AP Calculus AB FRQ Room

Advanced Analysis of an Oscillatory Function

Consider the function $$ f(x)= \begin{cases} \sin(1/x), & x\ne0 \\ 0, & x=0 \end{cases} $$.

Algebraic Simplification and Limit Evaluation of a Log-Exponential Function

Consider the function $$z(x)=\ln\left(\frac{e^{3*x}+e^{2*x}}{e^{3*x}-e^{2*x}}\right)$$ for $$x \neq

Analyzing a Discontinuous Function with a Sequence Component

The function is given by $$f(x) = \frac{\sin(\pi x)}{\pi (x - 1)}$$ for $$x \neq 1$$ (with f(1) unde

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

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 Analysis of a Rational Function

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

Capstone Problem: Continuity and Discontinuity in a Compound Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-1}{x-1} & x<2 \\ \frac{x^2-4}{x-2} & x\ge2 \end

Combined Limit Analysis of a Piecewise Function

Consider the function $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & \text{if } x \neq 1, \\ c & \text{if

Continuity and Limit Comparison for Two Particle Paths

Two particles, A and B, travel along the same line. Their position functions are given by $$s_A(t)=

Continuity of a Composite Log-Exponential Function

Let $$f(x)=\begin{cases} \frac{\ln(1+e^{x})-\ln(2)}{x} & \text{if } x\neq 0, \\ C & \text{if } x=0.

Discontinuity in Acceleration Function and Integration

A particle’s acceleration is defined by the piecewise function $$a(t)= \begin{cases} \frac{1-t}{t-2}

Implicit Differentiation in an Exponential Equation

Consider the function defined implicitly by $$e^{x*y} + x - y = 0$$. Answer the following:

Intermediate Value Theorem Application

Consider the polynomial function $$f(x)=x^3-6*x^2+9*x+1$$ on the closed interval [0, 4].

Intermediate Value Theorem in Particle Motion

Consider a particle with position function $$s(t)= t^3 - 7*t+3$$. According to the Intermediate Valu

Inverse Function Analysis and Derivative

Let $$f(x)= x^3+2$$, defined for all real numbers.

Limit Analysis in Population Modeling

A population is modeled by the function $$P(t)= \frac{1000*t}{t+5}$$ where $$t \geq 0$$ (in years).

Limits at Infinity for Non-Rational Functions

Consider the function $$ h(x)=\frac{2*x+3}{\sqrt{4*x^2+7}} $$.

Limits Involving Radical Functions

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.

Logarithm Transformation and Limit Evaluation

Consider the function $$Y(x)=\ln\left(\frac{e^{2*x}+5}{e^{2*x}-5}\right)$$. Investigate the limits a

Mixed Function with Jump Discontinuity at Zero

Consider the function $$f(x)=\begin{cases} 1+x & x<0\\ 2 & x=0\\ \frac{\sin(x)}{x}+1 & x>0 \end{case

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 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

One-Sided Limits of a Piecewise Function

Let $$f(x)$$ be defined by $$f(x) = \begin{cases} x + 2 & \text{if } x < 3, \\ 2x - 3 & \text{if }

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

Piecewise Rational Function and Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$f(3)=2$$. A graph of this func

Rational Function Analysis

Consider the rational function $$f(x)=\frac{(x+3)*(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the

Removable Discontinuity and Direct Limit Evaluation

Consider the function $$f(x)=\frac{(x-2)(x+3)}{x-2}$$ defined for $$x\neq2$$. The function is not de

Removable Discontinuity and Limit

Consider the function $$ f(x)=\frac{x^2-9}{x-3} $$ for $$ x\ne3 $$, which is not defined at $$ x=3 $

Removable Discontinuity and Redefinition

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$. Note that f is undefined at $$x=2$$

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}$

Squeeze Theorem with an Oscillatory Term

Consider the function $$f(x) = x^2 \cdot \cos\left(\frac{1}{x^2}\right)$$ for $$x \neq 0$$, and defi

Squeeze Theorem with Trigonometric Function

Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:

Two-dimensional Particle Motion with Continuous Velocity Functions

A particle moves in the plane with velocity components given by $$v_x(t)= \frac{t^2-9}{t-3}$$ and $

Vertical Asymptote Analysis

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

Vertical Asymptotes and Horizontal Limits

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

Analyzing a Function with a Removable Discontinuity

Consider the function $$f(x)= \frac{x^2 - 4}{x-2}$$, defined for $$x \neq 2$$.

