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Absolute Value Limit Analysis II
Consider the function $$f(x)=\frac{x}{|x|}$$ for $$x \neq 0$$. Answer the following:
Analysis of a Rational Inflow Function with a Discontinuity
A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$
Continuity Assessment of a Rational Function with a Redefined Value
Consider the function $$r(x)= \begin{cases}\frac{x^2-9}{x-3}, & x \neq 3 \\ 7, & x=3\end{cases}$$.
Continuity in Piecewise Functions with Parameters
A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$
Continuity Involving a Radical Expression
Examine the function $$f(x)= \begin{cases} \frac{\sqrt{x+4}-2}{x} & x \neq 0 \\ k & x=0 \end{cases}$
End Behavior Analysis of a Rational Function
Consider the function $$f(x)=\frac{2 * x^3 - 5 * x + 1}{x^3+4 * x^2-x}$$. Answer the following:
Evaluating a Complex Limit for Continuous Extension
Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,
Inflow Function with a Vertical Asymptote
A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de
Investigating Limits at Infinity and Asymptotic Behavior
Given the rational function $$f(x)=\frac{5*x^2-3*x+2}{2*x^2+x-1}$$, answer the following: (a) Evalua
Limits and Continuity of Radical Functions
Examine the function $$f(x)= \frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$.
Limits Involving Trigonometric Functions
Consider the function $$q(x)=\frac{1-\cos(2*x)}{x^2}$$.
Manufacturing Process Tolerances
A manufacturing company produces components whose dimensional errors are found to decrease as each c
One-Sided Limits and Jump Discontinuities
Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en
Physical Applications: Temperature Continuity
A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^
Piecewise Function Continuity and Differentiability
Consider the piecewise function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\
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
Water Filling a Leaky Tank
A water tank is initially empty. Every minute, 10 liters of water is added to the tank, but due to a
Bacteria Culturing in a Bioreactor
In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5
Calculating Velocity and Acceleration from a Position Function
A car’s position along a straight road is given by the function $$s(t)= 0.5*t^3 - 3*t^2 + 4*t + 2$$
Car Motion and Critical Velocity
The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i
Chain Rule Verification with a Power Function
Let $$f(x)= (3*x+2)^4$$.
Chemical Mixing Tank
In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t
Complex Rational Differentiation
Consider the function $$f(x)=\frac{x^2+2}{x^2-1}$$. Answer the following:
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
Comprehensive Analysis of $$e^{-x^2}$$
The function $$f(x)=e^{-x^2}$$ is used to model temperature distribution in a material. Provide a co
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
Hot Air Balloon Altitude Analysis
A hot air balloon’s altitude is modeled by the function $$h(t)=5*\sqrt{t+1}$$, where $$h$$ is in met
Implicit Differentiation: Cost Allocation Model
A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1
Implicit Differentiation: Elliptic Curve
Consider the curve defined by $$2*x^2 + 3*x*y + y^2 = 20$$.
Implicit Differentiation: Mixed Exponential and Polynomial Equation
Consider the curve defined by $$x^3 + e^(y) = 3*x*y$$.
Investment Return Rates: Continuous vs. Discrete Comparison
An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.
Limit Definition of Derivative for a Rational Function
For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo
Manufacturing Production Rates
A factory produces items at a rate given by $$P_{in}(t)=\frac{200}{1+e^{-0.3*(t-4)}}$$ items per hou
Motion Along a Line
An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i
Oil Spill Containment
Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat
Optimization Using Derivatives
Consider the quadratic function $$f(x)=-x^2+4*x+5$$. Answer the following:
Piecewise Function and Discontinuity Analysis
Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2 \\ 3 & x = 2 \en
Plant Growth Rate Analysis
A plant’s height (in centimeters) after $$t$$ days is modeled by $$h(t)=0.5*t^3 - 2*t^2 + 3*t + 10$$
River Flow Dynamics
A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$
Secant and Tangent Lines: Analysis of Rate of Change
Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces
Secant and Tangent Slope Analysis
Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:
Secant vs. Tangent: Approximation and Limit Approach
Consider the function $$f(x)= \sqrt{x}$$. Use both a secant line approximation and the limit definit
Secants and Tangents in Profit Function
A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)
Tangent Line Approximation
Consider the function $$g(t)=t^2 - 4*t + 7$$. Answer the following parts to find the equation of the
Tangent Line to a Logarithmic Function
Consider the function $$f(x)= \ln(x+1)$$.
