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Analysis of a Piecewise Function with Multiple Definitions
Consider the function $$h(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x<3, \\ 2*x-1 & \text{if
Analysis of a Rational Inflow Function with a Discontinuity
A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$
Analyzing Limits of a Composite Function
Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:
Asymptotic Behavior and Horizontal Limits
Consider the function $$f(x)=\frac{2 * x^2 - x + 1}{x^2+1}$$. Answer the following questions regardi
Bacterial Growth Experiment
A laboratory experiment involves a bacterial culture whose population at hour $$n$$ is modeled by a
Composite Functions: Limits and Continuity
Let $$f(x)=x^2-1$$, which is continuous for all $$x$$, and let $$g(x)=f(\sqrt{x+1})$$.
Continuity Analysis of an Integral Function
Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{
Determining Continuity via Series Expansion
Consider the function $$f(x)= \frac{e^x - \ln(1+x) - x - 1}{x^2}$$ for $$x \neq 0$$ with $$f(0)=L$$.
End Behavior and Horizontal Asymptote Analysis
Consider the function $$f(x)=\frac{3*x^3-5*x+2}{2*x^3+4*x^2-1}$$. Answer the following:
End Behavior of an Exponential‐Log Function
Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as
Endpoint Behavior of a Continuous Function
Let $$m(x)=\sqrt{x+4}$$ be defined on the interval $$[-4,5]$$. Answer the following:
Establishing Continuity in a Piecewise Function
Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x
Evaluating a Limit with Algebraic Manipulation
Examine the function $$g(x)= \frac{\sqrt{x+9}-3}{x}$$ for $$x \neq 0$$.
Fuel Efficiency and Speed Graph Analysis
A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph
Graphical Analysis of Limits and Asymptotic Behavior
A graphical study titled 'Graph of Experimental Data' shows the measured concentration of a chemical
Horizontal Asymptote of a Rational Function
Consider the function $$h(x)=\frac{3x^2-x+2}{x^2+5}$$. Answer the following:
Indeterminate Limit with Exponential and Log Functions
Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$
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 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
Logarithmic Function Limits
Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:
Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$
Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin
One-Sided Infinite Limits in Rational Functions
Consider the function $$f(x)= \frac{1}{(x-2)^2}$$.
Oscillatory Functions and Discontinuity
Consider the function $$f(x)= \begin{cases} \sin\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0 \end{
Parameter Determination for Continuity
Let $$f(x)= \frac{x^2-1}{x-1}$$ for $$x \neq 1$$, and suppose that $$f(1)=m$$. Answer the following:
Physical Applications: Temperature Continuity
A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^
Piecewise Function Critical Analysis
Consider the piecewise function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} & \text{if } x \neq 2 \\
Rational Function Analysis with Removable Discontinuities
Consider the function $$f(x)=\frac{(x+3) * (x-1)}{(x-1)}$$ for $$x \neq 1$$. This function exhibits
Rational Function and Removable Discontinuity
Consider the function $$g(x) = \frac{(x+3)(x-2)}{(x-2)}$$, defined for $$x \neq 2$$, and suppose tha
Related Rates: Changing Shadow Length
A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol
Removable Discontinuity in a Rational Function
Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll
Trigonometric Rate Function Analysis
A pump’s output is modified by a trigonometric factor. The outflow rate is recorded as $$R(t)=\frac{
Acceleration and Jerk in Motion
The position of a car is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$t$$ is time in seconds and $$s(t
Analyzing a Removable Discontinuity in a Rational Function
Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e
Car Motion and Critical Velocity
The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i
Chemical Mixing Tank
In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t
Chemical Reaction Rate
In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=10 - 2*\ln(t+1)$$, wher
Composite Exponential-Log Function Analysis
Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch
Cost Optimization in Production: Derivative Application
A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu
Derivative of a Composite Function Using the Limit Definition
Consider the function $$h(x)=(2*x+3)^3$$. Use the limit definition of the derivative to answer the f
Derivatives of a Rational Function
Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol
Differentiability of an Absolute Value Function
Consider the function $$f(x) = |x|$$.
