AP Calculus BC FRQ Room

Absolute Value Function Limits

Examine the function $$f(x)=\frac{|x-2|}{x-2}$$.

Analysis of a Rational Inflow Function with a Discontinuity

A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$

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 Analysis in Road Ramp Modeling

A highway ramp is modeled by the function $$y(x)= \frac{(x-3)(x+2)}{x-3}$$ for $$x\neq3$$, where x i

Continuity Analysis of a Rational-Piecewise Function

Consider the function $$r(x)=\begin{cases} \frac{x^2-1}{x-1} & x<0, \\ 2*x+c & x\ge0. \end{cases}$$

Continuity in Piecewise Defined Functions

Consider the piecewise function $$f(x)= \begin{cases} x^2+1, & \text{if } x \leq 3 \\ 2*x+k, & \text

Continuity in Piecewise Functions with Parameters

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

Exponential Function Limit and Continuity

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

Exponential Function Limits at Infinity

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

Graphical Analysis of Discontinuities

A graph of a function is provided that shows multiple discontinuities, including a removable discont

Horizontal and Vertical Asymptotes of a Rational Function

Let $$h(x)=\frac{2*x^2-3*x+1}{x^2-1}.$$ Answer the following:

Limit Evaluation Involving Trigonometric Functions

Let $$f(x)=\frac{\sin(4*x)}{\tan(2*x)}$$ for $$x\neq0$$, with f(0) defined separately. Answer the

Limits Involving Trigonometric Functions and the Squeeze Theorem

Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Limits with Infinite Discontinuities

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

Parameterized Function Continuity and Differentiability

Let $$f(x)= \begin{cases} \frac{e^x - \ln(1+2x) - 1}{x} & x \neq 0 \\ k & x=0 \end{cases}.$$ Determi

Rational Function Analysis with Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3) * (x-1)}{(x-1)}$$ for $$x \neq 1$$. This function exhibits

Related Rates: Changing Shadow Length

A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol

Seasonal Temperature Curve Analysis

A graph represents the average daily temperature (in $$^\circ C$$) as a function of the day of the y

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

Understanding Behavior Near a Vertical Asymptote

For the function $$f(x)=\frac{1}{(x-2)^2}$$, answer the following: (a) Determine $$\lim_{x\to2} f(x)

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

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Circular Motion Analysis

An object moves along a circular path with angular position given by $$\theta(t)=2*t-0.1*t^2$$ (in r

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Use the limit definition of the derivative to find $$f'(x)$$.

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 in Biological Growth Models

In a biological experiment, the rate of resource consumption is modeled by $$R(t)=\frac{\ln(t^2+1)}{

Differentiation of an Exponential Function

Let $$f(x)=e^{2*x}$$. Answer the following:

Differentiation of Implicitly Defined Functions

An ellipse is defined by the equation $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$. Use implicit differenti

Estimating Instantaneous Acceleration from Velocity Data

An object's velocity (in m/s) is recorded over time as shown in the table below. Use the data to ana

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Graph Interpretation: Average vs Instantaneous Rates

A function is represented in the table below. Analyze the difference between average and instantaneo

Implicit Differentiation on an Ellipse

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

Implicit Differentiation: Cost Allocation Model

A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1

Instantaneous Rate of Change of a Polynomial Function

Consider the function $$f(x)=2*x^3 - 5*x^2 + 3*x - 7$$ which represents the position (in meters) of

Logarithmic Differentiation

Let $$T(x)= (x^2+1)^{3*x}$$ model a quantity with variable growth characteristics. Use logarithmic d

Logarithmic Differentiation in Temperature Modeling

The temperature distribution along a rod is modeled by the function $$T(x)=\ln(5*x^2+1)*e^{-x}$$. He

Logarithmic Differentiation: Equating Powers

Consider the equation $$y^x = x^y$$ that relates $$x$$ and $$y$$ implicitly.

Optimization and Tangent Lines

A rectangular garden is to be constructed along a river with 100 meters of fencing available for thr

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: Constant Area Rectangle

A rectangle maintains a constant area of $$A = l*w = 50$$ m², where the length l and width w vary wi

Secant Line Estimation for a Radical Function

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

Sediment Accumulation in a Dam

Sediment enters a dam reservoir at a rate of $$S_{in}(t)=5\ln(t+1)$$ kg/hour, while sediment is remo

Tracking a Car's Velocity

A car moves along a straight road according to the position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$,

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Calculating an Inverse Trigonometric Derivative in a Physics Context

A pendulum's angle is modeled by $$\theta = \arcsin(0.5*t)$$, where $$t$$ is time in seconds and $$\

