AP Calculus BC FRQ Room

Analyzing a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

Drainage Rate with a Removable Discontinuity

A drainage system is modeled by the function $$R_{out}(t)=\frac{t^2-2\,t-15}{t-5}$$ liters per minut

Estimating Limits from Tabulated Data

A function $$g(x)$$ is experimentally measured near $$x=2$$. Use the following data to estimate $$\l

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Evaluating Limits Involving Radical Expressions

Consider the function $$h(x)= \frac{\sqrt{4x+1}-3}{x-2}$$.

Graphical Analysis of a Continuous Polynomial Function

Consider the function $$f(x)=2*x^3-5*x^2+x-7$$ and its graph depicted below. The graph provided accu

Graphical Analysis of Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

Implicitly Defined Function and Differentiation

Consider the curve defined implicitly by the equation $$x*y + \sin(x) + y^2 = 10$$. Answer the follo

Intermediate Value Theorem Application with a Cubic Function

A function f(x) is continuous on the interval [-2, 2] and its values at certain points are given in

Intermediate Value Theorem in a Continuous Function

Consider the continuous function $$p(x)=x^3-3*x+1$$ on the interval $$[-2,2]$$. Answer the followi

Limits in Composite Functions Involving Absolute Values

Define the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1 and let $$g(x)=|x-1|.$$ Consider the com

Limits Involving Absolute Value Functions

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

Limits Involving Radicals

Consider the function $$f(x)=\frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$. Answer the following

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

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

One-Sided Limits and Jump Discontinuity Analysis

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

One-Sided Limits for a Piecewise Inflow

In a pipeline system, the inflow rate is modeled by the piecewise function $$R_{in}(t)= \begin{case

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 \

Pollution Level Analysis and Removable Discontinuity

A function $$f(x)$$ represents the concentration of a pollutant (in mg/L) in a river as a function o

Rational Function Analysis of a Drainage Rate

A drain’s outflow rate is given by $$R_{out}(t)=\frac{3\,t^2-12\,t}{t-4}$$ for \(t\neq4\). Answer th

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

Removable Discontinuity in a Rational Function

Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Squeeze Theorem with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Analysis of a Piecewise Function's Differentiability

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

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 a Function's Rate of Change from Graphical Data

A function has been experimentally measured and its values are represented by the following graph. U

Analyzing a Removable Discontinuity in a Rational Function

Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e

Chemical Reaction Rate Control

During a chemical reaction in a reactor, reactants enter at a rate of $$R_{in}(t)=\frac{10*t}{t+2}$$

Derivative of a Composite Function Using the Limit Definition

Consider the function $$h(x)=(2*x+3)^3$$. Use the limit definition of the derivative to answer the f

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Differentiation from First Principles

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

Differentiation in Exponential Growth Models

A population is modeled by $$P(t)=P_0e^{r*t}$$ with the initial population $$P_0=500$$ and growth ra

Differentiation in Polar Coordinates

Consider the polar curve defined by $$r(\theta)= 1+\cos(\theta).$$ (a) Use the formula for polar

Drug Concentration in Bloodstream: Differentiation Analysis

A drug's concentration in the bloodstream is modeled by $$C(t)= 50e^{-0.25t} + 5$$, where t is in ho

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Epidemiological Rate Change Analysis

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

Finding the Derivative of a Logarithmic Function

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

Higher-Order Derivatives

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

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Investment Return Rates: Continuous vs. Discrete Comparison

An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.

