AP Calculus AB FRQ Room

Algebraic Manipulation in Limit Calculations

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

Analyzing a Removable Discontinuity

Consider the function $$f(x) = \frac{x^2 - 4}{x - 2}$$ for $$x \neq 2$$. Notice that f is not define

Analyzing Continuity in a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2, & x < 2 \\ 4, & x = 2 \\ 3x - 2, & x > 2 \end{cases}

Analyzing Limits from Experimental Data (Table)

The table below shows measured values of a function $$f(x)$$ near $$x = 1$$. | x | f(x) | |-----

Applying the Squeeze Theorem with Trigonometric Function

Consider the function $$ f(x)= x^2 \sin(1/x) $$ for $$x\ne0$$, with $$f(0)=0$$. Use the Squeeze Theo

Asymptotic Behavior of a Logarithmic Function

Consider the function $$w(x)=\frac{\ln(x+e)}{x}$$ for $$x>0$$. Analyze its behavior as $$x \to \inft

Complex Rational Function with Removable and Essential Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^3-8}{x^2-4} & x\neq -2,2 \\ 4 & x=2 \end{cases}$$

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

Composite Limit Problem Involving Absolute Value

Consider the function $$f(x)=\frac{|x-3|}{x-3}$$ for \(x\neq3\). Answer the following:

Continuity in a Piecewise Function with Square Root and Rational Expression

Consider the function $$f(x)=\begin{cases} \sqrt{x+6}-2 & x<-2 \\ \frac{(x+2)^2}{x+2} & x>-2 \\ 0 &

Determining Horizontal Asymptotes for Rational Functions

Given the rational function $$R(x)= \frac{2*x^3+ x^2 - x}{x^3 - 4}$$, answer the following:

Direct Substitution in a Polynomial Function

Consider the polynomial function $$f(x)=2*x^2 - 3*x + 5$$. Answer the following parts concerning lim

Estimating Limits from a Data Table

A function f(x) is studied near x = 3. The table below shows selected values of f(x):

Evaluating Limits Involving Square Roots

Consider the function $$f(x)= \frac{\sqrt{x+4}-2}{x}$$. Answer the following:

Exponential Function Limits

Consider the function $$f(x) = \frac{e^x - 1}{x}$$ for $$x \neq 0$$, with the definition $$f(0) = 1$

Graph Analysis of a Discontinuous Function

A function f has been graphed below and exhibits a discontinuity at x = 1. Use the graph to answer t

Graphical Interpretation of Limits and Continuity

The graph below represents a function $$f(x)$$ defined by two linear pieces with a potential discont

Identifying Discontinuities in a Rational Function

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

Limit Involving an Exponential Function

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

Limits Involving Trigonometric Functions in Particle Motion

A particle moves along a line with velocity given by $$v(t)= \frac{\sin(2*t)}{t}$$ for $$t > 0$$. An

Limits of a Composite Particle Motion Function

A particle moves along a line with velocity function $$v(t)= \frac{\sqrt{t+5}-\sqrt{5}}{t}$$ for $$t

Modeling Population Growth with a Limit

A population P(t) is modeled by the function $$P(t) = \frac{5000}{1 + 40e^{-0.5*t}}$$ for t ≥ 0. Ans

One-Sided Limits and Continuity of a Piecewise Function

Consider the piecewise function $$w(x)= \begin{cases} \frac{e^{x}-1}{x} & \text{if } x<0, \\ \frac{\

One-Sided Limits and Vertical Asymptotes

Consider the function $$ f(x)= \frac{1}{x-4} $$.

One-Sided Limits in a Function Involving Logarithms

Define the function $$f(x)=\frac{e^{x}-1}{\ln(1+x)}$$ for $$x \neq 0$$ with a continuous extension g

Particle Motion with Vertical Asymptote in Velocity

A particle moves along a number line with velocity function $$v(t)= \frac{3*t}{t-1}$$ for $$t > 1$$.