Approximating the Instantaneous Rate of Change Using Secant Lines

A function $$f(t)$$ models the position of an object. The following table shows selected values of $

Behavior of the Derivative Near a Vertical Asymptote

Consider the function \(f(x)=\frac{1}{x+2}\) defined for \(x \neq -2\). Answer the following parts.

Chain Rule Application

Consider the composite function $$f(x)=\sqrt{1+4*x^2}$$, which may describe a physical dimension.

Comparing Average vs. Instantaneous Rates

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

Concavity and the Second Derivative

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

Critical Points of a Log-Quotient Function

Let $$f(x)= \frac{\ln(x)}{x}$$ for $$x > 0$$. This function is used to analyze efficiency in logarit

Derivative of a Logarithmic Function

Let $$g(x)= \ln(x)$$, a fundamental function in many natural logarithm applications.

Derivative of a Trigonometric Function

Let \(f(x)=\sin(2*x)\). Answer the following parts.

Derivative of an Exponential Decay Function

Consider the function $$f(t)=e^{-0.5*t}$$, which may represent the decay of a substance over time. A

Derivative using the Limit Definition for a Linear Function

For the linear function $$f(x)= 5*x - 3$$, perform an analysis of its derivative using the limit def

Derivatives and Optimization in a Real-World Scenario

A company’s profit is modeled by $$P(x)=-2*x^2+40*x-150$$, where $$x$$ represents the number of item

Derivatives in Economics: Cost Functions

A company's production cost is modeled by $$C(q)=500+20*q-0.5*q^2$$, where $$q$$ represents the quan

Derivatives on an Ellipse

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

Differentiability and Continuity

A function is defined piecewise as follows: $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x +

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

Finding the Derivative Using First Principles

Consider the function $$f(x)= 5*x^3 - 4*x + 7$$. Use the definition of the derivative to find the de

Finding the Tangent Line Using the Product Rule

For the function $$f(x)=(3*x^2-2)*(x+5)$$, which models a physical quantity's behavior over time (in

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

Implicit Differentiation in Motion

A particle’s motion is given by the implicit equation $$y^2 + x*y = 10$$, where x represents time (i

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 of Change from a Graph

A graph of a smooth curve (representing a function $$f(x)$$) is provided with a tangent line drawn a

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

Inverse Function Analysis: Rational Function

Consider the function $$f(x)=\frac{2*x+1}{x+3}$$ defined for all x except $$x=-3$$.

Linking Derivative to Kinematics: the Position Function

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

Logarithmic Differentiation of a Composite Function

Let $$f(x)= (x^{2}+1)*\sqrt{x}$$. Use logarithmic differentiation to find the derivative.

Marginal Profit Calculation

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

Mountain Stream Flow Adjustment

A mountain stream receives additional water from snowmelt at a rate of $$f(t)=4*t$$ (cubic feet/seco

Population Growth and Instantaneous Rate of Change

A town's population is modeled by $$P(t)= 2000*e^{0.05*t}$$, where $$t$$ is in years. Analyze the ch

Radioactive Decay Analysis

The amount of a radioactive substance is modeled by $$N(t)= 200*e^{-0.03*t}$$, where $$t$$ is in yea

Rate of Change for an Exponential Function

An amount of money grows according to the model $$A(t)=1000*e^{0.05*t}$$, where $$t$$ is measured in

Rate of Water Flow in a Rational Function Model

The water flow from a reservoir is modeled by $$F(t)= \frac{3*t}{t+2}$$, where $$t$$ is time in hour

Related Rates: Conical Tank Draining

A conical water tank drains so that its volume is given by $$V=\frac{1}{3}\pi r^2h$$. The radius r o

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 Approximation

A research experiment records temperature variations over time. The temperature function is approxim

Tangent Line to a Hyperbola

Consider the hyperbola defined by $$xy=20$$. Answer the following:

Tangent Lines and Local Linearization

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

Using the Limit Definition to Derive the Derivative

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

Analyzing a Function and Its Inverse

Consider the invertible function $$f(x)= \frac{x^3+1}{2}$$.