Logarithmic Differentiation: Equating Powers

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

Maclaurin Series for arctan(x) and Error Estimate

An engineer in signal processing needs the Maclaurin series for $$g(x)=\arctan(x)$$ and an understan

Piecewise Function and Discontinuity Analysis

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

Polar Coordinates and Tangent Lines

Consider the polar curve $$r(\theta)=1+\cos(\theta)$$. Answer the following:

Population Growth Rate

A population is modeled by $$P(t)=\frac{3*t^2 + 2}{t+1}$$, where $$t$$ is measured in years. Analyze

Population Growth Rates

A city’s population (in thousands) was recorded over several years. Use the data provided to analyze

Population of a Colony: Sum and Derivative Analysis

A colony of cells grows such that the number of cells on the nth day is given by $$a_n= 100(1.2)^{n-

Profit Rate Analysis in Economics

A firm’s profit function is given by $$\Pi(x)=-x^2+10*x-20$$, where $$x$$ (in hundreds) represents t

Radioactive Decay and Derivative

A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is

Rates of Change in Economics

A company’s demand function for a product is given by $$D(p) = 120 - 3*p^2,$$ where \(p\) is the

Renewable Energy Storage

A battery storage system experiences charging at a rate of $$C(t)=50+10\sin(0.5*t)$$ kWh and dischar

Secant and Tangent Slope Analysis

Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:

Tangent and Normal Lines to a Curve

Given the function $$p(x)=\ln(x)$$ defined for $$x > 0$$, analyze its rate of change at a specific p

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Temperature Function Analysis

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

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Velocity Function from a Cubic Position Function

An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a

Water Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

Water Treatment Plant Simulator

A water treatment plant receives contaminated water at a rate of $$R_{in}(t)=50e^{-0.1*t}$$ liters p

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

Composite Function with Hyperbolic Sine

A cable's displacement over time is modeled by $$s(t)= \sinh(\ln(t+1))$$, where $$t$$ is in seconds.

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

Composite Population Growth Function

A population model is given by $$P(t)= e^{3\sqrt{t+1}}$$, where $$t$$ is measured in years. Analyze

Continuity and Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca

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

Implicit Differentiation in a Circle

Consider the circle defined by $$ x^2+y^2=49 $$.

Implicit Differentiation in a Radical Equation

The relationship between $$x$$ and $$y$$ is given by $$\sqrt{x} + \sqrt{y} = 6$$.

Implicit Differentiation Involving Product and Logarithm

Consider the curve defined by $$x*y + \ln(y) = x^2$$. Answer the following parts:

Implicit Differentiation with Trigonometric Equation

Consider the curve defined implicitly by $$\sin(x*y) + x^2 = y^3$$. Answer the following parts:

Inverse Analysis via Implicit Differentiation for a Transcendental Equation

Consider the equation $$e^{x*y}+x-y=0$$ defining y implicitly as a function of x near a point where

Inverse Function Analysis for Exponential Functions

Let $$f(x)=e^{2*x}+1$$ and let g be the inverse function of f. Answer the following parts.

Inverse Function Differentiation Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

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 Function Differentiation in Exponential-Linear Model

Let $$f(x)= x + e^{-x}$$, which is invertible with inverse function $$g(x)$$. Use the inverse functi

Inverse Functions in Economic Modeling

Let the cost function be given by $$f(x)= 4*x + \sqrt{x}$$, where x represents the number of items p

Inverse Trigonometric Differentiation

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

Rocket Fuel Consumption Analysis

A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)

Tangent Line to an Ellipse

Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan

Trigonometric Composite Inverse Function Analysis

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

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

Bacterial Population Growth

The population of a bacterial culture is modeled by $$P(t)=1000e^{0.3*t}$$, where $$P(t)$$ is the nu

Circular Motion and Angular Rate

A point moves along a circle of radius 5 meters. Its angular position is given by $$\theta(t)=2*t^2-

City Population Migration

A city's population is influenced by immigration at a rate of $$I(t)=100e^{-0.2t}$$ (people per year

Concavity and Acceleration in Motion

A car’s position is modeled by $$s(t)= t^3 - 6*t^2 + 9*t+5$$ with time $$t$$ in seconds. Analyze the

Critical Points and Inflection Analysis

Consider the function $$f(x)= (x-2)^2*(x+5)$$ which models a physical quantity. Answer the following

Cubic Function with Parameter and Its Inverse

Examine the family of functions given by $$f(x)=x^3+k*x$$, where $$k$$ is a constant.