Piecewise Function Continuity Analysis

The function f is defined by $$ f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k, & x

Real-World Analysis of Vehicle Deceleration Using Data

A study measures the speed of a car (in m/s) as it approaches a stop sign. The recorded speeds at di

Related Rates: Shadow Length of a Moving Object

A 1.8 m tall person is walking away from a 3 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the

Removing a Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{x^2 - 1}{x - 1} & x \neq 1 \\ 3 & x=1 \end{cases}$$

Removing Discontinuities

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

Robotic Arm and Limit Behavior

A robotic arm moving along a linear axis has a velocity function given by $$v(t)= \frac{t^3-8}{t-2}$

Analyzing Rate of Change in Economics

The cost function for producing $$x$$ units of product (in dollars) is given by $$C(x)= 0.5*x^2 - 8*

Approximating Tangent Line Slopes

A curve is given by the function $$f(x)= \ln(x) + e^{-x}$$, modeling a physical measurement obtained

Car's Position and Velocity

A car’s position is modeled by \(s(t)=t^3 - 6*t^2 + 9*t\), where \(s\) is in meters and \(t\) is in

Comparing Average vs. Instantaneous Rates

Consider the function $$f(x)= x^3 - 2*x + 1$$. Experimental data for the function is provided in the

Cost Optimization and Marginal Analysis

A manufacturer’s cost function is given by $$C(q)=500+4*q+0.02*q^2$$, where $$q$$ is the quantity pr

Derivatives on an Ellipse

The ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ represents a race track. Answer the follo

Difference Quotient for a Cubic Function

Let \(f(x)=x^3\). Using the difference quotient, answer the following parts.

Differentiability of a Piecewise Function

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

Differentiation of Exponential Functions

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

Implicit Differentiation of a Circle

Consider the equation $$x^2 + y^2 = 25$$ representing a circle with radius 5. Answer the following q

Instantaneous and Average Velocity

A particle's position is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$s(t)$$ is in meters and $$t$$ is

Instantaneous Rate and Maximum Acceleration

An object’s position is given by $$s(t)=t^4-4t^3+2t^2$$ (in meters), where t is in seconds. Answer t

Inverse Function Analysis: Cubic with Linear Term

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

Inverse Function Analysis: Hyperbolic-Type Function

Consider the function $$f(x)=\sqrt{x^2+1}$$ defined for $$x\geq0$$.

Inverse Function Analysis: Quadratic Function

Consider the function $$f(x)=x^2$$ restricted to $$x\geq0$$.

Inverse Function Analysis: Quadratic Transformation

Consider the function $$f(x)=x^2+2*x+2$$ with the domain restricted to $$x\geq -1$$ so that f is one

Marginal Profit Calculation

A company’s profit (in thousands of dollars) is given by $$P(x)= -2*x^2 + 50*x - 100$$, where $$x$$

Medication Infusion with Clearance

A patient receives medication via an IV at a rate of $$f(t)=5*e^{-0.1*t}$$ mg/min, while the body cl

Physical Motion with Variable Speed

A car's velocity is given by $$v(t)= 2*t^2 - 3*t + 1$$, where $$t$$ is in seconds.

Position Function from a Logarithmic Model

A particle’s position in meters is modeled by $$s(t)= \ln(t+1)$$ for $$t \geq 0$$ seconds.

Product Rule Application in Economics

A company's cost function for producing $$x$$ units is given by $$C(x)= (3*x+2)*(x^2+5)$$ (cost in d

Related Rates: Balloon Surface Area Change

A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu

Relating Average and Instantaneous Velocity in a Particle's Motion

A particle’s position is modeled by $$s(t)=\frac{4}{t+1}$$, where $$s(t)$$ is in meters and $$t$$ is

Secant and Tangent Line Approximation in a Real-World Model

A temperature sensor records data modeled by $$g(t)= 0.5*t^2 - 3*t + 2$$, where $$t$$ is in seconds

Secant and Tangent Lines

Consider the function $$f(x)= x^2$$. Use graphical and algebraic methods to examine the behavior of

Tangent Line to a Cubic Function

The function $$f(x) = x^3 - 6x^2 + 9x + 1$$ models the height (in meters) of a roller coaster at pos

Tangent to an Implicit Curve

Consider the curve defined implicitly by \(x^2 + y^2 = 25\). Answer the following parts.

