AP Calculus AB FRQ Room

Advanced Analysis of a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \en

Analysis of One-Sided Limits and Jump Discontinuity

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

Analyzing Asymptotic Behavior in a Rational Function

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

Application of the Squeeze Theorem in Trigonometric Limits

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

Continuity Analysis of a Piecewise Function

Consider the function defined by $$ f(x)=\begin{cases}2x+1, & x<1,\\ x^2, & 1\le x\le 3,\\ 7-x, & x

Continuity Analysis with a Piecewise-defined Function

A particle’s displacement is described by the piecewise function $$s(t)= \begin{cases} t^2+1, & t <

Continuity of a Sine-over-x Function

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

Continuous Extension and Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Evaluating a Limit with Radical Expressions

Evaluate the limit $$\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$$. Answer the following:

Evaluating Limits Near Vertical Asymptotes

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

Factorization and Limit Evaluation

Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e

Factorization and Removable Discontinuity

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

Graph Analysis: Identify Limits and Discontinuities

A graph of a function f(x) is provided in the stimulus. The graph shows a removable discontinuity at

Graphical Interpretation of Limits and Continuity

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

Implicit Differentiation in an Exponential Equation

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

Intermediate Value Theorem in Context

Let $$f(x) = x^3 - 6x^2 + 9x + 2$$, which is continuous on the interval [0, 4]. Answer the following

Inverse Function and Limit Behavior Analysis

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

Investigation of Continuity in a Piecewise Log-Exponential Function

A function is defined by $$ f(x)=\begin{cases} \frac{\ln(e^{2*x}+3)-\ln(5)}{x-1} & x \neq 1, \\ D &

Jump Discontinuity Analysis

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

Limit Analysis in a Population Growth Model

Consider the function $$y(t)=\frac{e^{2*t}-e^{t}}{t}$$ for $$t \neq 0$$, and define $$y(0)=L$$ so th

Limit Evaluation with a Parameter in a Log-Exponential Function

Consider the function $$r(x)=\frac{e^{a*x} - e^{b*x}}{\ln(1+x)}$$ defined for $$x \neq 0$$, where $$

Limit Involving an Exponential Function

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

Limits at Infinity and Horizontal Asymptotes

Examine the function $$f(x)=\frac{3x^2+2x-1}{6x^2-4x+5}$$ and answer the following:

Limits Involving Radical Functions

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

Limits Near Vertical Asymptotes

Consider the function $$f(x) = \frac{1}{x - 4}$$. (a) Determine $$\lim_{x \to 4^-} f(x)$$. (b) Dete

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

Particle Motion with Squeeze Theorem Application

A particle moves along a line with velocity given by $$v(t)= t^2 \sin(1/t)$$ for $$t>0$$ and is defi

Rational Function Limits and Removable Discontinuities

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

Rational Function with Two Critical Points

Consider the function $$f(x)=\begin{cases} \frac{x^2+x-6}{x^2-9} & x\neq -3,3 \\ \frac{5}{6} & x=-3

Rational Functions with Removable Discontinuities

Examine the function $$f(x)= \frac{x^2 - 5x + 6}{x - 2}$$. (a) Factor the numerator and simplify th

Redefining a Function for Continuity

A function is defined by $$f(x) = \frac{x^2 - 1}{x - 1}$$ for $$x \neq 1$$, while $$f(1)$$ is left u

Removable Discontinuity and Limit Evaluation

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

Removal of Discontinuity by Redefinition

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

Squeeze Theorem Application

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

Squeeze Theorem for an Oscillatory Function

Define the function $$f(x)= x \cos(\frac{1}{x})$$ for x ≠ 0, and let f(0)= 0.

Vertical Asymptotes and Horizontal Limits

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

Analysis of Savings Account Growth

A savings account has a balance given by $$S(t)= 1000*(1.005)^t$$, where $$t$$ is the number of mont

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

Consider the function $$f(x)= x^2$$.