Differentiation of a Trigonometric Function
Let $$f(x)=\sin(x)+x*\cos(x)$$. Differentiate the function using the sum and product rules.
Epidemiological Rate Change Analysis
In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex
Exponential Growth and Its Derivative
A culture of bacteria grows according to the model $$P(t)= 100*e^{0.03*t},$$ where \(P(t)\) is th
Graphical Derivative Analysis
A series of experiments produced the following data for a function $$f(x)$$:
Implicit Differentiation in Circular Motion
A particle moves along the circle defined by $$x^2 + y^2 = 25$$. Answer the following parts.
Implicit Differentiation: Exponential-Polynomial Equation
Consider the curve defined by $$e^(x*y) + x^2 = y^2$$.
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
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
Logarithmic Differentiation Simplification
Consider the function $$h(x)=\ln\left( \frac{(x^2+1)^{3}*e^{2*x}}{\sqrt{x+2}} \right)$$.
Maclaurin Series for ln(1+x)
A scientist modeling logarithmic growth wishes to approximate the function $$\ln(1+x)$$ around $$x=0
Market Price Rate Analysis
The market price of a product (in dollars) has been recorded over several days. Use the table below
Optimization in a Chemical Reaction
The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r
Optimization Problem via Derivatives
A manufacturer’s cost in dollars for producing $$x$$ units is modeled by the function $$C(x)= x^3 -
Product of Exponential and Trigonometric Functions
Let $$f(x)=e^(2*x)*\sin(x)$$. This function models oscillating growth. Answer the following:
Radioactive Decay with Logarithmic Correction
A radioactive substance decays following the model $$A(t)=A_0*e^{-k*t}+\ln(t+1)$$, where $$t$$ is th
Related Rates: Draining Conical Tank
Water is draining from an inverted conical tank with a height of 6 m and a top radius of 3 m. The vo
Satellite Orbit Altitude Modeling
A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}
Tangent Line Estimation in Transportation Modeling
A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote
Taylor Series Expansion of ln(x) About x = 2
For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to
Taylor Series for Cos(x) in Temperature Modeling
An engineer uses the cosine function to model periodic temperature variations. Approximate $$\cos(x)
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 Function Analysis
Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in
Traffic Flow and Instantaneous Rate
The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$
Water Treatment Plant Simulator
A water treatment plant receives contaminated water at a rate of $$R_{in}(t)=50e^{-0.1*t}$$ liters p
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
Bacterial Culture: Nutrient Inflow vs Waste Outflow
In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste
Chain Rule with Nested Logarithmic and Exponential Functions
Consider the function $$f(x)= \sqrt{\ln(5*x + e^{x})}$$. Differentiate this function using the chain
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 Functions in Biological Growth
Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c
Composite Implicit Differentiation Involving Trigonometric and Polynomial Terms
Consider the relation $$\sin(x*y) + y^3 = x$$.
Differentiation in a Logistic Population Model
The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}
Differentiation of Inverse Trigonometric Functions
Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and
Fuel Tank Dynamics
A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+
Implicit Differentiation and Inverse Challenges
Consider the implicit relation $$x^2+ x*y+ y^2=10$$ near the point (2,2).
Implicit Differentiation in Economic Equilibrium
In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand
Implicit Differentiation in Trigonometric Equations
For the equation $$\cos(x*y) + x^2 - y^2 = 0$$, y is defined implicitly as a function of x.
Implicit Differentiation of a Composite Equation
Given the implicit relation $$x^2*y + \sin(y) = x$$, answer the following:
Implicit Differentiation of an Ellipse
The ellipse is given by $$4*x^2 + 9*y^2 = 36$$.
Implicit Differentiation on an Elliptical Curve
Consider the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ representing an object’s cross-section. Answe
Implicit Differentiation with an Exponential Function
Given the equation $$ e^{x*y}= x+y $$, use implicit differentiation.