Chain Rule and Implicit Differentiation in Radial Motion

Let $$r$$ be a function of time $$t$$ defined implicitly by the equation $$r^2 + (\ln(r))^2 = t$$, w

Chain Rule with Exponential and Polynomial Functions

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

Chain Rule with Multiple Nested Functions in a Physics Model

In a physics experiment, the displacement of a particle is modeled by the function $$s(t)=\sqrt{\cos

Chain Rule with Nested Trigonometric Functions

Consider the function $$f(x)= \sin(\cos(3*x))$$. This function involves nested trigonometric functio

Composite Differentiation of an Inverse Trigonometric Function

Let $$H(x)= \arctan(\sqrt{x+3})$$.

Composite Function in Biomedical Model

The concentration C(t) (in mg/L) of a drug in the bloodstream is modeled by $$C(t) = \sin(3*t^2)$$,

Composite Function via Chain Rule in a Financial Context

A company’s profit (in dollars) based on production level (in thousands of units) is modeled by the

Composite Function with Nested Exponential and Trigonometric Terms

Let $$f(x)= e^{\sin(4*x)}$$. This function combines exponential and trigonometric elements.

Composite Inverse Trigonometric Function Evaluation

Let $$f(x)= \tan(2*x)$$, defined on a restricted domain where it is invertible. Analyze this functio

Derivative of an Inverse Trigonometric Composite

Let $$k(x)=\arctan\left(\frac{\sqrt{x}}{1+x}\right)$$.

Differentiation of a Complex Implicit Equation

Consider the equation $$\sin(xy) + \ln(x+y) = x^2y$$.

Implicit Differentiation in a Cubic Relationship

Examine the curve defined by $$x^3 + y^3 = 6*x*y$$, which represents a complex relationship between

Implicit Differentiation in a Trigonometric Context

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

Implicit Differentiation in Circular Motion

Consider the circle defined by the equation $$x^2+y^2=100$$, which could represent the track of an o

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve the following:

Implicit Differentiation of Quadratic Curve

Consider the curve defined by $$x^2+xy+y^2=7$$. Use implicit differentiation to analyze the behavior

Implicit Differentiation with Logarithmic and Trigonometric Combination

Consider the equation $$\ln(x+y)+\cos(x*y)=0$$, where $$y$$ is an implicit function of $$x$$. Find $

Inverse Function Derivative for a Log-Linear Function

Let $$f(x)= x+ \ln(x)$$ for $$x > 0$$ and let g be the inverse of f. Solve the following parts:

Inverse Function Differentiation in Temperature Conversion

Consider the function $$f(x)= \frac{1}{1+e^{-0.5*x}}$$, which converts a temperature reading in Cels

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 Differentiation in a Geometry Problem

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

Inverse Trigonometric Function Differentiation

Consider the function $$y=\arctan(2*x)$$. Answer the following:

Inverse Trigonometric Function Differentiation

Let $$f(x)=\arcsin\left(\frac{2*x}{5}\right)$$, with the understanding that $$\left|\frac{2*x}{5}\ri

Projectile Motion and Composite Function Analysis

A projectile is launched and its height $$h(t)$$ (in meters) is recorded at various times t (in seco

Related Rates of a Shadow

A 2 m tall lamp post casts a shadow from a person who is 1.8 m tall. The person is moving away from

Second Derivative via Implicit Differentiation

Consider the ellipse given by $$\frac{x^2}{4}+\frac{y^2}{9}=1$$. Answer the following:

Balloon Inflation Related Rates

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

Blood Drug Concentration

In a pharmacokinetic study, the concentration of a drug in the bloodstream is modeled by $$D(t)=\fra

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 $$C(x)=20*x + 50$$,

Economics: Marginal Revenue Analysis

A firm’s revenue function is given by $$R(x)=\frac{100x}{x+5}$$ (in dollars), where $$x$$ represents

Evaluating an Indeterminate Limit using L'Hospital's Rule

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2x} - 1}{3x}$$.