Chain Rule in a Nested Composite Function

Consider the function $$f(x)= \sin\left(\ln((2*x+1)^3)\right)$$. Answer the following parts:

Chain Rule with Nested Logarithmic and Exponential Functions

Consider the function $$f(x)= \sqrt{\ln(5*x + e^{x})}$$. Differentiate this function using the chain

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

Composite Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

Composite Functions in Population Growth

Consider a population $$P(t) = f(g(t))$$ modeled by the functions $$g(t) = 2 + t^2$$ and $$f(u) = 10

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Design Optimization for a Cylindrical Can

A manufacturer wants to design a cylindrical can that holds a fixed volume of $$V = 1000$$ cm³. The

Differentiation in an Economic Cost Function

The cost of producing $$q$$ units is modeled by $$C(q)= (5*q)^{3/2} + 200*\ln(1+q)$$. Differentiate

Differentiation of Composite Exponential and Trigonometric Functions

Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe

Geometric Context: Sun Angle and Shadow Length Inverse Function

Consider the function $$f(\theta)=\tan(\theta)+\theta$$ for $$0<\theta<\frac{\pi}{2}$$, which models

Higher Order Implicit Differentiation in a Nonlinear Model

Assume that \(x\) and \(y\) are related by the nonlinear equation $$e^{x*y} + x - \ln(y) = 5$$ with

Implicit Differentiation in a Non-Standard Function

Consider the equation $$x^2*y + \sin(y) = x$$, which implicitly defines $$y$$ as a function of $$x$$

Implicit Differentiation Involving Inverse Trigonometric Functions

Consider the equation $$\theta = \arctan\left(\frac{y}{x}\right)$$, where $$y$$ is a differentiable

Implicit Differentiation with Exponential and Trigonometric Components

Consider the relation $$ (x^2 + y^2) * e^{y} = x $$. Answer the following:

Inverse Function Differentiation in Economics

In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan

Inverse Function Differentiation with a Logarithmic Function

Let $$ f(x)= \ln(x+3) $$. Consider its inverse function $$ f^{-1}(y) $$.

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Inverse Trigonometric Functions: Analysis and Application

Consider the function $$f(x) = \arctan(3*x)$$. Analyze its rate of change and the equation of the ta

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Related Rates: Ladder Sliding Down a Wall

A ladder of length $$10\, m$$ leans against a wall such that its position is governed by $$x^2 + y^2

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Second Derivative of an Implicit Function

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

Water Tank Composite Rate Analysis

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

Analysis of a Piecewise Function with Discontinuities

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

Analyzing a Production Cost Function

A company's cost function for producing goods is given by $$C(x)=x^3-12x^2+40x+100$$, where x repres

Application of L’Hospital’s Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 2x + 1}{3x^3 + 7}$$ using L’Hospital’s Rule.

Chemistry: Rate of Change in a Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)= 50e^{-0.2*t}+5$$, wher

Conical Tank Filling

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

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

Continuity in a Piecewise-Defined Function

Let $$g(x)= \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 2*x + k & \text{if } x \ge 1 \end{cases}$$.

Cooling Analysis using Newton’s Law of Cooling

An object cools in a room according to Newton's Law of Cooling, given by $$T(t)=T_{env}+ (T(0)-T_{en

Differentiation of a Product Involving Exponentials and Logarithms

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

Ellipse Tangent Line Analysis

Consider the ellipse given by the implicit equation $$9*x^2 + 4*y^2 = 36$$. Answer the following par

Engineering Applications: Force and Motion

A force acting on a 4 kg object is given by $$F(t)= 12*t - 3$$ (Newtons), where $$t$$ is in seconds.

Instantaneous vs. Average Rate of Change in Temperature

A rod's temperature along its length is modeled by $$T(x)=20\ln(x+1)+e^{-x}$$, where x (in meters) i

Integration Region: Exponential and Polynomial Functions

Let the region be bounded by the curves $$y = x^2$$ and $$y = e^x$$. Analyze the area of the region

Interpreting Derivatives from Experimental Concentration Data

An experiment records the concentration (in moles per liter) of a substance over time (in minutes).

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

Linearization Approximation Problem

Given the function $$f(x)=\sqrt{x+4}$$, use linearization to approximate the value of $$f(5.1)$$.

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Linearization of a Power Function

Let $$f(x)=x^4$$. Use linearization at $$x=4$$ with $$\Delta x=-0.02$$ to approximate $$(3.98)^4$$.