Temperature Function Analysis
Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in
Testing Differentiability at a Junction Point
Consider the function $$f(x)= \begin{cases} x+2 & x < 3 \\ 5 & x = 3 \\ 2*x-1 & x > 3 \end{cases}$$.
Velocity Function from a Cubic Position Function
An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a
Chain Rule Combined with Inverse Trigonometric Differentiation
Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:
Chain Rule in a Trigonometric Light Intensity Model
A light sensor records the intensity of light according to the function $$I(x) = \cos(\sqrt{3*x + 2}
Complex Composite and Implicit Function Analysis
Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ
Composite Function with Exponential and Radical
Consider the function $$ f(x)= \sqrt{e^{5*x}+x^2} $$.
Composite Functions in a Biological Growth Model
A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an
Composite Temperature Change in a Chemical Reaction
A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))
Continuity and Differentiability of a Piecewise Function
Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca
Differentiation of a Nested Trigonometric Function
Let $$h(x)= \sin(\arctan(2*x))$$.
Differentiation of a Product Involving Inverse Trigonometric Components
Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\
Differentiation of an Arctan Composite Function
For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$
Differentiation of Inverse Trigonometric Functions
Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and
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 and Concavity of a Logarithmic Curve
The curve defined implicitly by $$y^3 + x*y - \ln(x+y) = 5$$ is given. Use implicit differentiation
Implicit Differentiation and Inverse Challenges
Consider the implicit relation $$x^2+ x*y+ y^2=10$$ near the point (2,2).
Implicit Differentiation for an Elliptical Path
An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.
Implicit Differentiation on a Trigonometric Curve
Consider the curve defined implicitly by $$\sin(x+y) = x^2$$.
Implicit Differentiation with Trigonometric Components
Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti
Inverse Analysis of a Composite Exponential-Trigonometric Function
Let $$f(x)=e^x+\cos(x)$$. Analyze the behavior of its inverse function under appropriate domain rest
Inverse Function Differentiation with a Cubic Function
Let $$f(x)= x^3+ x + 1$$ be a one-to-one function, and let $$g$$ be its inverse function. Answer the
Inverse of a Piecewise Function
Consider the piecewise function $$f(x)=\begin{cases} x^2 & x \ge 0 \\ -x & x < 0 \end{cases}$$. Anal
Inverse of a Radical Function with Domain Restrictions
Consider the function $$f(x)=\sqrt{1-x^2}$$. Analyze its invertibility.