Using Derivative Rules on a Trigonometric Function

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

Using the Limit Definition to Derive the Derivative

Let $$f(x)= 3*x^2 - 2*x$$.

Chain Rule in an Economic Model

In an economic model, the cost function for producing a good is given by $$C(x)=(3*x+1)^5$$, where $

Chain Rule in an Economic Model

In a manufacturing process, the cost function is given by $$C(q)= (\ln(1+3*q))^2$$, where $$q$$ is t

Chain Rule in Particle Motion

A particle's position is given by $$s(t)=\cos(5*t^2)$$ in meters, where time $$t$$ is measured in se

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Chain Rule with Exponential and Polynomial Functions

Let $$h(x)= e^{3*x^2+2*x}$$ represent a function combining exponential and polynomial elements.

Combining Composite and Implicit Differentiation

Consider the equation $$e^{x*y}+x^2-y^2=7$$.

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Composite Log-Exponential Function Analysis

A function is defined by $$f(x)=\ln\left(e^(2*x^2) + 5\right)$$. This function is composed of an exp

Designing a Tapered Tower

A tower has a circular cross-section where the relationship between the radius r (in meters) and the

Differentiation Under Implicit Constraints in Physics

A particle moves along a path defined by the equation $$\sin(x*y)=x-y$$. This equation implicitly de

Implicit and Inverse Function Differentiation Combined

Suppose that $$x$$ and $$y$$ are related by the equation $$x^2+y^2-\sin(x*y)=4$$. Answer the followi

Implicit Differentiation in a Biochemical Reaction

Consider a biochemical reaction modeled by the equation $$x*e^{y} + y*e^{x} = 10$$, where $$x$$ and

Implicit Differentiation in Logarithmic Functions

Consider the equation $$\ln(x)+\ln(y)=1$$. Answer the following:

Implicit Differentiation Involving Sine

Consider the equation $$\sin(x*y)+x-y=0$$.

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x} + y = x + e^{y}$$ which relates $$x$$ and $$y$$ via exponential functi

Implicit Differentiation with Mixed Functions

Consider the relation $$x\cos(y)+y^3=4*x+2*y$$.

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ which is one-to-one on its domain. Its inverse function is denoted by $$g(x)$$.

Inverse Function Differentiation in a Biological Growth Curve

A biological measurement is modeled by the function $$f(t)= \frac{4*t}{t+2}$$, which is one-to-one o

Inverse Function Differentiation in Temperature Conversion

In a temperature conversion model, the function $$f(T)=\frac{9}{5}*T+32$$ converts Celsius temperatu

Inverse Trigonometric and Logarithmic Function Composition

Let $$y=\arctan(\ln(x))$$. Answer the following:

Inverse Trigonometric Differentiation in a Geometry Problem

Consider a scenario in which an angle $$\theta$$ in a geometric configuration is given by $$\theta=\

Related Rates in a Circular Colony

A circular microorganism colony expands such that its radius at time $$t$$ (in seconds) is given by

Related Rates: Shadow Length

A 1.8 m tall person is walking away from a street lamp that is 5 m tall at a speed of 1.2 m/s. Using

Second Derivative via Implicit Differentiation

Given the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$, find the second derivative $$\frac{d^2y}{dx^2}$

Analyzing a Nonlinear Rate of Revenue Change

A company's revenue in thousands of dollars is modeled by the function $$R(x)=100\ln(x+1) + 0.5x$$,

Analyzing Experimental Motion Data

The table below shows the position (in meters) of a moving object at various times (in seconds):

Analyzing Position Data with Table Values

A moving object’s position, given by $$x(t)$$ in meters, is recorded in the table below. Use the dat

