AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Computations

Let $$s(x)=\frac{x^3-8}{x-2}.$$ Answer the following:

Algebraic Method for Evaluating Limits

Consider the function $$h(x)=\frac{x^2-9}{x-3}.$$ Answer the following parts.

Algorithm Time Complexity

A recursive algorithm has an execution time that decreases with each iteration: the first iteration

Analysis of a Jump Discontinuity

Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.

Analyzing Limits of a Composite Function

Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:

Analyzing Limits Using Tabular Data

A function $$f(x)$$ is described by the following table of values: | x | f(x) | |------|------|

Asymptotic Behavior and Horizontal Limits

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

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Continuity in Composition of Functions

Let $$g(x)=\frac{x^2-4}{x-2}$$ for x ≠ 2 and undefined at x = 2, and let f(x) be a continuous functi

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

End Behavior of an Exponential‐Log Function

Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as

Evaluating Limits Involving Radical Expressions

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

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Exponential Function Limits at Infinity

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

Indeterminate Limit with Exponential and Log Functions

Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

Left-Hand and Right-Hand Limits for a Sign Function

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

Limit Evaluation Involving Radicals and Rationalization

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

Limits at a Point: Removable Discontinuity Analysis

Consider the function $$f(x)=\frac{(x+3)*(x-2)}{(x+3)*(x+5)}$$ which is not defined at $$x=-3$$. Ans

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$g(x)= \frac{4*x^3-x+2}{2*x^3+3*x^2-5}$$.

Manufacturing Cost Sequence

A company's per-unit manufacturing cost decreases by $$50$$ dollars each year due to economies of sc

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

One-Sided Infinite Limits in Rational Functions

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

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

Piecewise Function Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2}, & x\neq2\\ k, & x=2 \en

Radical Function Limit via Conjugate Multiplication

Consider the function $$f(x)=\frac{\sqrt{2*x+9}-3}{x}$$ defined for $$x \neq 0$$. Answer the followi

Real-World Temperature Sensor Analysis

A temperature sensor is modeled by the function $$T(t)=\frac{t^2-9}{t-3}$$ for t ≠ 3 (with t in minu

Reciprocal Function Behavior and Asymptotes

Examine the function $$f(x)= \frac{1}{x-1}$$.

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

Water Tank Flow Analysis

A water tank receives water from an inlet and drains water through an outlet. The inflow rate is giv

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Analysis of Concavity and Second Derivative

Let $$f(x)=x^4-4*x^3+6*x^2$$. Analyze the concavity of the function and identify any inflection poin

Analyzing a Function with an Oscillatory Component

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

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Differentiability of an Absolute Value Function

Consider the function $$f(x) = |x|$$.

Differentiation of a Rational Function

Consider the function $$f(x) = \frac{2*x^2+3*x}{x-1}$$, which is defined on its domain. Analyze the

Differentiation of an Exponential Function

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

Error Analysis in Approximating Derivatives

Consider the function $$f(x)= \ln(1+x)$$. (a) Write the Maclaurin series for \(f(x)\) up to and inc

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 with Inverse Functions

Suppose a differentiable function $$f$$ satisfies the equation $$f(x) + f^(-1)(x) = 2*x$$ for all x

Implicit Differentiation with Trigonometric Functions

Consider the curve defined by $$\sin(x*y) = x + y$$.

Implicit Differentiation: Mixed Exponential and Polynomial Equation

Consider the curve defined by $$x^3 + e^(y) = 3*x*y$$.

Instantaneous Rate of Change and Series Approximation for √(1+x)

A company models its cost using the function $$C(x)=\sqrt{1+x}$$. To understand small changes in cos

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)

Interpreting Derivative Notation in a Real-World Experiment

A reservoir's water level (in meters) is measured at different times (in minutes) as shown in the ta

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

Limit Definition of the Derivative for a Trigonometric Function

Consider the function $$f(x)= \cos(x)$$.

Manufacturing Cost Function and Instantaneous Rate

The total cost (in dollars) to produce x units of a product is given by $$C(x)= 0.2x^3 - 3x^2 + 50x

Optimization and Tangent Lines

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

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Profit Optimization via Derivatives

A company's profit function is given by $$P(x)=-2*x^2 + 40*x - 100$$, where $$x$$ represents the num

Projectile Trajectory: Rate of Change Analysis

The height of a projectile is given by $$h(t)= -4.9t^2 + 20t + 1.5$$ in meters, where t is in second

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: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ (in m³) and radius $$r$$ (in m) satis

Secant and Tangent Slope Analysis

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

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Tangent Line Approximation

Consider the function $$g(t)=t^2 - 4*t + 7$$. Answer the following parts to find the equation of the

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Temperature Change with Provided Data