Inverse Trigonometric Function Differentiation
Let $$y=\arctan(\sqrt{3*x+1})$$. Answer the following parts:
Investment Growth and Rate of Change
An investor makes monthly deposits that increase according to an arithmetic sequence. The deposit am
Lake Water Level Dynamics: Seasonal Variation
A lake's water inflow is modeled by the composite function $$I(t)=p(q(t))$$, where $$q(t)=0.5*t-1$$
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
Physics Lab: Logarithmic Chain Rule in a Kinetics Experiment
In a kinetics experiment, the reactant concentration is modeled by $$C(t)=\ln(3*e^{2*t}+4)$$, where
Population Dynamics in a Fishery
A lake is being stocked with fish as part of a conservation program. The number of fish added per da
Rate of Change in a Biochemical Process Modeled by Composite Functions
The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,
Related Rates: Temperature Change in a Moving Object
An object moves along a path where its temperature is given by $$T(x)= \ln(3*x + 2)$$ and its positi
Tangent Line to a Circle via Implicit Differentiation
Consider the circle defined by $$x^2 + y^2 = 25$$. At the point $$(3, -4)$$, determine the slope of
Taylor/Maclaurin Polynomial Approximation for a Logarithmic Function
Let $$f(x) = \ln(1+3*x)$$. Develop a second-degree Maclaurin polynomial, determine its radius of con
Temperature Control: Heating Element Dynamics
A room's temperature is controlled by a heater whose output is given by the composite function $$H(t
Water Tank Composite Rate Analysis
A water tank receives water from an inflow pipe where the inflow rate is given by the composite func
Air Conditioning Refrigerant Balance
An air conditioning system is charged with refrigerant at a rate given by $$I(t)=12-0.5t$$ (kg/min)
Analyzing Pollutant Concentration in a River
The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/
Analyzing Runner's Motion
A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and
Approximating Changes with Differentials
Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch
Conical Tank Filling
A conical water tank has a height of $$10$$ m and a top radius of $$4$$ m. The water in the tank for
Conical Tank Water Flow
Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan
Critical Points and Inflection Analysis
Consider the function $$f(x)= (x-2)^2*(x+5)$$ which models a physical quantity. Answer the following
Curvature Analysis in the Design of a Bridge
A bridge's vertical profile is modeled by $$y(x)=100-0.5*x^2+0.05*x^3$$, where $$y$$ is in meters an
Estimating Rates from Experimental Position Data
The table below represents experimental measurements of the position (in meters) of a moving particl
Firework Trajectory Analysis
A firework is launched and its height (in meters) is modeled by the function $$h(t)=-4.9t^2+30t+5$$,
Implicit Differentiation on an Ellipse
An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo
Inflating Spherical Balloon
A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1
L'Hôpital's Analysis
Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following part
L'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 of Trigonometric Implicit Function
Consider the implicit equation $$\tan(x + y) = x - y$$, which implicitly defines $$y$$ as a function
Logarithmic Differentiation and Asymptotic Behavior
Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:
Logarithmic Transformation and Derivative Limits
Consider the function $$f(x)=\ln\left(\frac{e^{3x}+1}{1+e^{-x}}\right)$$. Answer the following:
Motion with Non-Uniform Acceleration
A particle moves along a straight line and its position is given by $$s(t)= 2*t^3 - 9*t^2 + 12*t + 3
Optimizing Factory Production with Log-Exponential Model
A factory's production is modeled by $$P(x)=200x^{0.3}e^{-0.02x}$$, where x represents the number of
Particle Motion Along a Line with Polynomial Velocity
A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=
Pollution Accumulation in a Lake
A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)
Production Cost Analysis
A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa
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
Revenue Function and Marginal Revenue
A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)
Series Approximation in Population Dynamics
A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans
Spherical Balloon Inflation
A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d
Temperature Change in Coffee Cooling
The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$, where $$T(t)$$ is in °F a
Vehicle Motion on a Curved Path
A vehicle moving along a straight road has its position given by $$s(t)= 4*t^3 - 24*t^2 + 36*t + 5$$
Volume Measurement Inversion
The volume of a sphere is given by $$f(x)=\frac{4}{3}*\pi*x^3$$, where $$x$$ is the radius. Analyze
Water Tank Filling: Related Rates
A cylindrical tank has a fixed radius of 2 m. The volume of water in the tank is given by $$V=\pi*r^
Analysis of a Decay Model with Constant Input
Consider the concentration function $$C(t)= 30\,e^{-0.5t} + \ln(t+1)$$, where t is measured in hours
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 a Rational Function and Its Inverse
Consider the function $$f(x)= \frac{2*x+3}{x-1}$$ defined for $$x \neq 1$$. Answer the following par
Analyzing a Function with Implicit Logarithmic Differentiation
Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating
Area Enclosed by a Polar Curve
Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima
Bank Account Growth and Instantaneous Rate
A bank account balance is modeled by the function $$B(t) = 1000*e^{0.05*t}$$, where t (in years) rep
Concavity and Inflection Points Analysis
Consider the function \( f(x)=\ln(x) - x \) where \( x > 0 \). Answer the following parts:
Economic Production Optimization
A company’s cost function is given by $$C(x) = 0.5*x^3 - 3*x^2 + 4*x + 200$$, where x represents the
Elasticity Analysis of a Demand Function
The demand function for a product is given by $$Q(p) = 100 - 5*p + 0.2*p^2$$, where p (in dollars) i
Exponential Decay in Velocity
A particle’s velocity is modeled by the function $$v(t)=10e^{-0.5*t}-3$$ (in m/s) for $$t\ge0$$.