Application of Product Rule

Differentiate the function $$f(x)=(3x^2+2x)(x-4)$$ by two methods. Answer the following:

Approximating Derivative using Secant Lines

Consider the function $$f(x)= \ln(x)$$. A student records the following data: | x | f(x) | |---

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

Differentiability of an Absolute Value Function

Consider the function $$f(x)=|x-3|$$, representing the error margin (in centimeters) in a calibratio

Differentiating a Product of Linear Functions

Let $$f(x) = (2*x^2 + 3*x)\,(x - 4)$$. Use the product rule to find $$f'(x)$$.

Differentiation of a Log-Linear Function

Consider the function $$f(x)= 3 + 2*\ln(x)$$ which might model a process with a logarithmic trend.

Evaluating Limits and Discontinuities in a Piecewise Function

Consider the function given by $$ f(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x\neq 3, \\

Inverse Function Analysis: Cosine and Linear Combination

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

Inverse Function Analysis: Exponential Function

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

Inverse Function Analysis: Square Root Function

Consider the function $$f(x)=\sqrt{4*x+1}$$ defined for $$x \geq -\frac{1}{4}$$.

Marginal Cost Function in Economics

A company’s cost function is given by $$C(x)=200+8*x+0.05*x^2$$, where $$C(x)$$ is in dollars and $$

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

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

Product and Chain Rule Combined

Let \(f(x)=(3*x+1)^2 * \cos(x)\). Answer the following parts.

Product and Quotient Rule Combination

Given $$u(x)=3*x^2+2$$ and $$v(x)=x-4$$, consider the function $$F(x)=\frac{u(x)*v(x)}{x+1}$$. Answe

Profit Function Analysis

A company's profit function is given by $$P(x)=-2x^2+12x-5$$, where x represents the production leve

Quotient Rule Application

Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone

Quotient Rule Challenge

For the function $$f(x)= \frac{3*x^2 + 2}{5*x - 7}$$, find the derivative.

Rate of Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by \(C(t)=10*e^{-0.3*t}\), where \

Rates of Change from Experimental Data

A chemical experiment yielded the following measurements of a substance's concentration (in molarity

Related Rates: Expanding Ripple Circle

Water droplets create circular ripples on a surface. The area of a ripple is given by $$A = \pi * r^

Sand Pile Growth with Erosion Dynamics

A sand pile is growing as sand is added at a rate of $$f(t)=8+0.3*t$$ (kg/min) and simultaneously lo

Secant and Tangent Lines Analysis

Consider the function $$g(t)=t^3-6*t^2+9*t+2$$ modeling the height (in meters) of a ball at time $$t

Tangent Line and Differentiability

Let $$h(x)=\frac{1}{x+2}$$, modeling the concentration of a substance in a chemical solution over ti

Tangent Line Approximation for a Cubic Function

Let $$f(x)=2*x^3 - 7*x + 1$$. At $$x=1$$, determine the equation of the tangent line and use it to a

Using the Limit Definition of the Derivative

Consider the function $$g(x)=3*x^3-2*x+5$$, which models the cost (in dollars) of manufacturing $$x$

Analyzing Composite Functions Involving Inverse Trigonometry

Let $$y=\sqrt{\arccos\left(\frac{1}{1+x^2}\right)}$$. Answer the following:

Analyzing Motion in the Plane using Implicit Differentiation

A particle moves in the xy-plane along a path defined implicitly by $$x^2+x*y+y^2=7$$. Determine the

Chain Rule in Angular Motion

An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$

Chain Rule Involving Inverse Trigonometric Functions

Differentiate the function $$f(x)=\arctan(\sqrt{3*x+4})$$ using the chain rule and the derivative fo

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 Exponential and Trigonometric Functions

A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq

Chain Rule with Nested Trigonometric Functions

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

Composite and Product Rule Combination

The function $$F(x)= (3*x^2+2)^{4} * \cos(x^3)$$ arises in modeling a complex system. Answer the fol

Composite Functions in Population Dynamics

The population of a species is modeled by the composite function $$P(t) = f(g(t))$$, where $$g(t) =

Derivative of an Inverse Function

Let $$f$$ be a differentiable function with an inverse function $$g$$ such that $$f(2)=5$$ and $$f'(

Derivative of an Inverse Trigonometric Composite

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

Differentiating an Inverse Trigonometric Function

Let $$y = \arcsin\left(\frac{2*x}{1+x^2}\right)$$.