The temperature at different times after midnight is modeled by $$T(t)=5*\ln(t+1)+20$$, with $$t$$ i

Temperature Change: Secant vs. Tangent Analysis

A scientist recorded the temperature $$T$$ (in °C) at various times $$t$$ (in seconds) as shown in t

Urban Population Flow

A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

Warehouse Inventory Management

A warehouse receives shipments at a rate of $$I(t)=100e^{-0.05*t}$$ items per day and ships items ou

Composite Differentiation in Biological Growth

A biologist models the temperature $$T$$ (in °C) of a culture over time $$t$$ (in hours) by the func

Composite Function Analysis

Consider the function $$f(x)= \sqrt{3*x^2+2*x+1}$$ which arises in an experimental study of motion.

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 Function with Implicitly Defined Inner Function

Let the function $$h(x)$$ be defined implicitly by the equation $$h(x) - \ln(h(x)) = x$$, and consid

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

Differentiation of a Log-Exponential Composition with Critical Points

Consider the function $$k(x)=x*\ln(e^{x}+3)$$. Answer the following parts.

Differentiation of a Nested Trigonometric Function

Let $$h(x)= \sin(\arctan(2*x))$$.

Differentiation of an Inverse Trigonometric Function

Define $$h(x)= \arctan(\sqrt{x})$$. Answer the following:

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Implicit Differentiation in a Conic Section

Consider the curve defined by $$x^2 + x*y + y^2 = 9$$.

Implicit Differentiation in a Nonlinear Equation

Consider the equation $$x*y + y^3 = 10$$, which defines y implicitly as a function of x.

Implicit Differentiation in Mixed Function Equation

Consider the relation $$x^2*y+\sin(y)=5*x$$. Analyze this relation using implicit differentiation.

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:

Implicit Differentiation with Logarithmic Functions

Consider the equation $$\ln(x+y)= x - y$$.

Implicit Differentiation with Trigonometric Equation

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

Implicit Differentiation: Conic Section Analysis

Consider the conic section defined by $$x^2 + 3*x*y + y^2 = 5$$. Answer the following:

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

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 a Logarithmic Context

Let $$f(x)= \ln(x+2) - x$$, and let $$g$$ be its inverse function. Answer the following:

Inverse Function Differentiation in Thermodynamics

In a thermodynamics experiment, a differentiable one-to-one function $$f$$ describes the temperature

Inverse Function Differentiation with Combined Logarithmic and Exponential Terms

Let $$f(x)=e^{x}+\ln(x)$$ for $$x>1$$ and let g be its inverse function. Answer the following.

Inverse of a Shifted Logarithmic Function

Analyze the function $$f(x)=\ln(x-1)+2$$ defined for $$x>1$$ and its inverse.

Inverse Trigonometric Differentiation

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

Logarithmic and Composite Differentiation

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

Optimization with Composite Functions - Minimizing Fuel Consumption

A car's fuel consumption (in liters per 100 km) is modeled by $$F(v)= v^2 * e^{-0.1*v}$$, where $$v$

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Reservoir Levels and Evaporation Rates

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

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

Balloon Inflation and Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of $$12\;in^

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

Cycloid Tangent Line

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Estimating Rates from Experimental Position Data

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

Graphical Analysis of an Inverse Function

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

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

Implicit Differentiation in Astronomy

The trajectory of a comet is given by the ellipse $$x^2 + 4*y^2 = 16$$, where \(x\) and \(y\) (in as

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$

Implicit Differentiation on an Ellipse

An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo

Inflating Spherical Balloon

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

Inflating Spherical Balloon: A Related Rates Problem

A spherical balloon is being inflated so that its volume increases at a constant rate of $$12\; in^3

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

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

Maximizing Efficiency: Derivative Analysis in a Production Process

The efficiency of a production process is modeled by $$E(x)=50+10*\ln(x)-0.5*x$$, where $$x$$ repres

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

Optimal Dimensions of a Cylinder with Fixed Volume

A closed right circular cylinder must have a volume of $$200\pi$$ cubic centimeters. The surface are

Particle Motion Analysis

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

Related Rates in a Conical Water Tank

Water is being pumped into a conical tank at a rate of $$2\;m^3/min$$. The tank has a height of 6 m

Revenue Function and Marginal Revenue

A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)

Series Analysis in Acoustics

The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^

Series Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

Series Differentiation in Heat Transfer Analysis

A heat transfer rate is modeled by $$H(t)= \sum_{n=0}^{\infty} \frac{(-1)^n (0.5t)^{2*n}}{(2*n)!}$$,

Series Solution of a Drug Concentration Model

The drug concentration in the bloodstream is modeled by $$C(t)= \sum_{n=0}^{\infty} \frac{(-t)^n}{n!

