AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

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

Algebraic Simplification and Limit Evaluation

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

Continuity Analysis from Table Data

The water level (in meters) in a reservoir is recorded at various times as shown in the table below.

Continuity in Piecewise Functions with Parameters

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

Continuity of Log‐Exponential Function

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - 1}{x}$$ for $$x \neq 0$$, with $$f(0)=c$$. Dete

Environmental Pollution Modeling

In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural

Evaluating a Limit with Algebraic Manipulation

Examine the function $$g(x)= \frac{\sqrt{x+9}-3}{x}$$ for $$x \neq 0$$.

Factorable Discontinuity Analysis

Let $$q(x)=\frac{x^2-x-6}{x-3}.$$ Answer the following:

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Horizontal Asymptote of a Rational Function

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

Identifying and Removing Discontinuities in a Traffic Flow Model

A model for traffic flow during rush hour is given by $$C(t)= \frac{t^2-9}{t-3}$$ for $$t \neq 3$$.

Investigating a Function with a Removable Discontinuity

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

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

Limit Definition of the Derivative for a Polynomial Function

Let $$f(x)=3*x^2-2*x+1$$. Use the limit definition of the derivative to find $$f'(2)$$.

Limits at Infinity and Horizontal Asymptotes

Examine the function $$h(x)=\frac{2*x^3-5*x+1}{4*x^3+3*x^2-2}$$.

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

One-Sided Limits and Jump Discontinuities

Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en

Pendulum Oscillations and Trigonometric Limits

A pendulum’s angular displacement from the vertical is given by $$\theta(t)= \frac{\sin(2*t)}{t}$$ f

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

Rational Function Limit and Continuity

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

Squeeze Theorem with an Oscillatory Factor

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

Water Tank Inflow with Oscillatory Variation

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

Analysis of Derivatives: Tangents and Normals

Consider the curve defined by $$y = x^3 - 6*x^2 + 9*x + 2.$$ (a) Compute the derivative $$y'$$ an

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

Biochemical Reaction Rates: Derivative from Experimental Data

The concentration of a reactant in a chemical reaction is modeled by $$C(t)= 8 - 5t + t^2$$ (in M) w

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Derivative Estimation from a Graph

A graph of a function $$f(x)$$ is provided in the stimulus. Using the graph, answer the following pa

Differentiability of a Piecewise Function

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

Differentiation from First Principles

Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Exploration of the Definition of the Derivative as a Limit

Consider the function $$f(x)=\frac{1}{x}$$ for $$x\neq0$$. Answer the following:

Implicit Differentiation for a Rational Equation

Consider the curve defined by $$\frac{x*y}{x+y} = 3$$.

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation: Square Root Equation

Consider the curve defined by $$\sqrt{x} + \sqrt{y} = \sqrt{10}$$, where $$x, y \ge 0$$.

Pollutant Levels in a Lake

A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre

Population Model Rate Analysis

A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base

Related Rates in Circle Expansion

A circular oil spill is expanding such that its radius increases at a constant rate of $$0.5\,m/s$$.

Related Rates: Sweeping Spotlight

A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit

Satellite Orbit Altitude Modeling

A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}

Secant Line Estimation for a Radical Function

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

Second Derivative and Concavity Analysis

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

Tangent Line to a Curve

Consider the function $$f(x)=\sqrt{x+4}$$ modeling a physical quantity. Analyze the behavior at $$x=

Taylor Series for sin(x) Approximation

A researcher studying oscillatory phenomena wishes to approximate the function $$f(x)=\sin(x)$$ for

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Using the Limit Definition for a Non-Polynomial Function

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

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)

Chain Rule for Inverse Trigonometric Functions in Optics

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

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Composite Exponential Logarithmic Function Analysis

Consider the function $$f(x)=\ln(2*e^{3*x}+5)$$ which models a logarithmic transformation of an expo

Composite Function Rates in a Chemical Reaction

In a chemical reaction, the concentration of a substance at time $$t$$ is given by $$C(t)= e^{-k*(t+

Composite Functions in a Biological Growth Model

A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an

Composite Functions in a Biological Model

In a biological model, the concentration of a substance is given by $$P(x)=e^{-\sqrt{x^2+1}}$$, wher

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Derivative of an Inverse Function with a Quadratic

Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of

Differentiation of an Inverse Trigonometric Composite Function

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

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Exponential Composite Function Differentiation

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

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Implicit Differentiation in a Cost-Production Model

In an economic model, the relationship between the production level $$x$$ (in units) and the average

Implicit Differentiation in Circular Motion

Consider the circle described by $$x^2+y^2=49$$, representing a particle's path. Answer the followin

Implicit Differentiation Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Implicit Differentiation of a Product Equation

Consider the equation $$ x*y + x + y = 10 $$.