Differentiation of a Complex Implicit Equation

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

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

Expanding Spherical Balloon

A spherical balloon has its volume given by $$V=\frac{4}{3}\pi r^3$$. The radius of the balloon incr

Finding Second Derivative via Implicit Differentiation

Given the curve defined by $$x^2+y^2+ x*y=7$$, answer the following:

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

Implicit Differentiation for an Ellipse

Consider the ellipse defined by the equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. This equation re

Implicit Differentiation in a Financial Model

An implicit relationship between revenue $$R$$ (in thousands of dollars) and price $$p$$ (in dollars

Implicit Differentiation in a Population Growth Model

Consider the model $$e^{x*y} + x - y = 5$$ that relates time \(x\) to a population scale value \(y\)

Implicit Differentiation in an Exponential Context

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

Implicit Differentiation of an Ellipse in Navigation

A flight path is modeled by the ellipse $$\frac{x^2}{16}+\frac{y^2}{9}=1$$.

Implicit Differentiation with Chain and Product Rules

Consider the curve defined implicitly by $$e^{xy} + x^2y = 10$$. Assume that the point $$(1,2)$$ lie

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Mixed Functions

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

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Implicit Differentiation with Trigonometric Components

Consider the equation $$\sin(x) + \cos(y) = x*y$$, which implicitly defines $$y$$ as a function of $

Implicit Differentiation with Trigonometric Terms

Consider the implicit equation $$\sin(x*y)+x^2=y^2$$. Find the derivative $$\frac{dy}{dx}$$.

Implicitly Defined Inverse Relation

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

Inverse Analysis in Exponential Decay

A radioactive substance decays according to $$N(t)= N_0*e^(-0.5*t)$$, where N(t) is the quantity at

Inverse Function Derivative

Suppose that $$f$$ is a differentiable and one-to-one function. Given that $$f(4)=10$$ and $$f'(4)=2

Inverse Function Differentiation for a Log Function

Let $$f(x)=\ln(3*x+2)$$, and assume that $$f$$ is invertible with inverse function $$g$$. Find the d

Inverse Function Differentiation for an Exponential Function

Let $$f(x)= e^{2*x} + 1$$. This function involves an exponential model shifted upward.

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 Trigonometric Function Differentiation

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

Related Rates in a Circular Colony

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

Second Derivative via Implicit Differentiation

Consider the curve defined by $$x^2+x*y+y^2=7$$. Answer the following parts.

Analysis of a Function Combining Polynomial and Exponential Terms

The concentration of a substance over time t (in hours) is modeled by $$C(t)= t^2 e^{-0.5*t} + 5$$.

Analysis of Wheel Rotation

Consider a wheel whose angular position is given by $$\theta(t) = 2t^2 + 3t$$, in radians, where $$t

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

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 Analysis

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{10}{1+e^{0.5t}}$$,

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Determining the Tangent Line

Consider the function $$f(x)=\ln(x)+ x$$. The graph of the function is provided for reference.

Error Estimation in Pendulum Period

The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length

Expanding Circular Ripple in a Pond

A circular ripple in a pond has its area increasing at a constant rate of 10 square meters per secon

Falling Object's Velocity Analysis

A rock is thrown upward from the top of a building with a velocity function $$v(t)= 20 - 9.8*t$$ (in

FRQ 5: Coffee Cooling Experiment

A cup of coffee cools according to the function $$T(t) = 70 + 50e^{-0.1*t}$$, where T is the tempera

FRQ 12: Airplane Climbing Dynamics

An airplane’s altitude is modeled by the equation $$y = 0.1*x^2$$, where x (in km) is the horizontal

Inflating Balloon: Rate of Change of Radius

A spherical balloon is being inflated so that its volume increases at a constant rate of $$\frac{dV}

Kinematics on a Straight Line

A particle moves along a straight line with a position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, wher

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

Linearization for Function Estimation

Use linear approximation to estimate the value of $$\ln(4.1)$$. Let the function be $$f(x)=\ln(x)$$

Linearization of a Machine Component's Length

A machine component's length is modeled by $$L(x)=x^4$$, where x is a machine setting in inches. Use

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

Motion Analysis from Velocity Function

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

Motion Analysis of a Particle on a Line

A particle’s position is modeled by $$s(t)=3t^3-6t^2+2t+1$$, where s is in meters and t in seconds.

