AP Calculus BC FRQ Room

Comparing Methods for Limit Evaluation

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

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Continuity Involving a Radical Expression

Examine the function $$f(x)= \begin{cases} \frac{\sqrt{x+4}-2}{x} & x \neq 0 \\ k & x=0 \end{cases}$

Electricity Consumption Rate Analysis

A table provides the instantaneous electricity consumption, $$E(t)$$ (in kW), at various times durin

End Behavior and Horizontal Asymptote Analysis

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

Evaluating a Logarithmic Limit

Given the limit $$\lim_{x \to 2} \frac{\ln(x-1)}{x^2-4} = k$$, find the value of $$k$$ using algebra

Graphical Analysis of Volume with a Jump Discontinuity

A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer

Implicitly Defined Curve and Its Tangent Line

Consider the circle defined by the equation $$x^2+y^2=16$$. Answer the following:

Implicitly Defined Function and Differentiation

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

Infinite Limits and Vertical Asymptotes

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

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 a Function with a Removable Discontinuity

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

Limit and Continuity with Parameterized Functions

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

Limits with Infinite Discontinuities

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

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

One-Sided Limits and Jump Discontinuity Analysis

Consider the piecewise function $$ f(x)= \begin{cases} x+2, & x < 1 \\ 3-x, & x \ge 1 \end{cases} $

Oscillatory Behavior and Non-Existence of a Limit

A water sensor records the inflow rate in a canal as $$R(t)=5+\sin(1/t)$$ for \(t>0\). The function

Population Growth and Limits

The population $$P(t)$$ of a small town is recorded every 10 years as shown in the table below. Assu

Radioactive Material Decay with Intermittent Additions

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

Rate of Change in a Chemical Reaction (Implicit Differentiation)

In a chemical reaction the concentration C (in M) of a reactant is related to time t (in minutes) by

Rational Function Limit and Continuity

Consider the function $$f(x)=\frac{x^2+5*x+6}{x+3}$$ defined for $$x\neq -3$$. Notice that the funct

Squeeze Theorem with an Oscillatory Factor

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

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 Tank Inflow with Oscillatory Variation

A water tank is equipped with a sensor that records the inflow rate with a slight oscillatory error.

Analysis of a Piecewise Function

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

Analysis of Concavity and Second Derivative

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

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

Composite Function Behavior

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

Comprehensive Analysis of $$e^{-x^2}$$

The function $$f(x)=e^{-x^2}$$ is used to model temperature distribution in a material. Provide a co

Cooling Tank System

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

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Derivatives of Inverse Functions

Let $$f(x)=\ln(x)$$ with inverse function $$f^{-1}(x)=e^x$$. Answer the following parts.

Differentiation of a Trigonometric Function

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

Differentiation of Parametric Equations

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

Epidemic Spread Rate: Differentiation Application

The number of infected individuals in an epidemic is modeled by $$I(t)= \frac{200}{1+e^{-0.5(t-5)}}$

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:

Exploration of Derivative Notation and Higher Order Derivatives

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

Exponential Population Growth in Ecology

A certain species in a reserve is observed to grow according to the function $$P(t)=1000*e^{0.05*t}$

Graphical Estimation of Tangent Slopes

Using the provided graph of a function g(t), analyze its rate of change at various points.

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

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 in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

Implicit Differentiation on a Circle

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

Implicit Differentiation with Exponential and Trigonometric Functions

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

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Interpreting Graphical Slope Data

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

Logarithmic Differentiation in Temperature Modeling

The temperature distribution along a rod is modeled by the function $$T(x)=\ln(5*x^2+1)*e^{-x}$$. He

Manufacturing Cost Function and Instantaneous Rate

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

Optimization Problem via Derivatives

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

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Population Growth Approximation using Taylor Series

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

Population Growth Rates

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

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Product and Quotient Rules in Economic Modeling

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x)= (x+2)(x-1)$$ where

Product of Exponential and Trigonometric Functions

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

Secant to Tangent Convergence

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

Second Derivative and Concavity Analysis

Consider the function $$f(x)=x^3-6*x^2+12*x-5$$. Answer the following:

Tangent Line Approximation

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

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Taylor Series for Cos(x) in Temperature Modeling

An engineer uses the cosine function to model periodic temperature variations. Approximate $$\cos(x)

Taylor Series of ln(x) Centered at x = 1

A researcher studies the natural logarithm function $$f(x)=\ln(x)$$ by constructing its Taylor serie

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

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

Calculating an Inverse Trigonometric Derivative in a Physics Context

A pendulum's angle is modeled by $$\theta = \arcsin(0.5*t)$$, where $$t$$ is time in seconds and $$\

Chain Rule Combined with Inverse Trigonometric Differentiation

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

Chain Rule for Inverse Trigonometric Functions in Optics

In an optics experiment, the angle of incidence $$\theta(t)$$ (in radians) is modeled by $$\theta(t)

Chain Rule with Trigonometric Composite Function

Examine the function $$ h(x)= \sin((2*x^2+1)^2) $$.