Implicit Differentiation with Logarithmic Functions

Let $$x$$ and $$y$$ be related by the equation $$\ln(x*y) + x - y = 0$$.

Implicit Differentiation: Second Derivative of Exponential-Trigonometric Equation

Consider the equation $$e^{x*y} + \sin(y) - x^2 = 0$$ where $$y$$ is defined implicitly as a functio

Implicit Equation with Logarithmic and Exponential Terms

The relation $$\ln(x+y)+e^{x-y}=3$$ defines y implicitly as a function of x. Answer the following pa

Inverse Function Derivative for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Inverse Function Differentiation in Economics

A product’s demand is modeled by a one-to-one differentiable function $$Q = f(P)$$, where $$P$$ is t

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Rainwater Harvesting System

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

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

Taylor/Maclaurin Polynomial Approximation for a Logarithmic Function

Let $$f(x) = \ln(1+3*x)$$. Develop a second-degree Maclaurin polynomial, determine its radius of con

Temperature Modeling and Composite Functions

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

Water Tank Composite Rate Analysis

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

Analyzing Concavity through the Second Derivative

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

Analyzing Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

Analyzing Motion on an Inclined Plane

A sled moves along an inclined plane with displacement given by $$s(t)=5*t^2-0.5*t^3$$, where $$s$$

Analyzing Rate of Approach in a Pursuit Problem

Two cars are traveling on perpendicular roads. Car A is moving east at 60 km/h and is 3 km from the

Bacterial Culture Dynamics

In a bioreactor, bacteria are introduced at a rate given by $$I(t)=200e^{-0.1t}$$ (cells per minute)

Chemical Reaction Temperature Change

In a laboratory experiment, the temperature T (in °C) of a reacting mixture is modeled by $$T(t)=20+

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Differentiation and Concavity for a Non-Motion Problem: Water Filling a Tank

The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Inflating Balloon: Radius and Surface Area

A spherical balloon is being inflated such that its volume increases at a constant rate of 12 cm³/s.

Interpreting the Derivative in Straight Line Motion

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

Inversion in a Light Intensity Decay Model

A laboratory experiment records the decay of light intensity over time, modeled by $$f(t)=80*e^{-0.2

L'Hôpital's Analysis

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

L'Hôpital's Rule in Inverse Function Context

Consider the function $$f(x)=x+e^{-x}$$. Although its inverse cannot be expressed in closed form, an

L'Hospital's Rule for Indeterminate Limits

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ using L'Hospita

Limits and L'Hôpital's Rule Application

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

Linearization Approximation

Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Particle Motion Analysis

A particle moves along a straight line and its position at time $$t$$ seconds is given by $$s(t)= t^

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Pollution Decay and Inversion

A model for pollution decay is given by the function $$f(t)=\frac{100}{1+t}$$ where $$t\ge0$$ repres

Population Decline Modeled by Exponential Decay

A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the

Population Growth and Change: A Nonlinear Model

The population of a bacterial culture is modeled by $$P(t)=\frac{500e^{0.3*t}}{1+e^{0.3*t}}$$, where

Population Growth Rate

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

Population Growth: Rate of Change Analysis

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

Road Trip Distance Analysis

During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is

Sliding Ladder

A 10 m long ladder rests against a vertical wall. Let $$x$$ be the distance from the foot of the lad

Spherical Balloon Inflation

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

Water Filtration Plant Analysis

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

Air Pollution Control in an Enclosed Space

In an enclosed environment, contaminated air enters at a rate of $$I(t)=15-\frac{t}{2}$$ m³/min and

Analysis of a Function with Oscillatory Behavior

Consider the function $$f(x)=x+\sin(x)$$. Answer the following parts:

Application of Rolle's Theorem

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

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima

Chemical Reaction Rate

During a chemical reaction, the concentration of a reagent (in M) is measured over time (in minutes)

Derivatives and Inverses

Consider the function $$f(x)=\ln(x)+x$$ for x > 0, and let g(x) denote its inverse function. Answer

Energy Consumption Rate Model

A household's energy consumption rate (in kW) is modeled by $$E(t) = 2*t^2 - 8*t + 10$$, where t is

Error Approximation using Linearization

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

Evaluating an Improper Integral using Series Expansion

The function $$I(x)=\sum_{n=0}^\infty (-1)^n * \frac{(2*x)^{n}}{n!}$$ converges to a known function.