Bacterial Growth Analysis

The number of bacteria in a culture is given by $$P(t)=500e^{0.2*t}$$, where $$t$$ is measured in ho

Balloon Inflation Analysis

A spherical balloon inflates such that its volume increases at a constant rate of 10 cubic inches pe

Balloon Inflation Related Rates

A spherical balloon is being inflated, and its volume is increasing at a constant rate of $$12$$ cub

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is given by $$C(t)=100e^{-0.05*t}$$ mg/L, wh

Chemical Reaction Rate Analysis

A chemical reaction follows the concentration model $$c(t)=\frac{100}{1+5e^{-0.3t}}$$, where c is in

Chemistry Reaction Rate

The concentration of a chemical in a reaction is given by $$C(t)= \frac{100}{1+5*e^{-0.3*t}}$$ (in m

Coffee Cooling Analysis Revisited

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where $$t$$ is ti

Cooling Coffee Temperature Analysis

A cup of coffee cools according to the function $$T(t)=80+20e^{-0.3t}$$ (in °F), where $$t$$ is meas

Critical Points and Concavity Analysis

Consider the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ modeling the position of an

Depth of a Well: Related Rates Problem

A bucket is being lowered into a well, and its depth is modeled by $$d(t)= \sqrt{t + 4}$$, where $$t

Designing a Flower Bed: Optimal Shape

A landscape designer wants to create a rectangular flower bed with a fixed area of 200 square meters

Differentiability of a Piecewise Function

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

Economics and Marginal Analysis: Revenue and Cost Differentiation

A company’s revenue and cost functions are given by $$R(x)=100*x - 0.5*x^2$$ and $$C(x)=20*x + 50$$,

Estimating Function Change Using Differentials

Let $$f(x)=x^{1/3}$$. Use differentials to approximate the change in $$f(x)$$ when $$x$$ increases f

Expanding Balloon: Related Rates with a Sphere

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

Expanding Circular Oil Spill

An oil spill on water forms a circular shape. The area of the spill is increasing at a rate of $$200

Expanding Oil Spill

The area of an oil spill is modeled by $$A(t)=\pi (2+t)^2$$ square kilometers, where $$t$$ is in hou

FRQ 1: Vessel Cross‐Section Analysis

A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^

FRQ 17: Water Heater Temperature Change

The temperature of water in a heater is modeled by $$T(t) = 20 + 80e^{-0.05*t}$$, where t is in minu

FRQ 20: Market Demand Analysis

In an economic market, the demand D (in thousands of units) and the price P (in dollars) satisfy the

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

Linear Approximation of Natural Logarithm

Estimate $$\ln(1.05)$$ using linear approximation for the function $$f(x)=\ln(x)$$ at $$a=1$$.

Linearization and Differentials Approximation

A function $$f(x)= x^3$$ models the volume (in cubic centimeters) of liquid in a container as a func

Linearization in Medicine Dosage

A drug’s concentration in the bloodstream is modeled by $$C(t)=\frac{5}{1+e^{-t}}$$, where $$t$$ is

Marginal Analysis in Economics

A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the

Marginal Profit Analysis

A company's profit in thousands of dollars is given by $$P(x)= -0.5*x^2+20*x-50$$, where $$x$$ (in h

Maximizing the Area of an Enclosure with Limited Fencing

A farmer has 240 meters of fencing available to enclose a rectangular field that borders a river (th

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Motion along a Straight Line: Changing Direction

A runner's position is modeled by $$s(t)= t^4 - 8*t^2 + 16$$, where $$s(t)$$ is in meters and $$t$$

Optimization in Packaging

An open-top box with a square base is to be constructed so that its volume is fixed at $$1000\;cm^3$

Particle Motion Analysis

A particle moves along a straight line with displacement given by $$s(t)=t^3-6t^2+9t+2$$ for $$0\le

Population Growth Model and Asymptotic Limits

A population is modeled by $$P(t) = \frac{5000e^{0.1t}}{1 + e^{0.1t}}$$, where $$P(t)$$ is the popul

Population Growth Rate Analysis

A town's population is modeled by the exponential function $$P(t) = 500e^{0.03t}$$, where $$t$$ is i

Population Growth with Changing Rates

A population is modeled by the piecewise function $$P(t)=\begin{cases}50e^{0.1t}&t<10\\500e^{0.05t}&

Rate of Change of Temperature

The temperature of a cooling liquid is modeled by $$T(t)= 100*e^{-0.5*t} + 20$$, where $$t$$ is in m

Related Rates in a Conical Tank

Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.

