interva

Average Velocity of a Car

A car's velocity is given by $$v(t)=20-4*\ln(t+1)$$ (in m/min) for $$t$$ in minutes on the interval

Center of Mass of a Rod with Variable Density

A thin rod of length 10 meters lies along the x-axis from $$x=0$$ to $$x=10$$. Its density is given

Comparing Average and Instantaneous Rates of Change

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

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

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

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. 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

Net Cash Flow Analysis

A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont

Piecewise Velocity Analysis

A particle moves along a straight line with velocity given by the following piecewise function: $$v

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

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

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 Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

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

Work Done by a Variable Force

A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as

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 given by $$F(x)= 2*x + 3$$ (in Newtons) is applied to an object as it moves along a

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Analyzing a Looping Parametric Curve

The curve is defined by the equations $$x(t)=t^3-3t$$ and $$y(t)=t^2$$ for \(-2\le t\le 2\). Due to

Analyzing a Walker's Path: A Vector-Valued Function

A pedestrian's path is modeled by the vector function $$\vec{r}(t)= \langle t^2 - 4, \sqrt{t+5} \ran

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 Polar Curve

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

Comparing Arc Lengths in Parametric and Polar Systems

Consider the curve given in parametric form by $$x(t)=\cos(2*t)$$ and $$y(t)=\sin(2*t)$$ for $$0\le

Conversion Between Polar and Cartesian Coordinates

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

Conversion from Polar to Cartesian Coordinates

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

Conversion to Cartesian and Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.

Converting Polar to Cartesian and Computing Slope

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

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,

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

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

Lissajous Figures and Their Properties

A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Oscillatory Motion in a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

Parametric Curves and Intersection Points

Two curves are defined by $$C_1: x(t)=t^2,\, y(t)=2*t+1$$ and $$C_2: x(s)=4-s^2,\, y(s)=3*s$$. Find

Parametric Equations of a Cycloid

A cycloid is generated by a point on the circumference of a circle of radius $$r$$ rolling along a s

Parametric Particle Motion

A particle moves along a path described by the parametric equations: $$x(t)=t^2-2*t$$ and $$y(t)=3*t

Parametric Representation of Circular Motion

An object moves along a circle of radius $$5$$, with its position given by $$x(t)=5\cos(t)$$ and $$y

Particle Motion in the Plane

A particle's position is given by the parametric equations $$x(t)=3*t^2-2*t$$ and $$y(t)=4*t^2+t-1$$

Particle Motion with Logarithmic Component

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

Polar Coordinates and Dynamics

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

Polar Differentiation and Tangent Lines

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

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Projectile Motion via Parametric Equations

A projectile is launched with initial speed $$v_0 = 20\,m/s$$ at an angle of $$45^\circ$$. Its motio

Vector-Valued Fourier Series Representation

The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c

Vector-Valued Motion: Acceleration and Maximum Speed

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

Wind Vector Analysis in Navigation

A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)