Temperature Change of Cooling Coffee

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

Analysis of a Quartic Function as a Perfect Power

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

Analysis of a Rational Function and the Mean Value Theorem

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$. Answer the following parts.

Analyzing a Function with Implicit Logarithmic Differentiation

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

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

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:

Arc Length Approximation

Let $$f(x) = \sqrt{x}$$ be defined on the interval [1,9].

Concavity and Inflection Points

The function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$ models a certain process. Use the second derivative to

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.

Determining the Meeting Point of Two Functions

Consider the functions $$f(x)= e^x$$ and $$g(x)= 3 + \ln(x)$$ representing two different processes.

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 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-

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Linear Approximation and Differentials

Let \( f(x) = \sqrt{x} \). Use linear approximation to estimate \( \sqrt{10} \). Answer the followin

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a circle of radius $$5$$. Determine the dimensions of the rectangle that

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)$

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Projectile Trajectory: Parametric Analysis

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

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

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

Relative Extrema and Critical Points of a Cubic Polynomial

Consider the function $$f(x)=x^3 - 6*x^2 + 9*x + 2$$. Use the analytical techniques of differentiati

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

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

Staircase Design for a Building

A staircase is being designed for a building. The first step has a height of 7 inches, and each subs

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Accumulated Displacement from a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

Applying the Fundamental Theorem of Calculus

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

Approximating an Exponential Integral via Riemann Sums

Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below

Approximating Area Under f(x) = √x Using Riemann Sums

Consider the function $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Divide the interval into 3 equal sub

Area Between Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x - 3$$. These curves intersect and enclose a region.

Area Between Two Curves

Given the functions $$f(x)= x^2$$ and $$g(x)= 4*x$$, determine the area of the region bounded by the

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

Continuity and Integration of a Sinc-like Function

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

Convergence of an Improper Integral

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

Cross-Sectional Area of a River Using Trapezoidal Rule

The depth $$h(x)$$ (in meters) of a river’s cross-section is measured at various points along a hori

Cyclist's Displacement from Variable Acceleration

A cyclist's acceleration is given by $$a(t)= 7 - 3*t$$ (in m/s²). At time $$t=0$$, the cyclist has a

Displacement and Distance from a Velocity Function

A particle moves along a straight line with its velocity given by $$v(t)=3\sin(t)$$ (in m/s) for $$t

Drug Absorption Modeling

The rate of drug absorption into the bloodstream is modeled by $$C'(t)= 2*e^{-0.5*t}$$ mg/hr, with a

Drug Concentration in a Bloodstream

A patient receives an intravenous drug infusion at a rate $$R_{in}(t)=3e^{-0.1*t}$$ mg/min for $$0 \

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

Estimating Area Under a Curve from Tabular Data

A function $$f(t)$$ is sampled at discrete time points as given in the table below. Using these data

Exploring Riemann Sums and Discontinuities from Graphical Data

A graph of a function f(x) is provided that shows a smooth curve with a removable discontinuity (a h

Integration by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Integration of a Piecewise-Defined Function

Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \

Integration Using U-Substitution

Evaluate the definite integral $$\int_{0}^{2} (3*x+1)^{4} dx$$ using u-substitution. Answer the foll

Integration via Substitution and Numerical Methods

Evaluate the integral $$\int_0^2 \frac{2*x}{\sqrt{1+x^2}}\,dx$$.

Marginal Cost and Total Cost in Production

A company's marginal cost function is given by $$MC(q)=12+2*q$$ (in dollars per unit) for $$q$$ in t

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Power Series Analysis and Applications

Consider the function with the power series representation $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n x^{

Recovering Accumulated Change

A company’s revenue rate changes according to $$R'(t)=8*t-12$$ (in dollars per day). If the revenue

Revenue Accumulation and Constant of Integration

A company's revenue is modeled by $$R(t) = \int_{0}^{t} 3*u^2\, du + C$$ dollars, where t (in years)

Riemann Sum Approximation with Irregular Intervals

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

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+

Temperature Change Analysis

A series of temperature readings (in °C) are recorded over the day as shown in the table. Analyze th

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 Rule Error Estimation

Given the function $$f(x)=\ln(x)$$ on the interval $$[1,4]$$, answer the following:

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.