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)

Composite Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Differentiation Involving an Inverse Function and Logarithms

Let $$f(x)= x^2+ 3*x+ 2$$, and let y be defined by $$y= \ln(f^{-1}(e^{x}))$$, where $$f^{-1}$$ is th

Differentiation of an Inverse Function

Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a

Exponential Composite Function Differentiation

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

Higher Order Implicit Differentiation in a Nonlinear Model

Assume that \(x\) and \(y\) are related by the nonlinear equation $$e^{x*y} + x - \ln(y) = 5$$ with

Implicit Differentiation in a Circle

Consider the circle defined by $$ x^2+y^2=49 $$.

Implicit Differentiation in a Nonlinear Equation

Consider the equation $$x*y + y^3 = 10$$, which defines y implicitly as a function of x.

Implicit Differentiation in an Economic Model

A company’s production is modeled by the implicit relationship $$x*y^2 + \ln(x+y) = 10$$, where $$x$

Implicit Differentiation in an Elliptical Orbit

An orbit of a satellite is modeled by the ellipse $$4*x^2 + 9*y^2 = 36$$. At the point $$\left(1, \f

Implicit Differentiation of a Circle

The equation of a circle is given by $$x^2+y^2=25$$. Answer the following parts:

Implicit Differentiation with Exponential and Trigonometric Mix

Consider the equation $$e^{x*y} + \sin(x) - y = 0$$. Differentiate implicitly with respect to $$x$$

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

Inverse Function Analysis for Exponential Functions

Let $$f(x)=e^{2*x}+1$$ and let g be the inverse function of f. Answer the following parts.

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 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 Logarithmic Context

Let $$f(x)= \ln(x+2) - x$$, and let $$g$$ be its inverse function. Answer the following:

Inverse Function Differentiation in a Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Inverse Function Differentiation in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Investigating the Inverse of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$ with $$x \neq -3$$. Analyze its inverse.

Logarithmic Differentiation of a Variable Exponent Function

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

Rainwater Harvesting System

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

Revenue Model and Inverse Analysis

A company's revenue (in millions) is modeled by $$R(x)=\sqrt{x+4}+3$$, where x represents production

Tangent Line for a Parametric Curve

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

Temperature Modeling and Composite Functions

A weather balloon ascends and the temperature at altitude x (in kilometers) is modeled by $$T(x) = \

Trigonometric Composite Inverse Function Analysis

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

Water Tank Composite Rate Analysis

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

Analyzing a Production Cost Function

A company's cost function for producing goods is given by $$C(x)=x^3-12x^2+40x+100$$, where x repres

Analyzing Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

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 $$s$$

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

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.

Approximating Function Values Using Linearization

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

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 $

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

Compound Interest Rate Change

An investment grows according to $$A(t)=5000e^{0.07t}$$, where t is measured in years. Answer the fo

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Fuel Consumption in a Truck

A truck’s fuel tank is refilled at a constant rate of $$I(t)=10$$ (gallons per minute) while fuel is

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

Implicit Differentiation: Tangent to a Circle

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

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

L'Hôpital's Analysis

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following part

Limits and L'Hôpital's Rule Application

Consider the function $$f(x)=\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following:

Maximizing a Rectangular Enclosure Area

A farmer has 100 m of fencing to enclose a rectangular area. Answer the following:

Population Growth Rate

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

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 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

Tangent Lines in Motion Analysis

A particle's position is given by $$s(t)=t^3 - 6t^2 + 9t + 5$$. Analyze the tangent lines to the gra

Water Filtration Plant Analysis

A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an

Analysis of an Absolute Value Function

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

Analysis of Critical Points for Increasing/Decreasing Intervals

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

Average vs. Instantaneous Profit Rate

A company’s profit is modeled by the function $$P(t)= 0.2*t^3 - 3*t^2 + 10*t$$, where $$t$$ is the t

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

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

Cumulative Angular Displacement Analysis

A rotating wheel has an angular acceleration given by $$\alpha(t)=4-0.6*t$$ (in rad/s²), with an ini

Curve Sketching Using Derivatives

For the function $$f(x)=\frac{x}{x^2+1}$$, use its derivatives to sketch a rough graph of the functi

Epidemic Infection Model

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

Error Approximation using Linearization

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

Expanding Oil Spill - Related Rates

A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

Function Behavior Analysis

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

Graph Interpretation of a Function's Second Derivative

Using the provided graph of the second derivative, analyze the concavity of the original function $$

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.