Friction and Motion: Finding Instantaneous Rates

A block slides down an inclined plane. The height of the plane at a horizontal distance $$x$$ is giv

FRQ 7: Conical Water Tank Filling

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

Inflating Spherical Balloon

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

Inflection Points and Concavity in Business Forecasting

A company's profit is modeled by $$P(x)= 0.5*x^3 - 6*x^2 + 15*x - 10$$, where $$x$$ represents a pro

Interpretation of the Derivative from Graph Data

The graph provided represents the position function $$s(t)$$ of a particle moving along a straight l

Inverse Function Analysis in a Real-World Model

Consider the function $$f(x)=x^3+1$$ which models a certain transformation of measurable quantities.

Limit Evaluation Using L'Hôpital's Rule

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

Linear Approximation for Function Values

Consider the function $$f(x) = x^3$$. Use linearization at $$x = 4$$ to approximate the value of $$f

Linear Approximation in Production Cost Estimation

A company's cost function is given by $$C(x)=0.02x^2+10x+500$$, where $$x$$ (in thousands) is the nu

Linear Approximation of ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

Linear Approximations: Estimating Function Values

Let $$f(x)=x^4$$. Use linear approximation to estimate $$f(3.98)$$. Answer the following:

Linearization in Medicine Dosage

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

Logarithmic Differentiation in Exponential Functions

Let $$y = (2x^2 + 3)^{4x}$$. Use logarithmic differentiation to find $$y'$$.

Maximizing Enclosed Area

A rancher has 120 meters of fencing to enclose a rectangular pasture along a straight river (the sid

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Optimization & Linearization in Engineering Design

A material's strength is modeled by the function $$S(x)= 50*x^2 - 3*x^3$$, where $$x$$ (in centimete

Optimization of Production Costs

A manufacturing company’s cost function for producing $$x$$ units per hour is given by $$C(x)=\frac{

Optimization: Minimizing Surface Area of a Box

An open-top box with a square base is to have a volume of 500 cubic inches. The surface area (materi

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Related Rates: The Expanding Ripple

Ripples form in a pond such that the radius of a circular ripple increases at a constant rate. Given

Shadow Length: Related Rates

A 15-foot tall lamp post casts light on a 6-foot tall man walking away from the lamp at 4 ft/sec. Le

Temperature Change in a Cooling Process

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

Temperature Rate Change in Cooling Coffee

A cup of coffee cools following the model $$x(t)=70+50e^{-0.1t}$$, where x is in degrees Fahrenheit

Train Motion Analysis

A train’s acceleration is given by $$a(t)=\sin(t)+0.5$$ (m/s²) for $$0 \le t \le \pi$$ seconds. The

Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= t^2 - 3*t$$ and $$y(t)= 2*t^3 - 9*t^2 + 12*t$$. Analyze th

Analyzing Differentiability of a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x^2, & \text{if } x \le 1, \\ 2*x - 1, &

Application of the Mean Value Theorem on a Piecewise Function

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

Area and Volume: Polynomial Boundaries

Let $$f(x)= x^2$$ and $$g(x)= 4 - x^2$$. Consider the region bounded by these two curves.

Average Value of a Function and Mean Value Theorem for Integrals

Consider the function $$f(x)= e^{-x}$$ on the interval $$[0, 2]$$. Answer the following:

Behavior Analysis of a Logarithmic Function

Consider the function $$f(x)= \frac{\ln(x)}{x}$$ for $$x>0$$. Analyze the critical points and concav

Car Speed Analysis via MVT

A car's position is given by $$f(t) = t^3 - 3*t^2 + 2*t$$ (in meters) for $$t$$ in seconds on the cl

Critical Numbers and Concavity in a Polynomial Function

Analyze the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ by determining its critical

Estimating Total Revenue via Riemann Sums

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

Evaluating Rate of Change in Electric Current Data

An electrical engineer recorded the current (in amperes) in a circuit over time. The table below sho

FRQ 2: Daily Temperature Extremes and the Extreme Value Theorem

A function modeling the ambient temperature (in $$^\circ C$$) during the first 6 hours of a day is g

FRQ 5: Concavity and Points of Inflection for a Cubic Function

For the function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$, analyze its concavity.