Related Rates: Inflating Balloon

A spherical balloon is being inflated such that its volume increases at a rate of $$15\;cm^3/s$$. Th

Related Rates: Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$20$$ cu

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

Vehicle Deceleration Analysis

A vehicle’s position is given by $$s(t)=100t-5t^2$$ where $$s(t)$$ is in meters and $$t$$ in seconds

Vehicle Position and Acceleration

A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where

Analyzing Acceleration Functions Using Derivatives

For the position function $$s(t)= t^3 - 6*t^2 + 9*t + 1$$ (in meters), where \( t \) is in seconds,

Application of Rolle's Theorem to a Trigonometric Function

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

Area Bounded by $$\sin(x)$$ and $$\cos(x)$$

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$

Asymptotic Behavior in an Exponential Decay Model

Consider the model $$f(t)= 100*e^{-0.3*t}$$ representing a decaying substance over time. Answer the

Biological Growth and the Mean Value Theorem

In a bacterial culture, the population is modeled by $$P(t)= 4*t^2 + 3*t + 7$$ for $$t$$ in hours on

Chemical Reaction Rate and Exponential Decay

In a chemical reaction, the concentration of a reactant declines according to $$C(t)= C_0* e^{-k*t}$

Drag Force and Rate of Change from Experimental Data

Drag force acting on an object was measured at various velocities. The table below presents the expe

Estimating Total Revenue via Riemann Sums

A company’s marginal revenue (in thousand dollars per unit) is measured at various levels of units s

Exponential Bacterial Growth

A bacterial culture grows according to $$P(t)= P_0 * e^{k*t}$$, where $$t$$ is in hours. The culture

FRQ 1: Car's Motion and the Mean Value Theorem

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 5$$ (in meters) fo

FRQ 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If t

FRQ 19: Analysis of an Exponential-Polynomial Function

Consider the function $$f(x)= e^{-x}*x^2$$ defined for $$x \ge 0$$.

Inverse Analysis of a Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Inverse Analysis of a Logarithm-Exponential Hybrid Function

Consider the function $$f(x)=\ln(x+2)+e^(x)$$ defined for $$x>-2$$. Address the following regarding

Inverse Analysis: Logarithmic Ratio Function in Financial Context

Consider the function $$f(x)=\ln\left(\frac{x+4}{x+1}\right)$$ with domain $$x > -1$$. This function

Maximizing Revenue

A company’s revenue (in hundreds of dollars) is modeled by the function $$R(x)= 80*x - 2*x^3$$, wher

Motion Analysis via Derivatives

A particle moves along a straight line with its position described by $$s(t)= t^3 - 6*t^2 + 9*t + 5$

Motion Analysis: A Runner's Performance

A runner’s distance (in meters) is recorded at several time intervals during a race. Analyze the run

Oil Spill Cleanup

In an oil spill scenario, oil continues to enter an affected area while cleanup efforts remove oil.

Optimization of a Rectangle Inscribed in a Semicircle

A rectangle is inscribed in a semicircle of radius 5 (with the base along the diameter). The top cor

Pharmaceutical Drug Delivery

A drug is administered intravenously using an infusion device. The drug infusion rate is given by $$

Predicting Fuel Efficiency in Transportation

A vehicle’s performance was studied by recording the miles traveled and the corresponding fuel consu

Slope Analysis for Parametric Equations

A curve is defined parametrically by $$x(t)= t^2$$ and $$y(t)= t^3 - 3*t$$ for $$t$$ in the interval