Logistic Growth Model Analysis

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

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

Modeling Exponential Population Growth

A population is modeled by the function $$P(t)=500*e^{0.2*t}$$, where \(t\) is measured in years.

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Parameter Identification in a Log-Exponential Function

The function $$f(t)= a\,\ln(t+1) + b\,e^{-t}$$ models a decay process with t \(\geq 0\). Given that

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Rate of Change in a Logarithmic Temperature Model

A cooling process is modeled by the temperature function $$T(t)= 100 - 20\,\ln(t+1)$$, where t is me

Rational Function Discontinuities

Consider the rational function $$ R(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}.$$ Answer the following parts:

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

Related Rates: Draining Conical Tank

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

Reservoir Sediment Accumulation

A reservoir accumulates sediment at a rate of $$S_{in}(t)=3*t$$ tonnes/day but also loses sediment v

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

Tangent Line to a Parametric Curve

A curve is defined by the parametric equations $$x(t) = \cos(t)$$ and $$y(t) = \sin(t) + \frac{t}{2}

Taylor Series for $$\frac{1}{1-3*x}$$

Consider the function $$f(x)=\frac{1}{1-3*x}$$. Derive its Taylor series expansion about $$x=0$$, de

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

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

Verifying the Mean Value Theorem

Consider the function $$f(x) = x^3 - 4*x^2 + x + 6$$ defined on the interval [0,4].

Accumulated Change Prediction

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

Accumulated Population Change from a Growth Rate Function

A population changes at a rate given by $$P'(t)= 0.2*t^2 - 1$$ (in thousands per year) for t between

Accumulation Function Analysis

A function $$A(x) = \int_{0}^{x} (e^{-t} + 2)\,dt$$ represents the accumulated amount of a substance

Advanced U-Substitution with a Quadratic Expression

Evaluate the indefinite integral $$\int \frac{2*x}{\sqrt{x^{2}+1}}\,dx$$ using u-substitution.

Antiderivative with Initial Condition

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

Approximating an Exponential Integral via Riemann Sums

Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below

Area Between a Curve and Its Tangent

For the function $$f(x)=x^3-3*x^2+2*x$$, analyze the area between the curve and its tangent line at

Area Between Curves

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

Area Between the Curves: Linear and Quadratic Functions

Consider the curves $$y = 2*t$$ and $$y = t^2$$. Answer the following parts to find the area of th

Area Under an Even Function Using Symmetry

Consider the function $$f(x)=\cos(x)$$ on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

Comparing Riemann Sum Approximations for an Increasing Function

A function f(x) is given in the table below: | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x) | 3

Economic Surplus: Area between Supply and Demand Curves

In an economic model, the demand function is given by $$D(x)=10 - x^2$$ and the supply function by $

Estimating Area Under a Curve Using Riemann Sums

Consider the function $$f(x)$$ whose values on the interval $$[0,10]$$ are given in the table below.

Improper Integral Evaluation

Evaluate the integral $$\int_{1}^{\infty} \frac{1}{t^2}\, dt$$ by answering the following parts.

Integration by Parts: Logarithmic Function

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

Integration Using U-Substitution

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

Investigating Partition Sizes

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

Marginal Cost and Production

A factory's marginal cost function is given by $$MC(x)= 4 - 0.1*x$$ dollars per unit, where $$x$$ is

Midpoint Riemann Sum Estimation

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

Modeling a Car's Journey with a Time-Dependent Velocity

A car's velocity is modeled by $$ v(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 12, & 3 \le t \le 5, \en

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

Rainfall Accumulation and Runoff

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

Recovering Position from Velocity

A particle moves along a straight line with a velocity given by $$v(t)=6*t-2$$ (in m/s) for $$t\in [

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)

Signal Energy through Trigonometric Integration

A signal is described by $$f(t)=3*\sin(2*t)+\cos(2*t)$$. The energy of the signal over one period

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

Transportation Model: Distance and Inversion

A transportation system is modeled by $$f(t)= (t-1)^2+3$$ for $$t \ge 1$$, where \(t\) is time in ho

Trapezoidal Rule Error Estimation

Given the function $$f(x)=\ln(x)$$ on the interval $$[1,4]$$, answer the following:

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

Work Done by a Variable Force

A variable force given by $$F(x)= 3*x^2$$ (in Newtons) acts on an object as it moves along a straigh

Bacterial Growth with Time-Dependent Growth Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=(r_0+r_1*t)P$$, whe