FRQ 10: First Derivative Test for a Cubic Profit Function

A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the

Graphical Analysis Using First and Second Derivatives

The graph provided represents the function $$f(x)= x^3 - 3*x^2 + 2*x$$. Analyze this function using

Identification of Extrema and Critical Points

Let $$f(x)= x^3 - 6*x^2 + 9*x + 1$$ be defined on the interval $$[0,4]$$. Use your understanding of

Implicit Differentiation and Tangent Lines

Consider the curve defined implicitly by the equation $$x^2 + x*y + y^2= 7$$.

Inverse Analysis of a Composite Function

Consider the function $$f(x)=e^(x)+x$$. Although its inverse cannot be written in closed form, answe

Inverse Analysis of a Logarithmic Function

Consider the function $$f(x)=\ln(x-1)$$ defined for $$x>1$$. Answer the following questions about it

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

Investigating a Piecewise Function with a Vertical Asymptote

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

Optimal Production Level: Relative Extrema from Data

A manufacturer recorded profit (in thousands of dollars) at different levels of unit production. Use

Optimization of a Fenced Enclosure

A farmer wants to construct a rectangular garden using 120 meters of fencing along three sides, with

Optimizing an Open-Top Box from a Metal Sheet

A rectangular sheet of metal with dimensions 24 cm by 18 cm is used to create an open-top box by cut

Pharmaceutical Drug Delivery

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

Solving a Log-Exponential Equation

Solve the equation $$\ln(x)+x=0$$ for $$x>0$$. Answer the following:

Using Derivatives to Solve a Rate-of-Change Problem

A particle’s displacement is given by $$s(t) = t^3 - 9*t^2 + 24*t$$ (in meters), where \( t \) is in

Water Reservoir Net Change

A water reservoir receives water from a river at a rate $$R_{in}(t)=5+0.5*t$$ and discharges water a

Antiderivatives and the Constant of Integration in Modelling

A moving car has its velocity modeled by $$v(t)= 5 - 2*t$$ (in m/s). Answer the following parts to o

Antiderivatives of Trigonometric Functions

Evaluate the integral $$\int \sin(2*t)\,dt$$, and then use your result to compute the definite integ

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

Comparing Riemann Sum Methods for $$\int_1^e \ln(x)\,dx$$

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$. A table of approximate values is p

Environmental Modeling: Pollution Accumulation

The pollutant enters a lake at a rate given by $$P(t)=5*e^{-0.3*t}$$ (in kg per day) for $$t$$ in da

Estimating Displacement with a Midpoint Riemann Sum

A vehicle’s velocity is modeled by the function $$v(t) = -t^{2} + 4*t$$ (in meters per second) over

Estimating River Flow Volume

A river's flow rate (in cubic meters per second) has been measured at various times during an 8-hour

Estimating Total Biomass in an Ecosystem

An ecologist measured the population density (in kg/km²) of a species along an 8 km transect. Use th

Evaluating a Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

Exact Area Under a Parabolic Curve

Find the exact area under the curve $$f(x)=3*x^2 - x + 4$$ between $$x=1$$ and $$x=4$$.

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

FRQ16: Inverse Analysis of an Integral Function via U-Substitution

Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.

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.

Graphical Analysis of an Accumulation Function

Let $$f(t)$$ represent the rate of water flow (in $$m^3/hr$$) into a reservoir, and suppose the grap

Growth of Investment with Regular Contributions and Withdrawals

An investment account receives contributions at a rate of $$C(t)= 100e^{0.05t}$$ dollars per year an

Integration by Parts: Evaluating $$\int_1^e \ln(x)\,dx$$

Evaluate the integral $$\int_1^e \ln(x)\,dx$$ using integration by parts.