Graphical Analysis of a Differentiable Function

A function $$f(x)$$ is given, and its graph appears as shown in the stimulus. Answer the following p

Implicit Differentiation and Tangent to an Ellipse

Consider the ellipse defined by the equation $$4*x^2 + 9*y^2 = 36$$. Answer the following parts:

Increasing/Decreasing Intervals for a Rational Function

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

Integration of a Series Representing an Economic Model

An economist models the marginal cost by the power series $$MC(q)=\sum_{n=0}^\infty (-1)^n * \frac{q

Inverse Analysis for a Function with Multiple Transformations

Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.

Investigation of Extreme Values on a Closed Interval

For a particle moving along a path given by $$f(x)=x^3-6*x^2+9*x+5$$ where $$x\in[0,5]$$, analyze it

Logarithmic-Exponential Function Analysis

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

Numerical Integration using Taylor Series for $$\cos(x)$$

Approximate the integral $$\int_{0}^{0.5} \cos(x)\,dx$$ by using the Maclaurin series for $$\cos(x)$

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

Related Rates: Changing Shadow Length

A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ increases at a constant rate of $$\fr

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

Revenue Optimization in Business

A company’s price-demand function is given by $$P(x)= 50 - 0.5*x$$, where $$x$$ is the number of uni

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Taylor Series for $$e^{-x^2}$$

Consider the function $$f(x)=e^{-x^2}$$. In this problem, you will derive its Maclaurin series up to

Wireless Signal Attenuation

A wireless signal, originally at an intensity of 80 units, passes through a series of walls. Each wa

Accumulated Change via U-Substitution

Evaluate the definite integral representing the accumulated amount of a substance given by $$\int_{1

Antiderivative Application in Crop Growth

A crop field grows at a rate modeled by the function $$G'(t)=4*t-3$$ (in square meters per week). Th

Antiderivative with Initial Condition

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

Applying the Fundamental Theorem of Calculus

Consider the function $$f(x)=2*x$$. Use the Fundamental Theorem of Calculus to evaluate the definite

Approximating Water Volume Using Riemann Sums

A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a

Area and Volume for an Exponential Function Region

Consider the curve $$y=e^{-x}$$ for $$x\ge0$$. Answer the following:

Area Between the Curves f(x)=x² and g(x)=2x+3

Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete

Area Under a Piecewise-Defined Curve with a Jump Discontinuity

Consider the function $$ g(x)= \begin{cases} 2x+1 & \text{if } 0 \le x < 2, \\ 3x-2 & \text{if } 2 \

Average Value of an Exponential Function

For the function $$f(x)= x*e^{-x}$$, determine the average value on the interval $$[0,2]$$. Answer t

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Car Motion: From Acceleration to Distance

A car has an acceleration given by $$a(t)= 3 - 0.5*t$$ m/s² for time t in seconds. The initial velo

Continuous Antiderivative for a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,

Convergence of an Improper Integral

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{x^{p}}\,dx$$, where $$p$$ is a positive

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Finding Area Between Two Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.

Finding the Area Between Curves

Find the area of the region bounded by the curves $$y=4-x^2$$ and $$y=x$$.

Integration Involving Trigonometric Functions

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{2}} \cos(2*t)\,dt.$$

Integration of a Complex Trigonometric Function

Evaluate the integral $$\int_{0}^{\pi/2} 4*\cos^3(t)*\sin(t) dt$$.