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Capacitor Charging with Leakage

A capacitor is being charged by a constant current source of $$5$$ A, but it also leaks charge at a

Chemical Reaction in a Closed System

The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential

Chemical Reaction Kinetics

A first-order chemical reaction has its concentration $$C(t)$$ (in mol/L) governed by the differenti

Chemical Reaction Rate

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

Constructing and Interpreting a Slope Field

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

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

Differential Equation in a Gravitational Context

Consider the differential equation $$\frac{dv}{dt}= -G\,\frac{M}{(R+t)^2}$$, which models a simplifi

Direction Fields and Phase Line Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=(y-2)(3-y)$$. Answer the following par

Euler's Method Approximation

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

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

Implicit Differentiation and Homogeneous Equation

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

Mixing Problem in a Tank

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

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Optimization in Construction: Minimizing Material for a Container

A manufacturer is designing an open-top cylindrical container with fixed volume $$V$$. The material

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

RC Circuit Differential Equation

In an RC circuit, the capacitor charges according to the differential equation $$\frac{dQ}{dt}=\frac

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Separable Differential Equation and Slope Field Analysis

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

Advanced Parameter-Dependent Integration Problem

Consider the function $$g(x)=e^{-a*x}$$, where $$a>0$$ and $$x$$ lies within $$[0,b]$$. The average

Analyzing a Motion Graph from Data

The following table represents the instantaneous velocity (in m/s) of a vehicle over a 6-second inte

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

Area Enclosed by a Cardioid in Polar Coordinates

Consider the polar curve given by $$r(\theta)=1+\cos(\theta)$$.

Area Under an Exponential Decay Curve

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

Average Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Temperature Calculation

The temperature (in $$^\circ C$$) in City A is recorded at specific times over a 12-hour period. Est

Average Value of a Trigonometric Function

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

Average Value of a Velocity Function

A particle moves along a line with its velocity given by $$v(t)= 2*\cos(t) + \sin(t)$$ for $$t \in [

Center of Mass of a Nonuniform Rod

A thin rod extends from $$x=0$$ to $$x=3$$ and has a linear density given by $$\delta(x)=1+x$$ (in k

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

Displacement from a Velocity Graph

A runner’s velocity is given by $$v(t)=8-0.5*t$$ (m/s) for $$0\le t\le 12$$ seconds. A graph of this

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

Drone Motion Analysis

A drone’s vertical acceleration is modeled by $$a(t) = 6 - 2*t$$ (in m/s²) for time $$t$$ in seconds

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

Implicit Differentiation with Exponential Terms

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

Motion Analysis on a Particle with Variable Acceleration

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

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

Profit-Cost Area Analysis

A company’s profit (in thousands of dollars) is modeled by $$P(x) = -x^2 + 10*x$$ and its cost by $$

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

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

Total Charge in an Electrical Circuit

In an electrical circuit, the current is given by $$I(t)=5*\cos(0.5*t)$$ (in amperes), where \(t\) i

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=5*x$$ (in Newtons), where $$x$$ is

Analysis of a Polar Rose

Examine the polar curve given by $$ r=3*\cos(3\theta) $$.

Arc Length of a Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

Area Between Polar Curves

In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t

Concavity and Inflection Points of a Parametric Curve

For the curve defined by $$x(t)=e^{t}-t$$ and $$y(t)=\ln(1+t^2)$$ for $$t \ge 0$$, answer the follow

Conversion of Polar to Cartesian Coordinates

Consider the polar curve $$ r=4*\cos(\theta) $$. Analyze its Cartesian equivalent and some of its pr

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

Cycloid and Its Arc Length

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

Inner Loop of a Limaçon in Polar Coordinates

The polar curve given by \(r=1+2\cos(\theta)\) forms a limaçon with an inner loop. Answer the follow

Intersection Points of Polar Curves

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

Modeling Projectile Motion with Parametric Equations

A projectile is launched with an initial speed of \(20\) m/s at an angle of \(45^\circ\) above the h

Parameter Elimination in Logarithmic and Quadratic Relationships

Given the parametric equations $$x(t)= \ln(t)$$ and $$y(t)= t^2 - 4*t + 3$$ for $$t > 0$$, eliminate

Parametric Spiral Curve Analysis

The curve defined by $$x(t)=t\cos(t)$$ and $$y(t)=t\sin(t)$$ for $$t \in [0,4\pi]$$ represents a spi

Particle Motion in the Plane

A particle moves in the plane with its position described by the parametric equations $$x(t)=3*\cos(

Polar Coordinates and Dynamics

A point moves along a spiral defined by the polar equation $$r=3\theta$$, where $$\theta$$ is given

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Projectile Motion with Air Resistance: Parametric Analysis

A projectile is launched with air resistance, and its motion is modeled by the parametric equations:

Self-Intersection in a Parametric Curve

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

Spiral Motion in Polar Coordinates

A particle moves in polar coordinates with \(r(\theta)=4-\theta\) and the angle is related to time b

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 Fourier Series Representation

The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c

Vector-Valued Function Analysis

Let the vector-valued function be given by $$\vec{r}(t)=<e^{t},\, \sin(t),\, \cos(t)>$$ for $$0\leq

Vector-Valued Functions and Kinematics

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