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Net Change Calculation

The net change in a quantity $$Q$$ is modeled by the rate function $$\frac{dQ}{dt}=e^{t}-1$$ for $$0

Net Displacement and Total Distance Calculation

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

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). T

Particle Trajectory in the Plane

A particle moves in the plane with its velocity components given by $$v_x(t)=\cos(t)$$ and $$v_y(t)=

Piecewise-Defined Function and Discontinuities

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

U-Substitution in a Trigonometric Integral

Evaluate the integral $$\int \sin(2*x) * \cos(2*x)\,dx$$ using u-substitution.

Volume of a Solid of Revolution Using the Disk/Washer Method

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane bounded by the curves $$y=x$$ and $$y=x^2$$. Cross sections perpe

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=3*x^2+2*x$$ where x is in meters and F in Newtons. Th

Analysis of an Autonomous Differential Equation

Consider the autonomous differential equation $$\frac{dy}{dx}=y(4-y)$$ with the initial condition $$

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Area Under a Differential Equation Curve

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

Bacterial Culture with Antibiotic Treatment

A bacterial culture grows at a rate proportional to its size, but an antibiotic is administered cont

Bacterial Growth under Logistic Model

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

Bernoulli Differential Equation via Substitution

Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ

Chemical Reactor Temperature Profile

In a chemical reactor, the temperature $$T$$ (in °C) is recorded over time (in minutes) as shown. Th

Combined Cooling and Slope Field Problem

A cooling process is modeled by the equation $$\frac{d\theta}{dt}=-0.07\,\theta$$ where $$\theta(t)=

Comparative Population Decline

A population declines according to two models. Model 1 follows simple exponential decay: $$\frac{dN}

Cooling of a Hot Beverage

According to Newton's Law of Cooling, the temperature $$T(t)$$ of a hot beverage satisfies $$\frac{d

Differential Equation with Substitution using u = y/x

Consider the differential equation $$\frac{dy}{dx}= \frac{y}{x}+\sqrt{\frac{y}{x}}$$. Use the substi

Direction Fields and Integrating Factor

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

Drug Concentration Model

The concentration $$C(t)$$ (in mg/L) of a drug in a patient's bloodstream is modeled by the differen

Drug Elimination with Infusion

A drug is administered continuously to a patient. Its blood concentration $$C(t)$$ (in mg/L) satisfi

Falling Object with Air Resistance

A falling object experiences air resistance proportional to the square of its velocity. Its velocity

Heating and Cooling in an Electrical Component

An electronic component experiences heating and cooling according to the differential equation $$\fr

Implicit Differential Equation and Asymptotic Analysis

Consider the differential equation $$\frac{dy}{dx}= \frac{y(1-y)}{x}$$ for $$x > 0$$ with the initia

Implicit Differentiation of a Transcendental Equation

Consider the equation $$e^{x*y} + y^3= x$$. Answer the following:

Logistic Growth in a Population

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt}=0.5P\lef

Logistic Population Growth Model

A fish population in a lake is modeled by the logistic differential equation $$\frac{dP}{dt} = 0.3\,

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

Mixing Problem with Constant Flow

A tank initially contains 200 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Mixing Tank Problem

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

Modeling Orbital Decay

A satellite’s altitude $$h(t)$$ decreases over time due to atmospheric drag, following $$\frac{dh}{d

Nonlinear Cooling of a Metal Rod

A thin metal rod cools in an environment at $$15^\circ C$$ according to the differential equation $$

Population Dynamics with Harvesting

A fish population is governed by the differential equation $$\frac{dP}{dt} = 0.4*P\left(1-\frac{P}{1

Population Growth with Harvesting

A fish population in a lake grows according to $$\frac{dP}{dt}=0.08*P-50$$, where $$P(t)$$ represent

Qualitative Analysis of a Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=1-y^2$$.