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:
Indoor Air Quality Control
In a controlled laboratory environment, the rate of fresh air introduction is modeled by the composi
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 with Composite Trigonometric Functions
Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$
Inverse Trigonometric Function Differentiation
Let $$y=\arctan(\sqrt{3*x+1})$$. Answer the following parts:
Inverse Trigonometric Functions in Navigation
A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =
Modeling with Composite Functions: Pollution Concentration
The pollutant concentration in a lake is modeled by $$C(t) = \sqrt{100 - 2*e^{-0.1*t}}$$, where $$t$
Pipeline Pressure and Oil Flow
In an oil pipeline, the driving pressure is modeled by the composite function $$P(t)=r(s(t))$$, wher
Tangent Line for a Parametric Curve
A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.
Approximating Changes with Differentials
Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch
Comparison of Series Convergence and Error Analysis
Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in
Cooling Analysis using Newton’s Law of Cooling
An object cools in a room according to Newton's Law of Cooling, given by $$T(t)=T_{env}+ (T(0)-T_{en
Derivative of Concentration in a Chemical Reaction
The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{8e^{-0.5t}}{1+e^{-
Differentials in Engineering: Beam Stress Analysis
The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^
Implicit Differentiation on an Ellipse
An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo
Implicit Differentiation: Tangent to a Circle
Consider the circle given by $$x^2 + y^2 = 25$$.
Integration Region: Exponential and Polynomial Functions
Let the region be bounded by the curves $$y = x^2$$ and $$y = e^x$$. Analyze the area of the region
Interpreting the Derivative in Straight Line Motion
A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a
L'Hospital's Rule for Indeterminate Limits
Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ using L'Hospita
Linearization in Finance
The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i
Linearization of Trigonometric Implicit Function
Consider the implicit equation $$\tan(x + y) = x - y$$, which implicitly defines $$y$$ as a function
Maximizing Revenue in a Business Model
A company’s revenue as a function of price is given by $$R(p)= p*(100 - 2*p)$$, where $$p$$ is the p
Mixing a Saline Solution: Related Rates
A tank contains a saline solution with a constant volume of 50 liters. Salt is added at a rate of 2
Optimization of a Rectangular Enclosure
A rectangular enclosure is to be built adjacent to a river. Only three sides of the enclosure requir
Particle Motion Along a Line with Polynomial Velocity
A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=
Population Decline Modeled by Exponential Decay
A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the
Production Cost Analysis
A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa
Reactant Flow in a Chemical Reactor
In a chemical reactor, a reactant is introduced at a rate $$I(t)=15\sin(\frac{t}{2})$$ (grams per mi
Related Rates in a Circular Pool
A circular pool is being filled such that the surface area increases at a constant rate of $$10$$ ft
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
Related Rates: Inflating Spherical Balloon with Exponential Volume Rate
A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.
Revenue and Marginal Analysis
A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit
Series Analysis in Acoustics
The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^
Series Approximation of a Temperature Function
The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n
Series Integration for Work Calculation
A force along a displacement is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n+2}$$ (in Ne
Series Integration in Fluid Flow Modeling
The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$
Analyzing a Function with Implicit Logarithmic Differentiation
Consider the implicit equation $$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
Area Enclosed by a Polar Curve
Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima
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.
Curve Sketching Using Derivatives
For the function $$f(x)=\frac{x}{x^2+1}$$, use its derivatives to sketch a rough graph of the functi
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
Echoes in an Auditorium
In an auditorium, an audio signal produces echoes. The first echo has an intensity that is 70% of th
Extreme Value Theorem in Temperature Variation
A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x
Finding and Interpreting Inflection Points in a Complex Function
Analyze the function $$f(x)= e^{-x}\,\ln(x)$$ for $$x > 0$$. Investigate the points of inflection an
Investigation of Extreme Values on a Closed Interval
For a particle moving along a path given by $$f(x)=x^3-6*x^2+9*x+5$$ where $$x\in[0,5]$$, analyze it
Linear Approximation of a Radical Function
For the function $$f(x)= \sqrt{x+1}+x$$, find its linear approximation at $$x=3$$ and use it to appr
Logistic Growth Model Analysis
Consider the logistic growth model given by $$P(t)=\frac{100}{1+9e^{-0.5*t}}$$. Answer the following
Optimization in a Log-Exponential Model
A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr
Optimizing Material for a Container
An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.