Implicit Differentiation and Tangent Slope
Consider the curve defined implicitly by $$x^2 + x*y + y^2 = 7$$. Answer the following parts:
Intervals of Increase and Decrease in Vehicle Motion
A vehicle’s position along a straight road is given by the function $$s(t) = t^3 - 6*t^2 + 9*t + 10$
Inverse Function in a Physical Context
Suppose $$f(t)= t^3 + 2*t + 1$$ represents the displacement (in meters) of an object over time t (in
Logarithmic-Quadratic Combination Inverse Analysis
Consider the function $$f(x)= \ln(x^2+1)$$ for $$x \ge 0$$. Answer the following parts.
Mean Value Theorem with a Trigonometric Function
Let $$f(x)=\sin(x)$$ be defined on the interval $$[0,\pi]$$. Answer the following parts:
Optimization in Particle Routing
A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe
Optimizing Fencing for a Field
A farmer has $$100$$ meters of fencing to construct a rectangular field that borders a river (no fen
Optimizing Fencing for a Rectangular Garden
A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee
Particle Motion on a Curve
A particle moves along a straight-line path with its position given by \( s(t)=t^3 - 6*t^2 + 9*t + 1
Projectile Motion Analysis
A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th
Projectile Trajectory: Parametric Analysis
A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)
Rate of Change in a Chemical Reaction
The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in
Region Area and Volume: Polynomial and Linear Function
A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo
Retirement Savings with Diminishing Deposits
Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th
Taylor Polynomial for $$\ln(x)$$ about $$x=1$$
For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,
Taylor Series for $$\arcsin(x)$$
Derive the Maclaurin series for $$f(x)=\arcsin(x)$$ up to the $$x^5$$ term, determine the radius of
Taylor Series for $$\ln(1+3*x)$$
Let $$f(x)=\ln(1+3*x)$$. Develop its Maclaurin series up to the 3rd degree, determine the radius of
Taylor Series for $$\sqrt{1+x}$$
Consider the function $$f(x)=\sqrt{1+x}$$. In this problem, compute its 3rd degree Maclaurin polynom
Temperature Change in a Weather Balloon
A weather balloon’s temperature and altitude are related by the implicit equation $$T*e^{z} + z = 50
Volume Using Cylindrical Shells
The region bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is revolved about the y-axis to form a solid.
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
Accumulation Function in an Investment Model
An investment has an instantaneous rate of return given by $$r(t)=0.05*t+0.02$$ (per year). The accu
Antiderivative Application in Crop Growth
A crop field grows at a rate modeled by the function $$G'(t)=4*t-3$$ (in square meters per week). Th
Antiderivatives and the Fundamental Theorem
Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i
Approximating Energy Consumption Using Riemann Sums
A household’s power consumption (in kW) is recorded over an 8‐hour period. The following table shows
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
Chemical Reaction Rates
A chemical reaction in a vessel occurs at a rate given by $$R(t)= 8*e^{-t/2}$$ mmol/min. Determine t
Cost Function Accumulation
A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x
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
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
Differentiation and Integration of a Power Series
Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.
Error Bound Analysis for the Trapezoidal Rule
For the function $$f(x)=\ln(x)$$ on the interval $$[1,2]$$, the error bound for the trapezoidal rule
Evaluating an Integral Involving an Exponential Function
Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.
Evaluating an Integral via U-Substitution
Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.