Radioactive Decay

A radioactive substance decays according to $$\frac{dN}{dt} = -\lambda N$$. Initially, there are 500

Radioactive Decay Differential Equation

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

Radioactive Decay Differential Equation

A radioactive substance decays according to the differential equation $$\frac{dM}{dt}=-k*M$$. If the

RC Circuit Discharge

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

Reaction Kinetics in a Tank

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

Salt Mixing in a Tank

A tank initially contains 100 L of water with 5 kg of salt dissolved. Brine with a concentration of

Separable Differential Equation

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

Solving a Differential Equation by Substitution

Solve the differential equation $$\frac{dy}{dx}=\frac{2xy}{x^2-4}$$. (a) Separate the variables to

Solving a Differential Equation Using the SIPPY Method

Solve the differential equation $$\frac{dy}{dx}=4*x^3*e^{-y}$$ with the initial condition $$y(1)=0$$

Tumor Growth with Allee Effect

The growth of a tumor is modeled by the differential equation $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\

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

Area Between Cost Functions in a Business Analysis

A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C

Average and Instantaneous Rates in a Cooling Process

A cooling process is modeled by the function $$T(t)= 100*e^{-0.05*t}$$ (in degrees Fahrenheit), wher

Average Concentration in a Reaction

A chemical reaction has its concentration modeled by the function $$C(t)=50e^{-0.2*t}+5$$ (in mg/L)

Average Concentration in Medical Dosage

A patient’s bloodstream concentration of a medication is modeled by the function $$C(t)=\frac{100}{1

Average Reaction Rate Determination

A chemical reaction’s rate is modeled by the function $$r(t)=k*e^{-t}$$, where $$t$$ is in seconds a

Average Value of a Trigonometric Function

Consider the function $$f(x)=\sin(x)+1$$ defined on the interval $$[0,\pi]$$. This function models a

Average Voltage in a Physics Experiment

In a physics experiment, the voltage across a resistor is modeled by $$V(t)=5+3*\cos\left(\frac{\pi*

Average vs. Instantaneous Value of a Function

Consider the function $$f(x)= \sin(x) + 1$$ on the interval $$[0, \pi]$$.

Cooling Process Analysis

A cup of coffee cools in a room, and its temperature (in °C) is modeled by $$T(t)=30*e^{-0.1*t}+5$$

Designing an Open-Top Box

An open-top box with a square base is to be constructed with a fixed volume of $$5000\,cm^3$$. Let t

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

Exponential Decay Function Analysis

A lab experiment models the decay of a chemical concentration with the function $$f(t)=8*e^{-0.5*t}$

Free Workout Class Attendance

The attendance at a free workout class increases by a fixed number of people each session. The first

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

Medication Dosage Increase

A patient receives a daily medication dose that increases by a fixed amount each day. The first day'

Motion Analysis Using Integration of a Sinusoidal Function

A car has velocity given by $$v(t)=3*\sin(t)+4$$ (in $$m/s$$) for $$t \ge 0$$, and its initial posit

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 $$

Optimization of Average Production Rate

A manufacturing process has a production rate modeled by the function $$P(t)=50e^{-0.1*t}+20$$ (unit

Position and Velocity Relationship in Car Motion

A car's position along a highway is modeled by $$s(t)=t^3-6*t^2+9*t+2$$ (in kilometers) with time $$

Projectile Motion: Position, Velocity, and Maximum Height

A projectile is launched vertically upward with an initial velocity of $$20\,m/s$$ from a height of

Volume of a Solid of Revolution: Curve Raised to a Power

Consider the region under the curve $$f(x)=x^{1/3}$$ for $$x\in [0,8]$$. This region is revolved abo

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane given by the region bounded by the curves $$y=x$$ and $$y=\sqrt{x

Volume of a Solid with Square Cross Sections

A solid is formed over the region under the line $$f(x)=4-x$$ from $$x=0$$ to $$x=4$$ in the x-y pla

Volume with Semicircular Cross-Sections

A solid has a base on the interval $$[0,3]$$ along the x-axis, and its cross-sectional slices perpen

Washer Method with Logarithmic and Exponential Curves

Consider the region bounded by the curves $$f(x)=\ln(x+1)$$ and $$g(x)=e^{-x}$$ on the interval $$[0

Water Reservoir Inflow‐Outflow Analysis

A water reservoir receives water through an inflow pipe and loses water through an outflow valve. Th

Water Tank Filling with Graduated Inflow

A water tank is filled daily by adding a certain amount of water that increases by a fixed amount ea

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=3*x^2+2$$ (in Newtons) when an object is dis

Work to Pump Water from a Cylindrical Tank

A cylindrical tank with a radius of 3 m and a height of 10 m is completely filled with water (densit