Accumulated Bacteria Growth

A laboratory observes a bacterial colony whose rate of growth (in bacteria per hour) is modeled by t

Application of the Fundamental Theorem of Calculus

A particle moves along a straight line with an instantaneous velocity given by $$v(t)=3*t^2+2*t$$ (i

Area Between Curves

Consider the curves defined by $$f(x)=x^2$$ and $$g(x)=2*x$$. The region enclosed by these curves is

Area Between Two Curves

Consider the functions $$f(x)=x^2$$ and $$g(x)=2*x+3$$. They intersect at two points. Using the grap

Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined over the interval $$[1,7]$$ and its values are provided in the table

Area Under a Polynomial Curve

Consider the function $$f(x)=2*x^2-3*x+1$$ defined on the interval $$[0,4]$$. Answer the following p

Average Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Computing Accumulated Volume from a Filling Rate Function

A small pond is being filled at a rate given by $$r(t)=2*t + 3$$ (in $$m^3/hr$$), where $$t$$ is in

Cost Accumulation from Marginal Cost Function

A company's marginal cost (in dollars per unit) is given by $$MC(x)=0.2*x+50$$, where $$x$$ represen

Economic Accumulation of Revenue

The marginal revenue (MR) for a company is given by $$MR(x)=50*e^{-0.1*x}$$ (in dollars per item), w

Economic Analysis: Consumer Surplus

In a competitive market, the demand function is given by $$D(p)=100-2*p$$ and the supply function is

Estimating Work Done Using Riemann Sums

In physics, the work done by a variable force is given by $$W=\int F(x)\,dx$$. A force sensor record

Evaluating the Definite Integral of a Power Function

Let $$f(t)= 3*t^{1/3}$$. Evaluate the definite integral $$\int_{27}^{64} 3*t^{1/3}\,dt$$. Answer th

FRQ12: Inverse Analysis of a Temperature Accumulation Function

The cumulative temperature above freezing over the morning is modeled by $$ T(t)=\int_{0}^{t} (0.8*t

FRQ15: Inverse Analysis of a Quadratic Accumulation Function

Consider the function $$ Q(x)=\int_{0}^{x} (4*t+1)\,dt $$. Answer the following parts.

FRQ17: Inverse Analysis of a Biologically Modeled Accumulation Function

In a biological study, the net concentration of a chemical is modeled by $$ B(t)=\int_{0}^{t} (0.6*t

FRQ18: Inverse Analysis of a Square Root Accumulation Function

Consider the function $$ R(x)=\int_{1}^{x} \sqrt{t+1}\,dt $$. Answer the following parts.

Fuel Consumption for a Rocket Launch

During a rocket launch, fuel is consumed at a rate $$F_{cons}(t)=50-3t$$ kg/s while additional fuel

Integration of Exponential Functions with Shifts

Evaluate the integral $$\int_{0}^{2} e^{2*(x-1)}\,dx$$ using an appropriate substitution.

Modeling Savings with a Geometric Sequence

A person makes annual deposits into a savings account such that the first deposit is $100 and each s

Net Change in Population Growth

A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$

Net Surplus Calculation

A consumer's satisfaction is given by $$S(x)=100-4*x^2$$ and the marginal cost is given by $$C(x)=30

Particle Motion with Variable Acceleration

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

Piecewise-Defined Function and Discontinuities

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

Rainwater Collection in a Reservoir

Rainwater falls into a reservoir at a rate given by $$R(t)= 12e^{-0.5t}$$ L/min while evaporation re

Reservoir Accumulation Problem

A reservoir is filled at a rate given by $$R(t)=\frac{8}{1+e^{-0.5*t}}$$ cubic meters per minute, wh

Riemann Sum Approximation of f(x) = 4 - x^2

Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Temperature Change in a Room

The rate of change of the temperature in a room is given by $$\frac{dT}{dt}=0.5*t+1$$, where $$T$$ i

Temperature Cooling: An Initial Value Problem

An object cools according to the differential equation $$T'(t)=-0.2\,(T(t)-20)$$, where $$T(t)$$ is