Curvature Analysis in the Design of a Bridge

A bridge's vertical profile is modeled by $$y(x)=100-0.5*x^2+0.05*x^3$$, where $$y$$ is in meters an

Differentiating a Product: f(x)=x sin(x)

Let \(f(x)=x\sin(x)\). Analyze the behavior of \(f(x)\) near \(x=0\).

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:

Estimating Rates from Experimental Position Data

The table below represents experimental measurements of the position (in meters) of a moving particl

Exponential Relation

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

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

Implicit Differentiation in a Tank Filling Problem

A tank's volume and liquid depth are related by $$V=10y^3$$, where y (in meters) is the depth. Water

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Infrared Sensor Distance Analysis

An infrared sensor measures the distance to a moving target using the function $$d(t)=50*e^{-0.2*t}+

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Linearization in Finance

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

Linearization in Inverse Function Approximation

Let $$f(x)=x^5+2*x+1$$ be a one-to-one function. Although its inverse cannot be found explicitly, li

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

Modeling Cooling: Coffee Temperature with Logarithmic Decline

A cup of coffee cools according to the model $$T(t)= 90 - 20\ln(1+t)$$, where $$T$$ is in degrees Ce

Motion Model Inversion

Suppose that the position of a particle moving along a line is given by $$f(t)=t^3+t$$. Analyze the

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

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

Popcorn Sales Growth Analysis

A movie theater observes that the number of popcorn servings sold increases by 15% each week. Let $$

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

Revenue Concavity Analysis

A company’s revenue from sales is modeled by the function $$R(x)= 300*x - 2*x^2$$, where \(x\) repre

Revenue Concavity Analysis

A company's revenue over time is modeled by $$R(t)=100\ln(t+1)-2t$$. Answer the following:

Series Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Series Approximation of a Temperature Function

The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n

Series Integration in Fluid Flow Modeling

The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$

Temperature Change in Coffee Cooling

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$, where $$T(t)$$ is in °F a

Analysis of a Logarithmic Function

Consider the function $$q(x)=\ln(x)-\frac{1}{2}*x$$ defined on the interval [1,8]. Answer the follow

Analysis of an Absolute Value Function

Consider the function $$f(x)=|x^2-4|$$. Answer the following parts:

Analysis of Relative Extrema and Increasing/Decreasing Intervals

A particle moves along a line with position given by $$s(x)=x^3-6*x^2+9*x+4$$, where $$x$$ represent

Analyzing a Function with Implicit Logarithmic Differentiation

Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating

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

Car Depreciation Analysis

A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de

Epidemic Infection Model

In a community experiencing an epidemic, the rate of new infections is modeled by $$I(t)=\frac{200}{

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.

Exploring Inverses of a Trigonometric Transformation

Consider the function $$f(x)= 2*\tan(x) + x$$ defined on the interval $$(-\pi/4, \pi/4)$$. Answer th

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Garden Design Optimization

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

Implicit Differentiation in Economic Context

Consider the curve defined implicitly by $$x*y + y^2 = 12$$, representing an economic relationship b

Inverse Derivative Analysis of a Quartic Polynomial

Consider the function $$f(x)= x^4 - 4*x^2 + 2$$ defined for $$x \ge 0$$. Answer the following.

Inverse Function Derivative in a Constrained Domain

Let $$f(x)= \frac{x}{x+2}$$ defined for $$x > -2$$. Analyze the function and its inverse.