Maximizing Revenue in a Business Model

A company’s revenue as a function of price is given by $$R(p)= p*(100 - 2*p)$$, where $$p$$ is the p

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Mixed Quadratic Relation

Consider the implicit equation $$x^2 + x*y + y^2 = 7$$.

Optimization of Material Cost for a Pen

A rectangular pen is to be built against a wall, requiring fencing on only three sides. The area of

Production Cost Analysis

A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa

Quadratic Function Inversion with Domain Restriction

Let $$f(x)=x^2+4$$. Since quadratic functions are not one-to-one over all real numbers, consider an

Savings Account Dynamics

A bank account receives deposits at a rate of $$I(t)=50+10t$$ (dollars per month) and experiences wi

Series Approximation in Population Dynamics

A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans

Series Expansion in Vibration Analysis

A vibrating system has its displacement modeled by $$y(t)= \sum_{n=0}^{\infty} \frac{(-1)^n (2t)^{2*

Series-Based Analysis of Experimental Data

An experiment models a measurement function as $$g(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x/4)^n}{n+1

Tangent Line and Rate of Change Analysis

A scientist collected experimental data on the concentration of a chemical, and the graph below repr

Temperature Change of Coffee: Exponential Cooling

The temperature of a cup of coffee is modeled by the function $$x(t)= 70 + 50e^{-0.1*t}$$, where $$t

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Analysis of a Rational Function

Consider the function $$f(x)= \frac{x^2+4}{x+1}$$ defined for $$x\neq -1$$. Analyze its behavior.

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

Application of Rolle's Theorem

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

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:

Application of the Mean Value Theorem in Motion

A car's position on a straight road is given by the function $$s(t)=t^3-6*t^2+9*t+5$$, where t is in

Average and Instantaneous Velocity Analysis

A bird’s position is given by $$s(t)=2*t^2-t+1$$ (in meters) for $$t\in[0,3]$$ seconds.

Concavity and Inflection Points of an Exponential Log Function

Consider the function $$f(x)= x\,e^{-x} + \ln(x)$$ for $$x > 0$$. Analyze the concavity of f.

Concavity and Points of Inflection

Consider the function $$f(x)=x^3 - 6*x^2 + 9*x + 2$$. Analyze the concavity of the function using th

Determining Convergence and Error Analysis in a Logarithmic Series

Investigate the series $$L(x)=\sum_{n=1}^\infty (-1)^{n+1} * \frac{(x-1)^n}{n}$$, which represents a

Drug Dosage Infusion

A patient receives an intravenous drug infusion at a rate given by $$D(t)=4*\exp(-0.2*t)$$ mg/min. A

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Extreme Value Theorem for a Piecewise Function

Let $$h(x)$$ be defined on $$[-2,4]$$ as $$ h(x)= \begin{cases} -x^2+4 & \text{if } x \le 1, \\ 2x-

Function Behavior Analysis

Consider the function \( f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1 \). Answer the following parts:

Garden Design Optimization

A gardener wants to design a rectangular garden adjacent to a river, so that fencing is required for

Increasing/Decreasing Intervals for a Rational Function

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

Linear Particle Motion Analysis

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t\g

Maclaurin Approximation for $$\ln(1+2*x)$$

Consider the function $$f(x)=\ln(1+2*x)$$. In this problem, you will generate the Maclaurin series f

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

Optimization Problem: Designing a Box

A company needs to design an open-top box with a square base that has a volume of $$32\,m^3$$. The c

Optimization with a Combined Logarithmic and Exponential Function

A company's revenue is modeled by $$R(x)= x\,e^{-0.05x} + 100\,\ln(x)$$, where x (in hundreds) repre

Parameter Identification in a Log-Exponential Function

The function $$f(t)= a\,\ln(t+1) + b\,e^{-t}$$ models a decay process with t \(\geq 0\). Given that

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Radius of Convergence and Series Manipulation in Substitution

Let $$f(x)=\sum_{n=0}^\infty c_n * (x-2)^n$$ be a power series with radius of convergence $$R = 4$$.