Population Growth Modeling
A region's population (in thousands) is recorded over a span of years. Use the data provided to anal
Rolle's Theorem on a Cubic Function
Consider the cubic function $$f(x)= x^3-3*x^2+2*x$$ defined on the interval $$[0,2]$$. Verify that t
Second Derivative Test for Critical Points
Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.
Sign Chart Construction from the Derivative
Consider the function $$ f(x)=x^4-4x^3+6x^2.$$ Answer the following parts:
Staircase Design for a Building
A staircase is being designed for a building. The first step has a height of 7 inches, and each subs
Tangent Line to a Parametric Curve
A curve is defined by the parametric equations $$x(t) = \cos(t)$$ and $$y(t) = \sin(t) + \frac{t}{2}
Taylor Polynomial for $$\cos(x)$$ Centered at $$x=\pi/4$$
Consider the function $$f(x)=\cos(x)$$. You will generate the second degree Taylor polynomial for f(
Taylor Polynomial for $$\ln(x)$$ about $$x=1$$
For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,
Taylor Series in Differential Equations: $$y'(x)=y(x)\cos(x)$$
Consider the initial value problem $$y'(x)= y(x)\cos(x)$$ with $$y(0)=1$$. Assume a power series sol
Taylor Series in Economics: Cost Function
An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro
Antiderivative with Initial Condition
Find the general antiderivative of the function $$f(x)=5*x^3-2*x+6$$ and determine the particular an
Antiderivatives and the Fundamental Theorem
Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i
Arc Length Calculation
Find the arc length of the curve $$y=\frac{1}{3}x^{3/2}$$ from $$x=0$$ to $$x=9$$.
Area Between Inverse Functions
Consider the functions $$f(x)=\sqrt{x}$$ and $$g(x)=x-2$$.
Area Estimation with Riemann Sums
Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub
Area Under a Parametric Curve
A curve is defined parametrically by $$x(t)=t^2$$ and $$y(t)=t^3-3*t$$ for $$t \in [-2,2]$$.
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
Chemical Reactor Concentration
In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu
Comparing Integration Approximations: Simpson's Rule and Trapezoidal Rule
A student approximates the definite integral $$\int_{0}^{4} (x^2+1)\,dx$$ using both the trapezoidal
Determining Constant in a Height Function
A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity
Error Analysis in Riemann Sum Approximations
Consider approximating the integral $$\int_{0}^{2} x^3\,dx$$ using a left-hand Riemann sum with $$n$
Estimating Chemical Production via Riemann Sums
In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th
Finding Area Between Two Curves
Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.
Improper Integral Convergence
Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.
Integration of a Piecewise-Defined Function
Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \
Integration of a Trigonometric Function by Two Methods
Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me
Integration Using U-Substitution
Evaluate the integral $$\int (3*x+2)^5\,dx$$ using u-substitution.
Limit of a Riemann Sum as a Definite Integral
Consider the limit of the Riemann sum given by $$\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{6}{
Mechanical Systems: Total Change and Inverse Analysis
Consider the function \(f(x)= x^3 + 3*x\) defined for all real \(x\), modeling a mechanical system.
Modeling Water Inflow Using Integration
Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$
Population Increase from a Discontinuous Growth Rate
A sudden migration event alters the population growth rate. The growth rate (in individuals per year
Rainfall Accumulation Over Time
A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l
Total Rainfall Accumulation from a Discontinuous Rate Function
Rain falls at a rate (in mm/hr) given by $$ R(t)= \begin{cases} 3t, & 0 \le t < 2, \\ 5, & t = 2, \\
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.
U-Substitution Integration Challenge
Evaluate the integral $$\int_0^2 (2*x+1)\,(x^2+x+3)^5\,dx$$ using an appropriate u-substitution.