Midpoint Riemann Sum Estimation

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

Recovering Position from Velocity

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

Region Bounded by a Parabola and a Line: Area and Volume

Consider the region bounded by the curves $$y=x^{2}$$ and $$y=2*x+3$$. Answer the following:

Rewriting Functions for Integration

Consider the function $$f(x)=\frac{1}{\sqrt{x}} - \frac{1}{x+1}$$. Rewrite this function in a form s

Riemann Sum Approximation with Irregular Intervals

A set of experimental data provides the values of a function $$f(x)$$ at irregularly spaced points a

Riemann Sum Approximations: Midpoint vs. Trapezoidal

Consider the function $$f(x)=e^(-x)$$ on the interval [0, 3]. Compare the approximations for the def

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

Riemann Sums and Inverse Analysis from Tabular Data

Let $$f(x)= 2*x + 1$$ be defined on the interval $$[0,5]$$. Answer the following questions about $$f

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Tank Filling Problem

Water flows into a tank at a rate given by $$R(t)=8e^{-0.5*t}+2$$ (in liters per minute) for $$t\geq

Temperature Function Analysis with Inverses

A temperature profile over time is given by $$f(t)= \ln(2*t + 3)$$ for $$t \ge 0$$ (with temperature

U-Substitution Integration

Evaluate the definite integral $$\int_1^5 (2*x-3)^4 dx$$ using the method of u-substitution.

Volume of a Solid by the Shell Method

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

Water Accumulation in a Reservoir

A reservoir receives water at an inflow rate modeled by $$r(t)=\frac{20}{t+1}$$ (in cubic meters per

Bacterial Growth with Time-Dependent Growth Rate

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

Braking of a Car

A car decelerates according to the differential equation $$\frac{dv}{dt} = -k*v$$, where k is a posi

Chemical Reaction and Separable Differential Equation

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

Chemical Reaction Rate

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

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Cooling Coffee Data Analysis

A cup of coffee cools down in a room according to Newton's Law of Cooling. The temperature $$T(t)$$

Cooling of a Metal Rod

A metal rod cools according to the differential equation $$\frac{dT}{dt}=-k\,(T-25)$$ with an initia

Cooling of an Object Using Newton's Law of Cooling

An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto

Economic Model: Differential Equation for Cost Function

A company’s marginal cost is represented by $$MC(x)=\frac{dC}{dx}=3+2*x$$, where $$x$$ is the number

Environmental Modeling Using Differential Equations

The concentration $$C(t)$$ of a pollutant in a lake is modeled by the differential equation $$\frac{

Estimating Total Change from a Rate Table

A car's velocity (in m/s) is recorded at various times according to the table below:

Exponential Growth with Variable Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the

FRQ 9: Epidemiological Model Differential Equation

An epidemic evolves according to the differential equation $$\frac{dI}{dt}=r*I*(M-I)$$, where $$I(t)

FRQ 11: Linear Differential Equation via Integrating Factor

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

FRQ 18: Enzyme Reaction Rates

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

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

Integration Factor Method

Solve the linear differential equation $$\frac{dy}{dx}+ 2*y = \sin(x)$$ with the initial condition $

Logistic Growth in Populations

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

Logistic Model in Population Dynamics

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

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Mixing Problem: Salt Water Tank

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

Non-linear Differential Equation using Separation of Variables

Consider the differential equation $$\frac{dy}{dx}= \frac{x*y}{x^2+1}$$. Answer the following questi

Phase-Plane Analysis of a Nonlinear Differential Equation

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

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Projectile Motion with Drag

A projectile is launched horizontally with an initial velocity $$v_0$$. Due to air resistance, the h

Separable DE with Exponential Function

Solve the differential equation $$\frac{dy}{dx}=y\cdot\ln(y)$$ for y > 0 given the initial condition

Simplified Predator-Prey Model

A simplified model for a predator population is given by the differential equation $$\frac{dP}{dt} =

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Tumor Growth Under Chemotherapy

A tumor's size $$S(t)$$ (in cm³) grows at a rate proportional to its size, at $$0.08*S(t)$$, but che

Water Tank Inflow-Outflow Model

A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters

Arc Length of the Logarithmic Curve

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

Area Between Curves in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Area Under an Exponential Decay Curve

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

Average Population Density

In an urban study, the population density (in thousands per km²) of a city is modeled by the functio

Average Population in a Logistic Model

A population is modeled by a logistic function $$P(t)=\frac{500}{1+2e^{-0.3*t}}$$, where $$t$$ is me

Average Temperature in a City

An urban planner recorded the temperature variation over a 24‐hour period modeled by $$T(t)=10+5*\si

Car Motion Analysis

A car's acceleration is given by $$a(t) = 4 - 2 * t$$ (in m/s²) for $$0 \le t \le 4$$ seconds. The c