Filling a Tank: Antiderivative with Initial Value
Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial
Fuel Consumption Estimation with Midpoint Riemann Sums
A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The
Integration of a Piecewise-Defined Function
Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \
Integration of a Trigonometric Product via U-Substitution
Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.
Interpreting Area Under a Curve from a Graph
A graph displays the function $$f(x)=0.5*x+1$$ over the interval $$[0,6]$$.
Marginal Cost and Production
A factory's marginal cost function is given by $$MC(x)= 4 - 0.1*x$$ dollars per unit, where $$x$$ is
Midpoint Approximation Analysis
Let $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Answer the following:
Minimizing Material for a Container
A company wants to design an open-top rectangular container with a square base that must have a volu
Riemann Sum Estimation from Tabular Data
The following table lists values of a function $$f(x)$$ at selected points, which are used to approx
Signal Energy through Trigonometric Integration
A signal is described by $$f(t)=3*\sin(2*t)+\cos(2*t)$$. The energy of the signal over one period
Trapezoidal Approximation of a Definite Integral from Tabular Data
The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra
U-Substitution in Accumulation Functions
In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe
U-Substitution Integration Challenge
Evaluate the integral $$\int_0^2 (2*x+1)\,(x^2+x+3)^5\,dx$$ using an appropriate u-substitution.
Vehicle Motion and Inverse Time Function
A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se
Water Flow and the Trapezoidal Rule
Water flows into a reservoir at a rate given by $$R(t)$$ (in m³/hour) as provided in the table below
Work Done by a Variable Force
A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp
Work Done by an Exponential Force
A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\
Analysis of an Inverse Function from a Differential Equation Solution
Suppose a differential equation is solved to give an increasing function $$f(x)=\ln(2*x+3)$$ defined
Chemical Reaction and Separable Differential Equation
In a particular chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to t
Compound Interest with Continuous Payment
An investment account grows with a continuous compound interest rate $$r$$ and also receives continu
Depreciation Model of a Vehicle
A vehicle's value depreciates continuously over time according to the differential equation $$\frac{
Euler's Method Approximation
Consider the initial value problem $$\frac{dy}{dt}=t\sqrt{y}$$ with $$y(0)=1$$. Use Euler's method w
Existence and Uniqueness in an Implicit Differential Equation
Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y
Exponential Growth with Variable Rate
A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the
Exponential Population Growth and Doubling Time
A certain population is modeled by the differential equation $$\frac{dP}{dt} = k*P$$. This equation
FRQ 8: RC Circuit Analysis
In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$
FRQ 10: Cooling of a Metal Rod
A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th
FRQ 11: Linear Differential Equation via Integrating Factor
Solve the differential equation $$\frac{dy}{dx}+\frac{1}{x}y=\sin(x)$$ with the initial condition $$
Growth and Decay in a Bioreactor
In a bioreactor, the concentration of a chemical P (in mg/L) evolves according to the differential e
Inverse Function Analysis Derived from a Differential Equation Solution
Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e
Investment Growth Model
An investment account grows continuously at a rate proportional to its current balance. The balance
Logistic Equation with Harvesting
A fish population in a lake follows a logistic growth model with the addition of a constant harvesti
Logistic Growth in Population Dynamics
The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l
Mixing Problem with Constant Flow Rate
A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen
Mixing Problem with Time-Dependent Inflow
A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa
Modeling Ambient Temperature Change
The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT
Motion along a Line with a Separable Differential Equation
A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra
Motion Under Gravity with Air Resistance
An object falling under gravity experiences air resistance proportional to its velocity. Its motion
Newton's Law of Cooling
An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur
Newton's Law of Cooling
An object cools according to Newton's Law of Cooling, which is modeled by the differential equation
Picard Iteration for Approximate Solutions
Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic
Population Growth with Logistic Differential Equation
A population $$P(t)$$ is modeled by the logistic differential equation $$\frac{dP}{dt} = r\,P\left(1
Solving a Separable Differential Equation
Solve the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(0)=
Temperature Control in a Chemical Reaction Vessel
In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external
Analyzing Acceleration Data from Discrete Measurements
A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use
Arc Length in Polar Coordinates
Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\
Arc Length of a Logarithmic Curve
Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.