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
Bacteria Culture with Regular Removal
A bacterial culture has a population $$B(t)$$ that grows at a continuous rate of $$12\%$$ per hour,
Capacitor Discharge in an RC Circuit
In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio
Chemical Reaction in a Closed System
The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential
Chemical Reaction Rate
A chemical reaction causes the concentration $$A(t)$$ of a reactant to decrease according to the rat
Compound Interest with Continuous Payment
An investment account grows with a continuous compound interest rate $$r$$ and also receives continu
Differential Equation with Exponential Growth and Logistic Correction
Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a
Direction Field Analysis for Differential Equation
Consider the differential equation $$\frac{dy}{dx} = y\,(1-y)$$. A direction field for this equation
Direction Fields and Isoclines
Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying
Drug Concentration in the Bloodstream
A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif
Epidemic Spread Modeling
In a simplified epidemic model, the number of infected individuals $$I(t)$$ is modeled by the logist
Estimating Instantaneous Rate from a Table
A function $$f(x)$$ is defined by the following table of values:
Euler's Method Approximation
Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin
Euler's Method Approximation
Consider the initial value problem $$\frac{dy}{dt}=t\sqrt{y}$$ with $$y(0)=1$$. Use Euler's method w
Exponential Population Growth and Doubling Time
A certain population is modeled by the differential equation $$\frac{dP}{dt} = k*P$$. This equation
Flow Rate in River Pollution Modeling
A river system is modeled to study pollutant concentration $$C(t)$$ (in mg/L). Polluted water with c
FRQ 1: Slope Field Analysis for $$\frac{dy}{dx}=x$$
Consider the differential equation $$\frac{dy}{dx}=x$$. Answer the following parts.
FRQ 3: Population Growth and Logistic Model
A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt} = k*P\le
FRQ 7: Projectile Motion with Air Resistance
A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i
FRQ 12: Bacterial Growth with Limiting Resources
A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where
Loan Balance with Continuous Interest and Payments
A loan has a balance $$L(t)$$ (in dollars) that accrues interest continuously at a rate of $$5\%$$ p
Logistic Growth in Population Dynamics
The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l
Mixing Problem in a Tank
A tank initially contains 100 L of water with 5 kg of dissolved salt. Brine containing 0.1 kg of sal
Mixing Tank with Variable Inflow
A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s
Motion Under Gravity with Air Resistance
An object falling under gravity experiences air resistance proportional to its velocity. Its motion
Separable and Implicit Solution for $$\frac{dy}{dx}= \frac{x}{1+y^2}$$
Consider the differential equation $$\frac{dy}{dx}= \frac{x}{1+y^2}$$, which is defined for all real
Separable Differential Equation with Parameter Identification
A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -a*C$$, where $$C(t)$$
Slope Field and Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$
Consider the autonomous differential equation $$\frac{dy}{dx}= y*(1-y)$$. Answer the following:
Slope Field and Solution Curve Sketching
Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq
Temperature Control in a Chemical Reaction Vessel
In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external
Accumulated Change in a Population Model
A population of insects grows at a rate given by $$P'(t)=10e^{-0.2*t}$$, where $$t$$ is in days and
Analyzing a Reservoir's Volume Over Time
Water flows into a reservoir at a variable rate given by $$R(t)=50e^{-0.1*t}$$ m³/hour and simultane
Area Between Curves in Window Design
An architect is designing a decorative window whose outline is bounded by the curves $$y=5*x-x^2$$ a
Area Between Curves: Enclosed Region
The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:
Area Between Curves: Parabolic & Linear Regions
Consider the curves $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Answer the following questions regarding the re
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
Average Value of a Piecewise Function
Consider the piecewise function defined on $$[0,4]$$ by $$ f(x)= \begin{cases} x^2 & \text{for } 0
Average Value of a Temperature Function
A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t
Averaging Chemical Concentration in a Reactor
In a chemical reactor, the concentration of a substance is given by $$C(t)=100*e^{-0.5*t}+20$$ (mg/L
Bonus Payout: Geometric Series vs. Integral Approximation
A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5
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
Chemical Mixing in a Tank