Center of Mass of a Rod with Variable Density

A rod extending along the x-axis from $$x=0$$ to $$x=10$$ meters has a density given by $$\rho(x)=2+

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Force on a Submerged Plate

A vertical rectangular plate is submerged in water. The plate is 3 m wide and extends from a depth o

Motion Analysis of a Car

A car has an acceleration given by $$a(t)=2-0.5*t$$ for $$0\le t\le8$$ seconds. The initial velocity

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (in m

Particle on a Line with Variable Acceleration

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

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

Power Series Representation for ln(x) about x=4

The function $$f(x)=\ln(x)$$ is to be expanded as a power series centered at $$x=4$$. Find this seri

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

Rainfall Accumulation Analysis

A meteorological station recorded the following rainfall data (in mm) over a 24‐hour period. The rai

Salt Concentration in a Mixing Tank

A tank initially contains 50 L of water with 5 g of salt. A salt solution with a concentration of 0.

Volume of a Wavy Dome

An auditorium roof has a varying cross-sectional area given by $$A(x)=\pi*(1 + 0.1*\sin(x))^2$$ (in

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Water Tank Dynamics: Inflow and Outflow

A water tank receives water through an inflow at a rate given by $$I(t)=10+2*t$$ (liters per minute)

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

Work Done in Lifting a Cable

A cable of length 10 m with a uniform mass density of 5 kg/m hangs vertically from a winch. The winc

Work Done Pumping Water

A water tank is shaped like an inverted circular cone with a height of $$10$$ m and a top radius of

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

Catching a Thief: A Parametric Pursuit Problem

A police car and a thief are moving along a straight road. Initially, both are on the same road with

Circular Motion Analysis

A particle moves in a circle according to the vector-valued function $$\vec{r}(t)=<3\cos(t),\, 3\sin

Comparing Parametric, Polar, and Cartesian Representations

An object moves along a curve described by the parametric equations $$x(t)= \frac{t}{1+t^2}$$ and $$

Continuity Analysis of a Discontinuous Parametric Curve

Consider the parametric curve defined by $$x(t)= \begin{cases} t^2, & t < 1 \\ 2*t - 1, & t \ge 1 \

Curvature of a Vector-Valued Function

Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given

Designing a Race Track with Parametric Equations

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

Dynamics in Polar Coordinates

A particle moves such that its polar coordinates are given by $$ r(\theta)=1+\theta $$, where $$ \th

Enclosed Area of a Parametric Curve

A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Exponential Decay in Vector-Valued Functions

A particle moves in the plane with its position given by the vector-valued function $$\vec{r}(t)=\la

Helical Particle Motion

A particle travels along a helical path described by $$\vec{r}(t)= \langle \cos(t),\; \sin(t),\; t \

Integration of Vector-Valued Acceleration

A particle's acceleration is given by the vector function $$\mathbf{a}(t)=\langle 2*t,\; 6-3*t \rang

Motion Analysis via a Vector-Valued Function

An object's position is described by the vector function $$\mathbf{r}(t)= \langle e^{-t}, \; \ln(1+t

Motion on a Circle in Polar Coordinates

A particle moves along a circular path of constant radius $$r = 4$$, with its angle given by $$θ(t)=

Optimization in Parametric Projectile Motion

A projectile is launched from the ground with an initial speed of $$20\,m/s$$ at an angle $$\alpha$$

Parametric Intersection and Tangency

Two curves are given in parametric form by: Curve 1: $$x_1(t)=t^2,\, y_1(t)=2t$$; Curve 2: $$x_2(s

Particle Motion on an Elliptical Arc

A particle moves along a curve described by the parametric equations $$x(t)= 2*cos(t)$$ and $$y(t)=

Periodic Motion: A Vector-Valued Function

A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$

Reparameterization between Parametric and Polar Forms

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

Symmetry and Area in Polar Coordinates

Consider the polar curve given by $$r=2\cos(\theta)$$. Answer the following:

Vector-Valued Function and Derivatives

Consider the vector-valued function given by $$ r(t)=\langle e^t*\cos(t),\; e^t*\sin(t) \rangle $$ f

Vector-Valued Functions and Kinematics

A particle moves in space with its position given by the vector-valued function $$\vec{r}(t)= \langl

Vector-Valued Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$