Multi‐Phase Motion Analysis

A car's motion is described by a piecewise velocity function. For $$0 \le t < 2$$ seconds, the veloc

Optimization in a Manufacturing Process

A company designs an open-top container whose volume is given by $$V = x^2 y$$, where x is the side

Optimization of Production Costs

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

Optimizing Road Construction Costs

An engineer is designing a road that connects a point on a highway to a town located off the highway

Particle Motion Analysis

A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$, where $$t$$

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

Related Rates in Expanding Circular Oil Spill

An oil spill forms a circular patch. Its area is given by $$A= \pi*r^2$$. If the area is increasing

Related Rates: The Expanding Ripple

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

Relative Motion: Two Objects Approaching an Intersection

Two cars, A and B, are traveling toward an intersection. Car A is initially 100 m from the intersect

Revenue Sensitivity to Advertising

A firm's revenue (in thousands of dollars) is given by $$R(t)=50\sqrt{t+1}$$, where $$t$$ represents

Rocket Thrust: Analyzing Exponential Acceleration

A rocket’s velocity is modeled by $$v(t) = 100(1 - e^{-0.05t})$$, where $$t$$ is in seconds and $$v(

Temperature Change in a Cooling Process

A cup of coffee cools according to the function $$T(t)=70+50e^{-0.1*t}$$, where $$T$$ is in degrees

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

Using L'Hospital's Rule to Evaluate a Limit

Consider the limit $$L=\lim_{x\to\infty}\frac{5x^3-4x^2+1}{7x^3+2x-6}$$. Answer the following:

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 a Rate of Change in a Biological Growth Model

A bacterial culture's population is modeled by $$P(t)= 100*e^{0.3*t}$$, where \( P(t) \) is the numb

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

Applying the Mean Value Theorem and Analyzing Discontinuities

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

Capacitor Discharge in an RC Circuit

The voltage across a capacitor during discharge is given by $$V(t)= V_0*e^{-t/(RC)}$$, where $$t$$ i

Derivative and Concavity of f(x)= e^(x) - ln(x)

Consider the function $$f(x)= e^{x}-\ln(x)$$ for $$x>0$$. Answer the following:

Discontinuity in a Rational Function Involving Square Roots

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

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

FRQ 7: Maximizing Revenue in Production

A company’s revenue function is modeled by $$R(x)= -2*x^2 + 40*x$$ (in thousands of dollars), where

FRQ 12: Optimization in Manufacturing: Minimizing Cost

A company’s cost function is given by $$C(x)= 0.5*x^2 - 10*x + 125$$ (in dollars), where $$x$$ repre

FRQ 15: Population Growth and the Mean Value Theorem

A town’s population (in thousands) is modeled by $$P(t)= t^3 - 3*t^2 + 2*t + 50$$, where $$t$$ repre

FRQ 19: Analysis of an Exponential-Polynomial Function

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

Graph Analysis: Exponentially Modified Function

Consider the function $$f(x)= 2e^{x}-5\ln(x+1)$$ defined for $$x> -1$$. Answer the following:

Hydroelectric Dam Efficiency

A hydroelectric dam experiences water inflow and outflow that affect its efficiency. The inflow is g

Implicit Differentiation and Tangent Lines

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

Inflection Points and Concavity in a Real-World Cost Function

A company's cost function is given by $$C(x) = 0.5*x^3 - 6*x^2 + 20*x + 100$$, where \( x \) represe

Instantaneous Velocity Analysis via the Mean Value Theorem

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

Inverse Analysis of a Composite Trigonometric-Linear Function

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

Inverse Analysis of a Cooling Temperature Function

A cooling process is described by the function $$f(t)=20+80*e^{-0.05*t}$$, where t is the time in mi

Inverse Analysis of a Rational Function in a Work-Rate Context

Consider the function $$f(x)=\frac{4*x+5}{2*x-1}$$ which models a certain work-rate scenario. Analyz

Mean Value Theorem Applied to Car Position Data

A car’s position (in meters) is recorded at various times during a journey. Use the information prov

Modeling Disease Spread with an Exponential Model

In an epidemic, the number of infected individuals is modeled by $$I(t)= I_0 * e^{r*t}$$, where $$t$

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 Rectangular Enclosure

A rectangular pen is to be constructed along the side of a barn so that only three sides require fen

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Polynomial Rational Discontinuity Investigation

Consider the function $$ g(x) = \begin{cases} \frac{x^3 - 8}{x - 2}, & x \neq 2, \\ 5, & x = 2. \en

Predicting Fuel Efficiency in Transportation

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

Projectile Motion and Derivatives

A projectile is launched so that its height is given by $$h(t) = -4.9*t^2 + 20*t + 1$$, where $$t$$

Rate of Heat Loss in a Cooling Process

In a cooling experiment, the temperature of an object is recorded over time as it loses heat. Use th

Transcendental Function Analysis

Consider the function $$f(x)= \frac{e^x}{x+1}$$ defined for $$x > -1$$ and specifically on the inter

Accumulated Bacteria Growth

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

Accumulation and Inflection Points

Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ Answer the following parts:

Accumulation Function and Its Derivative

Define the function $$F(x)= \int_0^x \Big(e^{t} - 1\Big)\,dt$$. Answer the following parts related t

Antiderivatives and Initial Value Problems

Given that $$f'(x)=\frac{2}{\sqrt{x}}$$ for $$x>0$$ and $$f(4)=3$$, find the function $$f(x)$$.

Approximating the Area with Riemann Sums

Consider the linear function $$f(x) = 2*x + 1$$ on the interval $$[1,5]$$. Use Riemann sums to appro

Calculating Total Distance Traveled from a Changing Velocity Function

A particle moves with a velocity given by $$v(t)=t^2 - 4*t + 3$$ (in m/s) for $$0 \le t \le 5$$. Not

Car Fuel Consumption Analysis

A car engine’s fuel dynamics are modeled such that fuel is consumed at a rate of $$f(t)=0.1t^2$$ L/m

Chemical Reactor Conversion Process

In a chemical reactor, the instantaneous reaction rate is given by $$R(t)=4t^2-t+3$$ mol/min, while

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 Cost Function Analysis

A company's marginal cost (in dollars per unit) is recorded at various production levels. Use the da

Electric Charge Accumulation

An electrical circuit records the current (in amperes) at various times during a brief experiment. U

Error Estimates in Numerical Integration

Suppose a function $$f(x)$$ defined on an interval $$[a,b]$$ is known to be concave downward. Consid

Estimating Distance Traveled Using the Trapezoidal Rule from Speed Data

During a car journey, the speed (in km/hr) is recorded at regular intervals. The table below shows s

FRQ4: Inverse Analysis of a Trigonometric Accumulation Function

Let $$ H(x)=\int_{0}^{x} (\sin(t)+2)\,dt $$ for $$ x \in [0,\pi] $$, representing a displacement fun

Integration of Exponential Functions with Shifts

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

Midpoint Riemann Sum for Temperature Data

A weather station records temperature (in degrees Celsius) at hourly intervals. The data for a 4-hou

Motion Analysis from Velocity Data

A particle moves along a straight line with the following velocity data (in m/s) recorded at specifi

Optimizing Fencing Cost for a Garden Adjacent to a River

A farmer plans to fence a rectangular garden adjacent to a river, so that no fence is required along

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

Population Accumulation in a Lake

A researcher is studying a fish population in a lake. The rate of change of the fish population is m

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

Trapezoidal Rule Application with Population Growth

A biologist recorded the instantaneous growth rate (in thousands per year) of a species over several

Vehicle Distance Estimation from Velocity Data

A car's velocity (in m/s) is recorded at several time points during a trip. Use the table below for

Volume Accumulation in a Leaking Tank

Water leaks from a tank at a rate given by $$R(t)=3-0.5*t$$ (in liters per minute) for t in [0,6]. I

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

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

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-

Chemical Reaction Rate with Second-Order Decay

A chemical reaction follows the rate law $$\frac{d[A]}{dt}=-k[A]^2$$, where $$[A](t)$$ (in M) is the

Chemical Reactor Mixing

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

Differential Equation in Business Profit

A company's profit $$P(t)$$ changes over time according to $$\frac{dP}{dt} = 100\,e^{-0.5t} - 3P$$.

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 for an Autonomous Equation

Consider the differential equation $$\frac{dy}{dx}=y^2-9$$. Analyze the behavior of its solutions.

Exponential Growth and Doubling Time

A bacterial culture grows according to the differential equation $$\frac{dy}{dt} = k * y$$ where $$y

Falling Object with Air Resistance

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

Falling Object with Air Resistance

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

Logistic Growth in a Population

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

Logistic Growth Model

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

Logistic Growth Model

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt} = rP\left(1-\frac{P}{K}\

Mixing Problem with Time-Dependent Inflow Concentration

A tank initially contains 100 liters of water with 8 kg of dissolved salt. Brine enters the tank at

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 $$\frac{dA}{dt}=rA$$

Motion Under Gravity with Air Resistance

An object is falling vertically under the influence of gravity and air resistance. Its velocity $$v(

Particle Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=3-2*t$$ (in m/s²). Its initial veloc

Pollutant Concentration in a Reservoir

An urban water reservoir contains 100,000 L of water and initially 2000 kg of pollutant. Polluted wa

Population Growth with Harvesting

A fish population in a lake grows at a rate proportional to its current size, but fishermen harvest

Radioactive Decay

A radioactive substance is measured over time. The activity $$A$$ (in grams) is recorded at several

Radioactive Decay and Half-Life

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

Radioactive Decay Model

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

Radioactive Isotope in Medicine

A radioactive isotope used in medical imaging decays according to $$\frac{dA}{dt}=-kA$$, where $$A$$

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

Seasonal Temperature Variation

The temperature $$T(t)$$ in a region is modeled by the differential equation $$\frac{dT}{dt} = -0.2\

Separable Differential Equation

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

Slope Field and Integrating Factor Analysis

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

Slope Field Sketching for $$\sin(x)$$ Model

Given the slope field for the differential equation $$\frac{dy}{dx} = \sin(x)$$, sketch a solution c

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Tumor Treatment with Chemotherapy

A patient's tumor cell population $$N(t)$$ is modeled by the differential equation $$\frac{dN}{dt}=r

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 a Parabolic Curve and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ on the interval $$[0,4]$$. The table below sh

Area Between Curves: Exponential vs. Linear

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=1-x$$. A table of approximate values is provided b

Arithmetic Savings Account

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

Average Value of a Polynomial Function

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

Consumer Surplus Calculation

The demand function for a certain product is given by $$D(p)=100-5*p$$ and the supply function by $$

Economics: Consumer Surplus Calculation

Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons

Electric Charge Accumulation

The current flowing into a capacitor is defined by $$I(t)=\frac{10}{1+e^{-2*(t-3)}}$$ (in amperes) f

Graduated Rent Increase

An apartment’s yearly rent increases by a fixed amount. The initial annual rent is $$1200$$ dollars

Integrated Motion Analysis

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

Motion Under Resistive Force

A particle’s acceleration in a resistive medium is modeled by $$a(t)=\frac{10}{1+t} - 2*e^{-t}$$ (in

Net Change in Concentration of a Chemical Reaction

In a chemical reaction, the rate of production of a substance is given by $$r(t)$$ (in mol/min). The

Optimization of Average Production Rate

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

Reconstructing Position from Acceleration Data

A particle traveling along a straight line has its acceleration given by the values in the table bel

Retirement Savings Auto-Increase

A person contributes to a retirement fund such that the monthly contributions form an arithmetic seq

Revenue Optimization via Integration

A company’s revenue is modeled by $$R(t)=1000-50*t+2*t^2$$ (in dollars per hour), where $$t$$ (in ho

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

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

Water Pumping from a Parabolic Tank

A water tank has ends shaped by the region bounded by $$y=x^2$$ and $$y=4$$, and the tank has a unif