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

Mean Value Theorem with Trigonometric Function

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

Population Growth Model Analysis

A population of organisms is modeled by the function $$P(t)= -2*t^2+20*t+50$$, where $$t$$ is measur

Projectile Trajectory: Parametric Analysis

A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)

Real-World Modeling: Radioactive Decay with Logarithmic Adjustment

A radioactive substance decays according to $$N(t)= N_0\,e^{-0.03t}$$. In an experiment, the recorde

Retirement Savings with Diminishing Deposits

Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th

Taylor Series for $$\cos(2*x)$$

Consider the function $$f(x)=\cos(2*x)$$. Construct its 4th degree Maclaurin polynomial, determine t

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

Trigonometric Function and its Inverse

Consider the function $$f(x)= \sin(x) + x$$ defined on the interval $$[-\pi/2, \pi/2]$$. Answer the

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

Accumulation Function from a Rate Function

The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

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

Area Under a Parametric Curve

Consider the parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ for $$t \in [0,3]$$. The area u

Average Temperature from a Continuous Function

Along a metal rod, the temperature is modeled by $$f(t)= t^3 - 3*t^2 + 2*t$$ (in $$^\circ C$$) for

Center of Mass of a Rod with Variable Density

A thin rod of length 10 m has a linear density given by $$\rho(x)= 2 + 0.3*x$$ (in kg/m), where x is

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

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

Estimating Chemical Production via Riemann Sums

In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th

Flow of Traffic on a Bridge

Cars cross a bridge at a rate modeled by $$R(t)=300+50*\cos\left(\frac{\pi*t}{6}\right)$$ vehicles p

Integration of a Piecewise Function

A function representing a rate is defined piecewise by $$f(t)= \begin{cases} 2*t, & 0 \le t \le 3 \

Integration of a Rational Function

Consider the function $$f(x)=\frac{1}{x^2+4}$$ on the interval $$[0,2]$$. Evaluate the area under th

Integration Using U-Substitution

Evaluate the integral $$\int (3*x+2)^5\,dx$$ using u-substitution.

Midpoint Approximation Analysis

Let $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Answer the following:

Modeling Bacterial Growth Through Accumulated Change

A bacteria population's growth rate is given by $$r(t)=\frac{2*t}{1+t^{2}}$$ (in thousands per hour)

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

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:

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

Total Cost from a Marginal Cost Function

A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre

Transportation Model: Distance and Inversion

A transportation system is modeled by $$f(t)= (t-1)^2+3$$ for $$t \ge 1$$, where \(t\) is time in ho

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 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.

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

Volume by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Combined Differential Equations and Function Analysis

Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y

Cooling and Mixing Combined Problem

A container holds 2 L of water initially at 80°C. Cold water at 20°C flows into the container at a r

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

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:

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

FRQ 2: Separable Differential Equation with Initial Condition

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

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 14: Dynamics of a Car Braking

A car braking is modeled by the differential equation $$\frac{dv}{dt} = -k*v$$, where the initial ve

FRQ 15: Cooling of a Beverage in a Fridge

A beverage cools according to Newton's Law of Cooling, described by $$\frac{dT}{dt}=-k(T-A)$$, where

Interpreting Slope Fields for a Differential Equation

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

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 50 liters of pure water. A brine solution with a salt concentration of $$3

Mixing Problem in a Tank

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

Mixing Problem with Differential Equations

A tank initially holds 100 L of a salt solution containing 5 kg of salt. Brine with a salt concentra

Modeling Disease Spread with Differential Equations

In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Separable Differential Equation and Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=2$$. A

Series Solution for a Second-Order Differential Equation

Consider the differential equation $$y'' - y = 0$$ with the initial conditions $$y(0)=1$$ and $$y'(0

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Temperature Change and Differential Equations

A hot liquid cools in a room at $$20^\circ C$$ according to the differential equation $$\frac{dT}{dt

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

Average Temperature Over a Day

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

Average Value and Critical Points of a Trigonometric Function

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

Average Value of a Piecewise Function

A function $$f(x)$$ is defined piecewise over the interval $$[0,6]$$ as follows: $$f(x)=\begin{case

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Center of Mass of a Lamina

A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x * y} + x^2 * y = y^3$$. Answer the following:

Inflow Rate to a Reservoir

The inflow rate of water into a reservoir is given by $$R(t)=\frac{100*t}{5+t}$$ (in cubic meters pe

Rainfall Accumulation Analysis

The rainfall rate (in cm/hour) at a location is modeled by $$r(t)=0.5+0.1*\sin(t)$$ for $$0 \le t \l

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

Volume of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Volume of a Solid using the Washer Method

Consider the region bounded by the curves $$y= x$$ and $$y= \sqrt{x}$$ for $$0 \le x \le 1$$. This r

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

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

Arc Length and Speed from Parametric Equations

Consider the curve defined by $$x(t)=e^t$$ and $$y(t)=e^{-t}$$ for $$-1 \le t \le 1$$. Analyze the a

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 curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2+2$$ for $$t \in [0,2]$$.

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Area Between Polar Curves: Annulus with a Hole

Two polar curves are given by \(R(\theta)=3\) and \(r(\theta)=2+\cos(\theta)\) for \(0\le\theta\le2\

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r=2+2\sin(\theta)$$. This curve is a cardioid. Answer the foll

Average Position from a Vector-Valued Function

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle \sin(t), \cos

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 and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer 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\

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

Implicit Differentiation and Curves in the Plane

The curve defined by $$x^2y + xy^2 = 12$$ describes a relation between $$x$$ and $$y$$.

Inflow and Outflow in a Water Tank

A water tank has water entering at a rate given by $$I(t)=5+\sin(t)$$ (liters per minute) and water

Intersection of Two Parametric Curves

Two curves are represented parametrically as follows: Curve A is given by $$x(t)=t^2, \; y(t)=2*t+1$

Length of a Polar Spiral

For the polar spiral defined by $$r=\theta$$ for $$0 \le \theta \le 2\pi$$, answer the following:

Limit Evaluation of a Trigonometric Piecewise Function

Define the function $$f(θ)= \begin{cases} \frac{1-\cos(θ)}{θ}, & θ \neq 0 \\ 0, & θ=0 \end{cases}$$

Modeling Circular Motion with Vector-Valued Functions

An object moves along a circle of radius $$3$$ with its position given by $$\mathbf{r}(t)=\langle 3\

Numerical Integration Techniques for a Parametric Curve

A curve is defined by the parametric equations $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

Parametric and Polar Conversion Challenge

Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for

Parametric Curve: Intersection with a Line

Consider the parametric curve defined by $$ x(t)=t^3-3*t $$ and $$ y(t)=2*t^2 $$. Analyze the proper

Parametric Equations and Tangent Slopes

Consider the parametric equations $$x(t)= t^3 - 3*t$$ and $$y(t)= t^2$$, for $$t \in [-2, 2]$$. Anal

Parametric Intersection and Enclosed Area

Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given

Parametric Slope and Arc Length

Consider the parametric curve defined by $$x(t)= t-\ln(t)$$ and $$y(t)= t\cdot\ln(t)$$ for $$t > 1$$

Parametric Tangent Line and Curve Analysis

For the curve defined by the parametric equations $$x(t)=t^{2}$$ and $$y(t)=t^{3}-3t$$, answer the f

Particle Motion in the Plane

A particle's position is given by the parametric equations $$x(t)=3*t^2-2*t$$ and $$y(t)=4*t^2+t-1$$

Polar Boundary Conversion and Area

A region in the polar coordinate plane is defined by $$1 \le r \le 3$$ and $$0 \le \theta \le \frac{

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 $$

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

Vector-Valued Functions and 3D Projectile Motion

An object's position in three dimensions is given by $$\mathbf{r}(t)=\langle 3t, 4t, 10t-5t^2 \rangl

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 Functions: Position, Velocity, and Acceleration

Let $$\textbf{r}(t)= \langle e^t, \ln(t+1) \rangle$$ represent the position of a particle in the pla

Velocity and Acceleration of a Particle

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