Car Engine Temperature Dynamics

The temperature $$T(t)$$ (in °C) of a car engine is modeled by the differential equation $$\frac{dT}

Chemical Reaction and Separable Differential Equation

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

Chemical Reaction Rate

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

Combined Differential Equations and Function Analysis

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

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Differential Equation Involving Logarithms

Consider the differential equation $$\frac{dy}{dx} = (y-1)*\ln|y-1|$$ with the initial condition $$y

FRQ 10: Cooling of a Metal Rod

A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th

FRQ 16: Harvesting in a Predator-Prey Model

A prey population $$P(t)$$ in a marine ecosystem is modeled by the differential equation $$\frac{dP}

FRQ 18: Enzyme Reaction Rates

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

Investment Growth Model

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

Logistic Equation with Harvesting

A fish population in a lake follows a logistic growth model with the addition of a constant harvesti

Logistic Growth Population Model

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

Mixing Problem with Differential Equations

A tank initially contains $$S(0)=S_0$$ grams of salt dissolved in a volume $$V$$ liters of water. Br

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

Mixing Problem: Salt Water Tank

A tank initially contains $$1000$$ liters of pure water with $$50$$ kg of salt dissolved in it. Brin

Mixing Tank with Variable Inflow

A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

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

Phase-Plane Analysis of a Nonlinear Differential Equation

Consider the logistic differential equation $$\frac{dy}{dt} = y(1-y)$$, which models a normalized po

Reservoir Contaminant Dilution

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

Separable Differential Equation and Maclaurin Series Approximation

Consider the differential equation $$\frac{dy}{dx} = e^{x} * \sin(y)$$ with the initial condition $$

Arc Length of the Logarithmic Curve

For the function $$f(x)=\ln(x)$$ defined on the interval $$[1,e]$$, determine the arc length of the

Average Temperature Analysis

A weather station records the temperature throughout a day. The temperature, in degrees Celsius, is

Average Temperature of a Day

In a certain city, the temperature during the day is modeled by a continuous function $$T(t)$$ for $

Average Value and Critical Points of a Trigonometric Function

Consider the function $$f(x)=\sin(2*x)+\cos(2*x)$$ on the interval $$\left[0,\frac{\pi}{2}\right]$$.

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

Bacterial Decay Modeled by a Geometric Series

A bacterial culture is treated with an antibiotic that reduces the bacterial population by 20% each

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

Economic Analysis: Consumer and Producer Surplus

In a market analysis, the demand and supply for a product are modeled by the equations: Demand: $$D(

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

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

Movement Under Variable Acceleration

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

Particle on a Line with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$

River Cross Section Area

The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$

Total Change in Temperature Over Time (Improper Integral)

An object cools according to the function $$\Delta T(t) = e^{-0.5*t}$$, where $$t\ge 0$$ is time in

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$

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Volume Using the Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ with $$x\ge0$$. This region is rotated about th

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=10 - 0.5*x$$ newtons for $$0 \le x \le 12$$

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 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 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

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

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 Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Area Enclosed by a Polar Curve: Lemniscate

The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.

Circular Motion in Vector-Valued Form

A particle moves along a circle of radius 5 with its position given by $$ r(t)=\langle 5*\cos(t),\;

Component-Wise Integration of a Vector-Valued Function

Given the acceleration vector $$\textbf{a}(t)= \langle 3\cos(t), -3\sin(t) \rangle$$, answer the fol

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

Designing a Parametric Curve for a Cardioid

A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.

Designing a Race Track with Parametric Equations

An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:

Exponential Growth in Parametric Representation

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

Inner Loop of a Limaçon in Polar Coordinates

The polar curve given by \(r=1+2\cos(\theta)\) forms a limaçon with an inner loop. Answer the follow

Parametric Equations and Tangent Lines

A curve is defined parametrically by $$x(t)=t^3-3t$$ and $$y(t)=t^2+2$$, where $$t$$ is a real numbe

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 Motion in the Plane

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

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Tangent Line to a 3D Vector-Valued Curve

Let $$\textbf{r}(t)= \langle t^2, \sin(t), \ln(t+1) \rangle$$ for $$t \in [0,\pi]$$. Answer the foll

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Tangent Lines to Polar Curves

Consider the polar curve $$r(\theta)= 3\sin(\theta)$$. Analyze the tangent line at a point correspo

Tangents and Normals of a Parametric Curve

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

Vector-Valued Function of Particle Trajectory

A particle in space follows the vector function $$\mathbf{r}(t)=\langle t, t^2, \sqrt{t} \rangle$$ f

Vector-Valued Functions in Motion

A particle's position is given by the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*

Work Done Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa