AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{|x-5|}{x-5} & x\neq5 \\ 0 & x=5 \end{cases}$$. Answ

Algebraic Manipulation in Limit Calculations

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

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 Velocity Function with Nested Discontinuities

A particle’s velocity along a line is given by $$v(t)= \frac{(t-1)(t+3)}{(t-1)*\ln(t+2)}$$ for $$t>0

Analyzing End Behavior and Asymptotes

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

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

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

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

Compound Interest and Geometric Series

A bank account accrues interest compounded annually at an annual rate of 10%. The balance after $$n$

Continuity of Constant Functions

Consider the constant function $$f(x)=7$$ for all x. Answer the following parts.

Direct Substitution in a Polynomial Function

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

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}

Error Analysis in Limit Calculation

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

Evaluating a Compound Limit Involving Rational and Trigonometric Functions

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

Evaluating a Limit with Radical Expressions

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

Factorization and Limit Evaluation

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

Graphical Analysis of Function Behavior from a Table

A real-world experiment recorded the concentration (in M) of a solution over time (in seconds) as sh

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:

Implicit Differentiation Involving Logarithms

Consider the curve defined implicitly by $$\ln(x) + \ln(y) = \ln(5)$$. Answer the following:

Intermediate Value Theorem Application

Consider the continuous function $$f(x)= x^3 - 4*x + 1$$. Answer the following parts.

Limit Evaluation in a Parametric Particle Motion Context

A particle’s position in the plane is given by the parametric equations $$x(t)= \frac{t^2-4}{t-2}, \

Limits Involving Absolute Value Functions

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

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

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 Bacterial Growth with a Geometric Sequence

A particular bacterial colony doubles in size every hour. The population at time $$n$$ hours is give

Modeling Temperature Change: A Real-World Limit Problem

A scientist records the temperature (in °C) of a chemical reaction during a 24-hour period using the

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

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

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

Squeeze Theorem Application

Consider the function $$f(x)=x^2\sin(\frac{1}{x})$$ for $$x\neq0$$ and $$f(0)=0$$. Answer the follow

Squeeze Theorem for an Exponential Damped Function

A physical process is modeled by the function $$h(x)= x*e^{-1/(x*x)}$$ for $$x \neq 0$$ and is defin

Squeeze Theorem with an Oscillatory Function

Consider the function $$f(x) = x \cdot \sin\left(\frac{1}{x}\right)$$ for $$x \neq 0$$ and define $$

Trigonometric Limits

Consider the functions $$g(x)=\frac{\sin(3*x)}{\sin(2*x)}$$ and $$h(x)=\frac{1-\cos(4*x)}{x^2}$$. An

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 $

Analysis of Temperature Change via Derivatives

The temperature in a chemical reactor is modeled by $$T(x)=x^3 - 6*x^2 + 9*x$$, where $$T(x)$$ is in

Analyzing a Function's Derivative from its Graph

A graph of a smooth function is provided. Answer the following questions:

Analyzing Function Behavior Using Its Derivative

Consider the function $$f(x)=x^4 - 8*x^2$$.

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

Derivation of $$h(x)= \ln(2*x+3)$$ Using the Chain Rule

Let $$h(x)= \ln(2*x+3)$$, a composition of a logarithmic and a linear function.

Derivative from First Principles: The Function $$f(x)=\sqrt{x}$$

Consider the function $$f(x) = \sqrt{x}$$. Use the definition of the derivative to find an expressio

Derivative of a Trigonometric Function

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

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

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, \\

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 Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

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

Instantaneous Rate of Change in Motion

A particle’s position along a straight line is given by $$s(t)= 4*t^3 - 12*t^2 + 9*t + 5$$, where $$

Inverse Function Analysis: Logarithmic Function

Consider the function $$f(x)=\ln(x+4)$$ defined for $$x>-4$$.

Inverse Function Analysis: Logarithmic Transformation

Consider the function $$f(x)=\ln(2*x+5)$$ defined for $$x> -\frac{5}{2}$$.

Limit Definition for a Quadratic Function

For the function $$h(x)=4*x^2 + 2*x - 7$$, answer the following parts using the limit definition of

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.

Mountain Stream Flow Adjustment

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

Quotient Rule Application

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

Secant Approximation Convergence and the Derivative

Consider the natural logarithm function $$f(x)= \ln(x)$$. Investigate its rate of change using the d

Secant Line Slope Approximations in a Laboratory Experiment

In a chemistry lab, the concentration of a solution is modeled by $$C(t)=10*\ln(t+1)$$, where $$t$$

Tangent and Normal Lines in Road Construction

A road is modeled by the quadratic function $$f(x)= \frac{1}{2}*x^2 + 3*x + 10$$.

Tangent Line Approximation

Suppose a continuous function $$f(x)$$ is differentiable with $$f(2)=8$$ and $$f'(2)=5$$. Use this i

Tangent Line to a Hyperbola

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

Tangent to an Implicit Curve

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

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:

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 in Temperature Model

A scientist models the temperature in a laboratory experiment by the function $$T(t)=\sqrt{3*t^2+2}$

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 Logarithmic and Radical Functions

Let $$R(x)=\sqrt{\ln(1+x^2)}$$.

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 Function in a Real-World Fuel Consumption Problem

A company models its fuel consumption with the function $$C(t)=\ln(5*t^2+7)$$, where $$t$$ represent

Composite Function with Inverse Trigonometric Outer Function

Consider the function $$H(x)=\arctan(\sqrt{x^2+1})$$. Answer the following parts.

Composite Functions with Multiple Layers

Let $$f(x)=\sqrt{\ln(5*x^2+1)}$$. Answer the following:

Derivative of an Inverse Trigonometric Composite

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

Differentiation Involving Inverse Sine and Exponentials

Let $$f(x)= \arcsin(e^(-x))$$, where the domain is chosen so that $$e^(-x)$$ is within [-1, 1]. Solv

Differentiation of a Complex Implicit Equation

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

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Differentiation of Inverse Trigonometric Function via Implicit Differentiation

Let $$y=\arcsin(\frac{x}{\sqrt{2}})$$. Answer the following:

Expanding Spherical Balloon

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

Implicit and Inverse Function Analysis

Consider the function defined implicitly by $$e^{x*y}+x^2=5$$. Answer the following parts.

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 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 Hyperbola

Consider the hyperbola defined by $$x*y=10$$. Answer the following parts.

Implicit Differentiation in an Elliptical Orbit

The orbit of a satellite is given by the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Answer the

Implicit Differentiation in Circular Motion

A runner is moving along a circular track described by the equation $$x^2+y^2=16$$, where $$x$$ and

Implicit Differentiation in Circular Motion

Given the circle defined by $$x^2 + y^2 = 16$$, analyze its differential properties.

Implicit Differentiation of a Logarithmic Equation

Given the equation $$\ln(x) + \ln(y) = \ln(10)$$, answer the following parts.

Implicit Differentiation with Exponentials and Logarithms

Consider the curve defined implicitly by $$x*e^(y) + \ln(y)= e$$. It is given that the point $$(1, 1

Intersection of Curves via Implicit Differentiation

Two curves are defined by the equations $$y^2= 4*x$$ and $$x^2+ y^2= 10$$. Consider their intersecti

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 an Exponential Context

Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:

Inverse Function Differentiation in Mixing Solutions

Let the function $$f(x)=2*x^3+x-5$$ model the concentration of a solution as a function of a paramet

Inverse Function Differentiation with an Exponential-Linear Function

Let $$f(x)=e^{2*x}+x$$ and assume it is invertible. Answer the following:

Inverse Function Differentiation with Exponentials and Trigonometry

Let $$f(x)=\sin(e^{x})+e^{x}$$ and assume that it is invertible. Answer the following:

Inverse Function Differentiation with Logarithmic Function

Let $$f(x) = x + \ln(x)$$ and let g denote its inverse function. Answer the following parts.

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Nested Trigonometric Function Analysis

A physics experiment produces data modeled by the function $$h(x)=\cos(\sin(3*x))$$, where $$x$$ is

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 and Composite Functions

A 10-foot ladder is leaning against a wall such that its bottom moves away from the wall according t

Related Rates in a Circular Colony

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

Temperature Reaction Rate Analysis

A chemical reaction experiment measures the temperature T (in °C) at various times t (in minutes) as

Analyzing Rate of Change in a Compound Interest Model

The amount in a bank account is modeled by $$A(t)= P e^{rt}$$, where $$P = 1000$$, r = 0.05 (per yea

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

Blood Drug Concentration

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

Chemical Reaction Rate Analysis

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

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

Economic Cost Analysis Using Derivatives

A company’s cost function for producing $$x$$ units is given by $$C(x)=0.05*x^3 - 2*x^2 + 40*x + 100

Estimating Small Changes using Differentials

In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame

Expanding Oil Spill: Related Rates Problem

An oil spill forms a circular patch on the water with area $$A = \pi r^2$$. The area is increasing a

FRQ 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

FRQ 6: Particle Motion Analysis on a Straight Line

A particle moving along a straight line has its velocity described by $$v(t) = 3*t^2 - 4*t + 2$$, wh

FRQ 14: Optimizing Box Design with Fixed Volume

A manufacturer wants to design an open-top box with a fixed volume of $$V = x^2*y = 32$$ cubic units

FRQ 20: Market Demand Analysis

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

Implicit Differentiation and Related Rates in Conic Sections

A point moves along the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$. At a certain inst

Interpretation of the Derivative from Graph Data

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

Interpreting the Graph of a Derivative

A graph is provided showing the derivative $$f'(x)$$ of an unknown function $$f(x)$$. Use the inform

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 of ln(1.05)

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

Linearization and Differential Approximations

Let $$f(x)=x^4$$. Use linearization to approximate $$f(3.98)$$ near $$x=4$$.

Linearization and Differentials Approximation

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

Logarithmic Profit Optimization

A company’s profit is modeled by $$P(x) = 50x \ln(x) - 100x$$, where $$x$$ (in thousands) is the num

Maximization of Profit

A company's revenue and cost functions are given by $$R(x)=-2x^2+120x$$ and $$C(x)=50+30x$$, respect

Medicine Dosage: Instantaneous Rate of Change

The concentration of a medicine in the bloodstream is given by $$C(t) = 25e^{-0.2t}+5$$, where $$t$$

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 Acceleration and Direction of Motion

A particle moves along a straight line with a velocity function given by $$v(t)=3*t^2-12*t+9$$, wher

Population Change Rate

The population of a town is modeled by $$P(t)= 50*e^{0.3*t}$$, where $$t$$ is in years and $$P(t)$$

Profit Optimization Analysis

The profit function for a company is given by $$P(x)=-2x^3+15x^2-40x+25$$, where x (in thousands) re

Projectile Motion and Maximum Height

A basketball is thrown such that its height (in meters) is modeled by $$h(t)= -4.9*t^2 + 14*t + 1.5$

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

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 Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (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

Analyzing Differentiability of a Piecewise Function

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

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

Continuous Compound Interest

An investment account is governed by the formula $$A(t)= A_0 * e^{r*t}$$, where $$r$$ is the continu

Discontinuity and Derivative in a Hybrid Piecewise Function

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

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 4: Intervals of Increase and Decrease Analysis

Examine the function $$f(x) = 2*x^3 - 9*x^2 + 12*x + 5$$.

FRQ 14: Projectile Motion – Determining Maximum Height

The height of a projectile (in meters) is modeled by $$h(t)= -4.9*t^2 + 20*t + 5$$, where $$t$$ is t

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

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 Logarithmic Function

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

Inverse Analysis of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases}2*x+1 & x\le 0,\\ 3*x+1 & x>0\end{cases}$$. Ans

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, &

Mean Value Theorem for a Piecewise Function

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

Mean Value Theorem in Analyzing Weather Data

A weather station recorded temperature (in $$^{\circ}C$$) at various times throughout the day. Analy

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

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

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Profit Analysis and Inflection Points

A company's profit is modeled by $$P(x)= -x^3 + 9*x^2 - 24*x + 10$$, where $$x$$ represents thousand

Rational Function Optimization

Consider the rational function $$f(x)= \frac{x^2 + 1}{x - 1}$$ defined on the interval $$[2,6]$$. An

Reservoir Sediment Accumulation

A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim

Temperature Analysis Over a Day

The temperature $$f(x)$$ (in $$^\circ C$$) at time $$x$$ (in hours) during the day is modeled by $$f

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

Accumulated Rainfall Estimation

A meteorological station recorded the rainfall rate (in mm/hr) at various times during a rainstorm.

Area Between Curves

An engineering design problem requires finding the area of the region enclosed by the curves $$y = x

Area Between Curves: $$y=x^2$$ and $$y=4*x$$

Consider the curves defined by $$f(x)= x^2$$ and $$g(x)= 4*x$$. Answer the following questions to de

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

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

Computing a Definite Integral Using the Fundamental Theorem of Calculus

Let the function be defined as $$f(x) = 2*x$$. Use the Fundamental Theorem of Calculus to evaluate t

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Definite Integral as an Accumulation Function

A generator consumes fuel at a rate given by $$f(t)=t*e^{-t}$$ (in liters per second). Answer the fo

Definite Integral Evaluation via U-Substitution

Consider the integral $$\int_{2}^{6} 3*(x-2)^4\,dx$$ which arises in a physical experiment. Evaluate

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Error Estimates in Numerical Integration

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

Evaluating a Radical Integral via U-Substitution

Evaluate the integral $$\int_{1}^{9}\sqrt{2*x+1}\,dx$$ using U-substitution. Answer the following pa

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

Finding the Area of a Parabolic Arch

An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans

FRQ2: Inverse Analysis of an Antiderivative Function

Consider the function $$ G(x)=\int_{0}^{x} (t^2+1)\,dt $$ for all real x. Answer the following parts

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

FRQ13: Inverse Analysis of an Investment Growth Function

An investment's accumulated value is given by $$ G(t)=\int_{0}^{t} \frac{1}{1+u}\,du $$ for t ≥ 0. A

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.

Function Transformations and Their Integrals

Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze

Fundamentals of Accumulation: Displacement and Total Distance

A cyclist's velocity is modeled by $$v(t)= 4 - |t-2|$$ (in m/s) for $$t$$ in the interval $$[0,4]$$.

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

Improving Area Approximations with Increasing Subintervals

Consider the function $$f(x)= \sqrt{x}$$ on the interval $$[0,4]$$. Explore how Riemann sums approxi

Marginal Cost and Total Cost

In a production process, the marginal cost (in dollars per unit) for producing x units is given by $

Modeling Water Volume in a Tank via Integration

A tank is being filled with water at a rate given by $$R(t)= \frac{50}{t+2}$$ cubic meters per minut

Net Change in Salaries: An Accumulation Function

A company models its annual bonus savings with the rate function $$B'(t)= 500*(1+\sin(t))$$ dollars

Particle Motion with Changing Direction

A particle moves along a line with its velocity given by $$v(t)=t*(t-4)$$ (in m/s) for $$0\le t\le6$

Population Growth: Accumulation through Integration

A certain population grows at a rate modeled by $$R(t)= 0.5*t^2 - 3*t + 10$$ (individuals per year),

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

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

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

Water Tank: Accumulation and Maximum Level

A water tank is being filled with water at a rate $$r_{in}(t) = 4 + \sin(t)$$ L/min and is simultane

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

Bacterial Culture with Antibiotic Treatment

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

Bernoulli Differential Equation Challenge

Consider the nonlinear differential equation $$\frac{dy}{dt} - y = -y^3$$ with the initial condition

Carbon Dating and Radioactive Decay

Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia

Chemical Reactor Mixing

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

Comparative Population Decline

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

Cooling of a Liquid

A liquid is cooling in a lab experiment. Its temperature $$T$$ (in °C) is recorded at several times

Ecosystem Nutrient Cycle

In a forest ecosystem, nitrogen is deposited from the atmosphere at a rate of $$2$$ kg/ha/year while

Epidemic Spread (Simplified Logistic Model)

In a simplified model of an epidemic, the number of infected individuals $$I(t)$$ (in thousands) is

Exact Differential Equation

Consider the differential equation $$ (2xy+y^2)\,dx+(x^2+2xy)\,dy=0$$. Answer the following question

Falling Object with Air Resistance

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

Falling Object with Air Resistance

A falling object experiences air resistance proportional to its velocity. Its motion is modeled by t