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Rate of Change in a Logarithmic Temperature Model

A cooling process is modeled by the temperature function $$T(t)= 100 - 20\,\ln(t+1)$$, where t is me

Relative Extrema Using the First Derivative Test

Consider the function $$ f(x)=e^{-x^2}.$$ Answer the following parts:

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Taylor Series for $$e^{\sin(x)}$$

Let $$f(x)=e^{\sin(x)}$$. First, obtain the Maclaurin series for $$\sin(x)$$ up to the $$x^3$$ term,

Wireless Signal Attenuation

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

Accumulation Function and the Fundamental Theorem of Calculus

Let $$F(x) = \int_{2}^{x} \sqrt{1 + t^3}\, dt$$. Answer the following parts regarding this accumulat

Arc Length of an Architectural Arch

An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len

Biomedical Modeling: Drug Concentration Dynamics

A drug concentration in the bloodstream is modeled by $$f(t)= 5\left(1 - e^{-0.3*t}\right)$$ for $$t

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 Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

Definite Integral via the Fundamental Theorem of Calculus

Consider the linear function $$f(x)=2*x+3$$ defined on the interval $$[1,4]$$. A graph of the functi

Determining Antiderivatives and Initial Value Problems

Suppose that $$F(x)$$ is an antiderivative of the function $$f(x)=5*x^4 - 2*x + 3$$, and that it is

Error Analysis in Riemann Sum Approximations

Consider approximating the integral $$\int_{0}^{2} x^3\,dx$$ using a left-hand Riemann sum with $$n$

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Integration by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Parametric Integral and Its Derivative

Let $$I(a)= \int_{0}^{a} \frac{t}{1+t^2}dt$$ where a > 0. This integral is considered as a function

Particle Motion with Changing Velocity Signs

A particle is moving along a line with its velocity given by $$v(t)= 6 - 4*t$$ (in m/s) for t betwee

Population Model Inversion and Accumulation

Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Series Convergence and Integration with Power Series

Consider the power series $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$, which represents $$

Signal Energy through Trigonometric Integration

A signal is described by $$f(t)=3*\sin(2*t)+\cos(2*t)$$. The energy of the signal over one period

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

Trapezoidal and Riemann Sums from Tabular Data

A scientist collects data on the concentration of a chemical over time as given in the table below.

Trapezoidal Approximation of a Definite Integral from Tabular Data

The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

U-Substitution Integration

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

Variable Interest Rate and Continuous Growth

An investment grows continuously with a variable interest rate given by $$r(t)=0.05+0.01*t$$. The in

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Analysis of a Piecewise Function with Potential Discontinuities

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

Capacitor Charging in an RC Circuit

In an RC circuit, when a capacitor is charging, the voltage across the capacitor, $$V(t)$$, satisfie

Capacitor Discharge in an RC Circuit

In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio

Chemical Reaction Rate

A chemical reaction causes the concentration $$A(t)$$ of a reactant to decrease according to the rat

Chemical Reaction Rate Modeling

In a chemical reaction, the concentration $$C(t)$$ (in moles per liter) of a reactant decreases acco

Coffee Cooling: Differential Equation Application

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

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

FRQ 4: Newton's Law of Cooling

A cup of coffee cools according to Newton's Law of Cooling, where the temperature $$T(t)$$ satisfies

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 12: Bacterial Growth with Limiting Resources

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where

Implicit Differentiation and Homogeneous Equation

Consider the differential equation $$\frac{dy}{dx}= \frac{x+y}{x-y}$$. Answer the following:

Implicit Solution of a Separable Differential Equation

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

Integrating Factor Application

Solve the first order linear differential equation $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ with the initi

Logistic Growth Model

A population P grows according to the logistic differential equation $$\frac{dP}{dt} = rP\left(1-\fr

Mixing Problem in a Tank

A tank initially contains $$100$$ liters of water with $$5$$ kg of dissolved salt. Brine with a salt

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie

Particle Motion with Damping

A particle moving along a straight line is subject to damping and its motion is modeled by the secon

Population Saturation Model

Consider the differential equation $$\frac{dy}{dt}= \frac{k}{1+y^2}$$ with the initial condition $$y

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Separable DE with Trigonometric Component

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

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx}= \frac{4*x}{y}$$ with the initial condition $$y(0

Simplified Predator-Prey Model

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

Slope Field and Solution Curve Sketching

Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq

Solution Curve from Slope Field

A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio

Solving a Separable Differential Equation

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

Tank Draining Problem

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

Accumulated Interest in a Savings Account

An investor’s savings account experiences continuous deposits and withdrawals. The deposit rate is g

Accumulated Rainfall

The rate of rainfall over a 12-hour storm is modeled by $$r(t)=4*\sin\left(\frac{\pi}{12}*t\right) +

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Analyzing the Inverse of an Exponential Function

Let $$f(x)=\ln(2*x+1)$$, defined for $$x\ge0$$.