The Accumulation Function for a Linear Rate Model

Consider the accumulation function defined by $$A(t)= \int_0^t (3*t' + 2)\,dt'$$, where $$t'$$ is a

Total Distance Traveled from Velocity Data

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for t in [0

Total Fuel Used Over a Trip

A car consumes fuel at a rate modeled by $$r(t) = 0.2*t + 1.5$$ (in gallons per hour) during a long

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=3*x^{2} - 2*x + 1$$ (in newtons) fo

Applying the SIPPY Method to $$dy/dx = \frac{4x}{y}$$

Solve the differential equation $$\frac{dy}{dx}=\frac{4x}{y}$$ with the initial condition $$y(0)=5$$

Area Under a Differential Equation Curve

Consider the differential equation $$\frac{dy}{dx} = y(1-y)$$, with a particular solution given by $

Bacterial Growth under Logistic Model

A bacterial culture grows according to the logistic differential equation $$\frac{dB}{dt}=rB\left(1-

Bacterial Growth with Constant Removal

A bacterial colony (in thousands) grows according to the differential equation $$\frac{dP}{dt}=0.4P-

Chemical Reactor Mixing

In a continuously stirred chemical reactor, a reactant is added at a rate that results in an inflow

Cooling with a Time-Dependent Coefficient

A substance cools according to $$\frac{dT}{dt} = -k(t)(T-25)$$ where the cooling coefficient is give

Environmental Pollution Model

Pollutant concentration in a lake is modeled by the differential equation $$\frac{dC}{dt}=\frac{R}{V

Exponential Growth: Separable Equation

Solve the differential equation $$\frac{dy}{dx} = \frac{y}{1+x^2}$$ with the initial condition $$y(0

Falling Object with Air Resistance

A falling object with mass $$m=70\,kg$$ is subject to gravity and a resistive force proportional to

Implicit Differentiation with Trigonometric Functions

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

Logistic Growth Model Analysis

A population $$y(t)$$ grows according to the logistic differential equation $$\frac{dy}{dt} = k * y

Logistic Population Growth

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\left(1

Mixing Problem: Salt in a Tank

A 100-liter tank initially contains 50 grams of salt. Brine with a salt concentration of $$0.5$$ gra

Mixing Tank Problem

A tank initially contains $$100$$ liters of pure water. A salt solution with a concentration of $$0.

Modeling Continuous Compound Interest

An account accrues interest continuously according to the differential equation $$\frac{dA}{dt}=rA$$

Newton's Law of Cooling with Variable Ambient Temperature

An object cools according to Newton's Law of Cooling, but the ambient temperature is not constant. I

Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=(1-y^2)\cos(x)$$ with the initial condition $$y(0

Population Dynamics with Harvesting

A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-kN$$. If the

Radioactive Decay

A radioactive substance decays according to $$\frac{dy}{dt} = -0.05\,y$$ with an initial mass of $$y

Radioactive Decay and Half-Life

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda N$$.

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

RC Circuit Charging

In an RC circuit, the charge $$Q(t)$$ on the capacitor satisfies the differential equation $$\frac{d

RC Circuit Discharge

In an RC circuit, the voltage across a capacitor discharging through a resistor follows $$\frac{dV}{

Related Rates: Conical Tank Filling

Water is pumped into a conical tank at a rate of $$3$$ m$^3$/min. The tank has a height of $$4$$ m a

Related Rates: Shadow Length

A 2 m tall lamp post casts a shadow of a 1.8 m tall person who is walking away from the lamp post at

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

Sketching Solution Curves on a Slope Field

Consider the differential equation $$\frac{dy}{dx}=x-y$$. A slope field for this equation is provide

Slope Field Analysis for $$\frac{dy}{dx}=\frac{y}{x}$$

Consider the differential equation $$\frac{dy}{dx}= \frac{y}{x}$$. A slope field for this equation i

Slope Field Analysis for $$dy/dx = x$$

Consider the differential equation $$dy/dx = x$$. A slope field representing this equation is provid