Area Between a Rational Function and Its Asymptote
Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter
Area Between Curves: Enclosed Region
The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. 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
A city's temperature during a day is modeled by $$T(t)=10+5*\sin\left(\frac{\pi*t}{12}\right)$$ for
Average Temperature Over a Day
A function modeling the temperature (in °F) throughout a day is given by $$T(t)= 3*\sin\left(\frac{\
Average Value of a Trigonometric Function
Let $$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 [
Bacterial Decay Modeled by a Geometric Series
A bacterial culture is treated with an antibiotic that reduces the bacterial population by 20% each
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
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
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.
Electric Charge Distribution Along a Rod
A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per
Moment of Inertia of a Thin Plate
A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$
Particle Motion: Position, Velocity, and Acceleration
A particle moves along a straight line with acceleration given by $$a(t)=3*t-4$$ (in m/s²). At time
Population Growth: Cumulative Increase
A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t
Volume about a Vertical Line using Two Methods
A region in the first quadrant is bounded by $$y=x$$, $$y=0$$, and $$x=2$$. This region is rotated a
Volume by Revolution: Washer Method
Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$. When this region is rotated about
Volume of a Hollow Cylinder Using the Shell Method
A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x
Volume of a Solid by the Disc Method
Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio
Volume of a Solid Rotated about y = -1
The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$
Volume of a Solid using the Shell Method
Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, $$x=1$$, and $$x=4$$. When this region is ro
Volume of a Solid Using the Shell Method
The region in the first quadrant bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is rotated about the y-axi
Volume of a Solid via Shell Method
Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved
Volume of a Solid with Equilateral Triangle Cross Sections
Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by
Volume of a Solid with Square Cross Sections
The base of a solid is the region in the plane bounded by $$y=x$$ and $$y=x^2$$ (with $$x$$ between
Acceleration Analysis in a Vector-Valued Function
Consider a particle whose position is given by $$ r(t)=\langle \sin(2*t),\; \cos(2*t) \rangle $$ for
Analysis of Vector Trajectories
A particle in the plane follows the path given by $$\mathbf{r}(t)=\langle \ln(t+1), \sqrt{t} \rangle
Area Between Two Polar Curves
Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of
Continuity Analysis of a Discontinuous Parametric Curve
Consider the parametric curve defined by $$x(t)= \begin{cases} t^2, & t < 1 \\ 2*t - 1, & t \ge 1 \
Curvature of a Vector-Valued Function
Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given
Designing a Parametric Curve for a Cardioid
A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.
Designing a Race Track with Parametric Equations
An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:
Distance Traveled in a Turning Curve
A curve is defined by the parametric equations $$x(t)=4*\sin(t)$$ and $$y(t)=4*\cos(t)$$ for $$0\le
Intersection Points of Polar Curves
Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:
Kinematics in Polar Coordinates
An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t
Modeling Periodic Motion with a Vector Function
A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle \cos(2*t),\;
Modeling with Polar Data
A researcher collects the following polar coordinate data for a phenomenon.
Motion Along a Parametric Curve
Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i
Parametric Curve Intersection
Two curves are defined parametrically as follows: For curve C1, $$x(t) = t^2$$ and $$y(t) = 2*t + 1$
Parametric Curve with a Loop and Tangent Analysis
Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An
Periodic Motion: A Vector-Valued Function
A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$
Polar to Parametric Conversion and Arc Length
A curve is defined in polar coordinates by $$r= 1+\sin(\theta)$$. Convert and analyze the curve.
Sensitivity Analysis and Linear Approximation using Implicit Differentiation
The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.
Symmetry and Self-Intersection of a Parametric Curve
Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \
Tangent Line to a 3D Vector-Valued Curve
Let $$\textbf{r}(t)= \langle t^2, \sin(t), \ln(t+1) \rangle$$ for $$t \in [0,\pi]$$. Answer the foll
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: Tangent and Normal Components
A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*
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