A tank initially contains 100 liters of water. A chemical solution with a concentration of 0.5 g/l f
Comparing Average and Instantaneous Rates of Change
For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its
Cost Analysis of a Water Channel
A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=
Distance Traveled versus Displacement
A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for $$t\in[
Electric Charge Accumulation
A circuit has a current given by $$I(t)=4e^{-t/3}$$ A for $$t$$ in seconds. Analyze the charge accum
Environmental Contaminant Spread Analysis
A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i
Fluid Force on a Submerged Plate
A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g
Implicit Differentiation with Trigonometric Function
Consider the equation $$\cos(x * y) + x = y$$. Answer the following:
Net Cash Flow Analysis
A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont
Optimizing the Shape of a Parabolic Container
A container is formed by rotating the region under the curve $$y=8 - x^2$$ for $$0 \le x \le \sqrt{8
Particle Motion: Position, Velocity, and Acceleration
A particle moves along a straight line with an acceleration given by $$a(t)= 4 - 2*t$$ (in m/s²). Th
Radioactive Decay Accumulation
The rate of decay of a radioactive substance is given by $$R(t)=100*e^{-0.3*t}$$ decays per day. Ans
River Cross Section Area
The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$
Surface Area of a Solid of Revolution
Consider the curve $$y=\sqrt{x}$$ on the interval $$[0,9]$$. When this curve is rotated about the x-
Volume of a Hollow Cylinder Using the Shell Method
A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x
Volume of Revolution between sin(x) and cos(x)
Consider the region bounded by $$y = \sin(x)$$ and $$y = \cos(x)$$ over the interval where they inte
Work Done by a Variable Force
A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi
Work Done Pumping Water
A water tank is shaped like an inverted circular cone with a height of $$10$$ m and a top radius of
Arc Length of a Parabolic Curve
The parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ models a portion of a parabolic path for
Arc Length of a Parametric Curve
Consider the parametric curve defined by $$x(t)= t^2$$ and $$y(t)= t^3$$ for $$0 \le t \le 1$$. Anal
Arc Length of an Elliptical Curve
The parametric equations $$x(t)= 4\cos(t)$$ and $$y(t)= 3\sin(t)$$, for $$0 \le t \le \frac{\pi}{2}$
Area Between Polar Curves
In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t
Area Enclosed by a Polar Curve: Lemniscate
The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.
Conversion Between Polar and Cartesian Coordinates
Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.
Conversion from Polar to Cartesian Coordinates
The polar equation $$r(\theta)=4*\cos(\theta)$$ represents a curve.
Curve Analysis and Optimization in a Bus Route
A bus follows a route described by the parametric equations $$x(t)=t^3-3*t$$ and $$y(t)=2*t^2-t$$, w
Enclosed Area of a Parametric Curve
A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t
Exponential Growth in Parametric Representation
A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,
Intersection and Area Between Polar Curves
Two polar curves are given by $$r_1(\theta)=2\sin(\theta)$$ and $$r_2(\theta)=1+\cos(\theta)$$.
Intersection of Parametric Curves
Consider two particles moving along different paths: Particle A: $$x_A(t)= t^2, \quad y_A(t)= 2t +
Parametric and Polar Conversion Challenge
Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for
Particle Motion with Uniform Angular Change
A particle moves in the polar coordinate plane with its distance given by $$r(t)= 3*t$$ and its angl
Polar Differentiation and Tangent Lines
Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$.
Reparameterization between Parametric and Polar Forms
A curve is described by the parametric equations $$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 $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),
Symmetry and Area in Polar Coordinates
Consider the polar curve given by $$r=2\cos(\theta)$$. Answer the following:
Symmetry and Self-Intersection of a Parametric Curve
Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \
Vector Fields and Particle Trajectories
A particle moves in the plane with velocity given by $$\vec{v}(t)=\langle \frac{e^{t}}{t+1}, \ln(t+2
Vector-Valued Fourier Series Representation
The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c
Vector-Valued Function and Particle Motion
Consider the vector-valued function $$\vec{r}(t)= \langle e^t, \sin(t), \cos(t) \rangle$$ representi
Vector-Valued Function with Constant Acceleration
A particle moves in the plane with its position given by $$\vec{r}(t)=\langle 5*t, 3*t+2*t^2 \rangle
Vector-Valued Functions in Motion
A particle's position is given by the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t
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