Implicit Differentiation Involving a Logarithmic Function

Consider the function defined implicitly by $$\ln(y) + x^2y = 7$$. Answer the following:

Integrating Factor Method

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

Logistic Population Growth

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt}=0.3P\Bigl(1-\frac{P}{100

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

Modeling Orbital Decay

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

Newton's Law of Cooling with Temperature Data

A thermometer records the temperature of an object cooling in a room. The object's temperature $$T(t

Population Growth in a Bacterial Culture

A bacterial culture has its population measured (in thousands) at various times (in hours). The tabl

Population Growth with Logistic Equation

A population grows according to the logistic differential equation $$\frac{dy}{dx} = 0.5*y\left(1-\f

Radioactive Decay and Half-Life

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

Reaction Rate Model: Second-Order Decay

The concentration $$C$$ of a reactant in a chemical reaction obeys the differential equation $$\frac

Separable Differential Equation

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

Separable Differential Equation: $$dy/dx = x*y$$

Consider the differential equation $$dy/dx = x*y$$ with the initial condition $$y(0)=2$$. Solve the

Separable Differential Equation: y and x

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

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 and Solution Curve Analysis

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

Temperature Regulation in a Greenhouse

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

Water Temperature Regulation in a Reservoir

A reservoir’s water temperature adjusts according to Newton’s Law of Cooling. Let $$T(t)$$ (in \(^{\

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 Damped Oscillatory Motion

A mass-spring system exhibits a damped oscillation modeled by $$f(t)=e^{-0.3*t}*\sin(t)$$ (in meters

Area Between a Cubic and a Linear Function

Consider the functions $$f(x)=x^3-3*x$$ and $$g(x)=x$$. Use integration to determine the area of the

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

Area Between Transcendental Functions

Consider the curves $$f(x)=\cos(x)$$ and $$g(x)=\sin(x)$$ on the interval $$[0,\frac{\pi}{4}]$$.

Average Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

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 Temperature of a Cooling Liquid

The temperature of a cooling liquid is modeled by $$T(t)=50*e^{-0.1*t}+20$$ (in $$^\circ C$$) for $$

Car Braking Analysis

A car decelerates with acceleration given by $$a(t)=-4e^{-t/2}$$ (in m/s²) and has an initial veloci

Charity Donations Over Time

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

Distance Traveled by a Jogger

A jogger increases her daily running distance by a fixed amount. On the first day she runs $$2$$ km,

Economic Profit Analysis via Area Between Curves

A company's revenue and cost are modeled by the linear functions $$R(x)=50*x$$ and $$C(x)=20*x+1000$

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

Filling a Container: Volume and Rate of Change

Water is being poured into a container such that the height of the water is given by $$h(t)=2*\sqrt{

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

Interpreting Integrated Quantities in a Changing System

A system is modeled by a rate function given by $$R(t)=t^2-4*t+6$$, where $$t$$ is in minutes. The c

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

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

Particle Motion on a Parametric Path

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

Pharmacokinetic Analysis

A drug concentration in the bloodstream is modeled by $$C(t)=15*e^{-0.2*t}+2$$, where $$t$$ is in ho

Pollutant Accumulation in a River

Along a 20 km stretch of a river, a pollutant enters the water at a rate described by $$p(x)=0.5*x+2

Radioactive Decay Accumulation

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

Reconstructing Position from Acceleration Data

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

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

Stress Analysis in a Structural Beam

A beam in a building experiences a stress distribution along its length given by $$\sigma(x)=100-15*

Temperature Increase in a Chemical Reaction

During a chemical reaction, the rate of temperature increase per minute follows an arithmetic sequen

Total Distance Traveled from a Velocity Profile

A particle’s velocity over the interval $$[0, 6]$$ seconds is given in the table below. Note that th

Volume of a Solid of Revolution: Disc Method

Consider the region R bounded by $$y= \sqrt{x}$$, the x-axis, and the vertical line $$x=4$$. When R

Volume of a Solid Using the Washer Method

Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev

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 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

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

Water Reservoir Inflow‐Outflow Analysis

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

Work Calculation from an Exponential Force Function

An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen

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=