Temperature Regulation in a Greenhouse

The temperature $$T$$ (in °F) inside a greenhouse is recorded over time (in hours) as shown. The war

Accumulated Rainfall Calculation

During a 12-hour storm, the rainfall rate is modeled by $$r(t)=3+\sin(t)$$ (in mm/hr) for $$0 \le t

Analysis of a Rational Function's Average Value

Consider the rational function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$[-1,1]$$. Analyz

Area Between \(\ln(x+1)\) and \(\sqrt{x}\)

Consider the functions $$f(x)=\ln(x+1)$$ and $$g(x)=\sqrt{x}$$ over the interval $$[0,3]$$.

Area Between a Parabola and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll

Arithmetic Savings Account

A person makes monthly deposits into a savings account such that the amount deposited each month for

Average Concentration Calculation

In a continuous stirred-tank reactor (CSTR), the concentration of a chemical is given by $$c(t)=5+3*

Calculation of Consumer Surplus

The demand function for a product is given by $$p(x)=20-0.5*x$$, where $$p$$ is the price (in dollar

Cell Phone Battery Consumption

A cell phone’s battery life degrades over time such that the effective battery life each month forms

Center of Mass of a Lamina with Variable Density

A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x

Charity Donations Over Time

A charity receives monthly donations that form an arithmetic sequence. The first donation is $$50$$

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

Electrical Charge Calculation

The current in an electrical circuit is given by $$I(t)=6*e^{-0.5*t} - 3*e^{-t}$$ (in amperes) for $

Estimating Instantaneous Velocity from Position Data

A car's position along a straight road is recorded over a 10-second interval as shown in the table b

Finding the Area Between Two Curves

Let the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ be given. Find the area of the region bounded by t

Funnel Design: Volume by Cross Sections

A funnel is designed by rotating the region bounded by the curve $$y=4-x^2$$ for -2 ≤ x ≤ 2 about th

Ice Rink Design: Volume and Area

An ice rink is designed with a cross-sectional profile given by $$y=4-x^2$$ (with y=0 as the base).

Kinematics with Variable Acceleration

A particle is moving along a straight path with an acceleration given by $$a(t)=10-6*t$$ (in m/s²) f

Loaf Volume Calculation: Rotated Region

Consider the region bounded by $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for 0 ≤ x ≤ 4. This region is ro

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

Net Change and Total Distance in Particle Motion

A particle has acceleration $$a(t)=12-8*t$$ (in $$m/s^2$$) for $$t \ge 0$$, with initial velocity $$

Net Change in Biological Population

A species' population changes at a rate given by $$P'(t)=0.5e^{-0.2*t}-0.05$$ (in thousands per year

Population Growth and Average Rate

A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in

Population Model Using Exponential Function

A bacteria population is modeled by $$P(t)=100*e^{0.03*t} - 20$$, where $$t$$ is measured in hours.

Radioactive Decay Accumulation

A radioactive substance decays at a rate given by $$r(t)= C*e^{-k*t}$$ grams per day, where $$C$$ an

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

River Discharge Analysis

The flow rate of a river is modeled by $$Q(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$ (in cubic met

Tank Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Viral Video Views

A viral video’s daily views form a geometric sequence. On day 1, the video is viewed 1000 times, and

Volume of a Solid of Revolution Rotated about a Line

Consider the region bounded by $$y=x^2$$ and $$y=x$$ for $$x\in [0,1]$$. This region is rotated abou

Volume of a Solid of Revolution Using the Washer Method

The region bounded by the curves $$x=\sqrt{y}$$ and $$x=\frac{y}{2}$$ for $$y\in[0,4]$$ is revolved

Volume of Solid of Revolution: Bottle Design

A region under the curve $$f(x)=\sqrt{x}$$ on the interval 0 ≤ x ≤ 9 is rotated about the x-axis to

Volume with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

Work Done by a Variable Force

A variable force $$F(x)$$ (in Newtons) acts on an object as it moves along a straight line from $$x=