Arc Length of a Curve

Consider the curve defined by $$y= \ln(\cos(x))$$ for $$0 \le x \le \frac{\pi}{4}$$. Determine the l

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

Area of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

Average and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

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

Cost Analysis of a Water Channel

A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=

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 $

Draining a Conical Tank Related Rates

Water is draining from a conical tank that has a height of $$8$$ meters and a top radius of $$3$$ me

Implicit Differentiation with Trigonometric Function

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

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 with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

Projectile Motion with Constant Acceleration

A ball is thrown upward and moves under the constant acceleration due to gravity $$a(t)=-9.8$$ (in m

Rainfall Accumulation Analysis

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

River Crossing: Average Depth and Flow Calculation

The depth of a river along a 100-meter cross-section is modeled by $$d(x)=2+\cos\left(\frac{\pi}{50}

Solid of Revolution via Disc Method

Consider the region bounded by the curve $$y = x^2$$ and the x-axis for $$0 \le x \le 3$$. This regi

Volume by the Washer Method: Between Curves

Consider the region between the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x$$ between their

Volume of a Solid by the Disc Method

Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Volume of a Solid with Square Cross Sections

Consider the region bounded by the curve $$f(x)= 4 - x^2$$ and the x-axis for $$x \in [-2,2]$$. A so

Volume of a Solid with Variable Cross Sections

A solid has a cross-sectional area perpendicular to the x-axis given by $$A(x)=4-x^2$$ for $$x\in[-2

Volume of an Arch Bridge Support

The arch of a bridge is modeled by $$y=12-\frac{x^2}{4}$$ for $$x\in[-6,6]$$. Cross-sections perpend

Work Done by a Variable Force

A force acting on an object is given by the function $$F(x)=3*x^2$$ (in Newtons). The object moves a

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=5+3*x$$ (in newtons), where $$x$$ is the displacement

Analysis of Particle Motion Using Parametric Equations

A particle moves in the plane with its position defined by $$x(t)=4*t-2$$ and $$y(t)=t^2-3*t+1$$, wh

Arc Length of a Cycloid

Consider the cycloid defined by the parametric equations $$x(t)= t - \sin(t)$$ and $$y(t)= 1 - \cos(

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t) = t^2$$ and $$y(t) = t^3$$ for $$0 \le t \le 2$$.

Arc Length of a Parametric Curve with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Area Between Polar Curves

Consider the polar curves defined by $$r_1= 4$$ and $$r_2= 2+2\cos(\theta)$$. Find the area of the r

Comparing Representations: Parametric and Polar

A curve is represented by the parametric equations $$x(t)=3\cos(t)-\sin(t)$$ and $$y(t)=3\sin(t)+\co

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

Conversion Between Polar and Cartesian Coordinates

Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.

Curvature of a Space Curve

Let the space curve be defined by $$r(t)= \langle t, t^2, \ln(t+1) \rangle$$ for $$t > -1$$.

Curve Analysis and Optimization in a Bus Route

A bus follows a route described by the parametric equations $$x(t)=t^3-3*t$$ and $$y(t)=2*t^2-t$$, w

Designing a Parametric Curve for a Cardioid

A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.

Intersection Analysis with the Line y = x

Given the parametric equations $$x(t)=\ln(t+2)$$ and $$y(t)=t^2-1$$ for $$t \ge 0$$, answer the foll

Intersection of Parametric Curves

Two curves are given by the parametric equations $$x_1(t)=t^2,\; y_1(t)=t^3$$ and $$x_2(s)=1-s^2,\;

Intersection Points of Polar Curves

Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:

Motion in the Plane: Logarithmic and Radical Components

A particle’s position in the plane is given by the vector-valued function $$\mathbf{r}(t)=\langle \l

Motion of a Particle in the Plane

A particle moves in the plane with parametric equations $$x(t)=t^2-4*t$$ and $$y(t)=2*t^3-6*t^2$$ fo

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

Parametric Equations and Intersection Points

Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(

Parametric Oscillations and Envelopes

Consider the family of curves defined by the parametric equations $$x(t)=t$$ and $$y(t)=e^{-t}\sin(k

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Polar Spiral: Area and Arc Length

Consider the polar spiral defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0\le\theta\le 2\pi$$. An

Slope of a Tangent Line for a Polar Curve

For the polar curve defined by \(r=3+\sin(\theta)\), determine the slope of the tangent line at \(\t

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Vector-Valued Function of Particle Trajectory

A particle in space follows the vector function $$\mathbf{r}(t)=\langle t, t^2, \sqrt{t} \rangle$$ f

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*

Vector-Valued Kinematics

A particle follows a path in space described by the vector-valued function $$r(t) = \langle \cos(t),

Velocity and Acceleration of a Particle

A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}

Wind Vector Analysis in Navigation

A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)