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Absolute Value Function Limits
Examine the function $$f(x)=\frac{|x-2|}{x-2}$$.
Algebraic Manipulation with Radical Functions
Let $$f(x)= \frac{\sqrt{x+5}-3}{x-4}$$, defined for $$x\neq4$$. Answer the following:
Analyzing a Composite Function Involving a Limit
Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven
Analyzing Limits of a Combined Exponential‐Log Function
Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and
Analyzing Limits of a Composite Function
Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:
Application of the Squeeze Theorem with Trigonometric Functions
Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior
Caffeine Metabolism in the Human Body
A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu
Continuity and the Intermediate Value Theorem in Temperature Modeling
A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ
Continuity in a Parametric Function Context
A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an
Continuity of Log‐Exponential Function
Consider the function $$f(x)= \frac{e^x - \ln(1+x) - 1}{x}$$ for $$x \neq 0$$, with $$f(0)=c$$. Dete
Economic Model of Depreciating Car Value
A car purchased for $$30,000$$ dollars depreciates in value by $$15\%$$ each year. The value of the
Environmental Pollution Modeling
In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural
Epsilon-Delta Style (Conceptual) Analysis
Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the
Evaluating a Logarithmic Limit
Given the limit $$\lim_{x \to 2} \frac{\ln(x-1)}{x^2-4} = k$$, find the value of $$k$$ using algebra
Graphical Analysis of a Continuous Polynomial Function
Consider the function $$f(x)=2*x^3-5*x^2+x-7$$ and its graph depicted below. The graph provided accu
Graphical Analysis of a Removable Discontinuity
Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans
Graphical Analysis of Limits and Asymptotic Behavior
A graphical study titled 'Graph of Experimental Data' shows the measured concentration of a chemical
Graphical Analysis of Volume with a Jump Discontinuity
A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer
Horizontal and Vertical Asymptotes of a Rational Function
Let $$h(x)=\frac{2*x^2-3*x+1}{x^2-1}.$$ Answer the following:
Intermediate Value Theorem Application
Suppose $$f(x)$$ is a continuous function on the interval $$[1, 5]$$ with $$f(1) = -2$$ and $$f(5) =
Investigating Infinite Limits: Vertical and Horizontal Asymptotes
Given the function $$f(x)=\frac{2*x}{x-3}$$, answer the following questions: (a) Determine $$\lim_{x
Limits Involving Infinity and Vertical Asymptotes
Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following:
Limits with Composite Logarithmic Functions
Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.
Modeling with a Removable Discontinuity
A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi
One-Sided Infinite Limits in Rational Functions
Consider the function $$f(x)= \frac{1}{(x-2)^2}$$.
Piecewise Function Critical Analysis
Consider the piecewise function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} & \text{if } x \neq 2 \\
Pollution Level Analysis and Removable Discontinuity
A function $$f(x)$$ represents the concentration of a pollutant (in mg/L) in a river as a function o
Saturation of Drug Concentration in Blood
A patient is given a drug with each dose containing 50 mg. However, due to metabolism, only 20% of t
Squeeze Theorem with a Log Function
Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$
Squeeze Theorem with Oscillatory Behavior
Examine the function $$s(x)=x^2*\sin(1/x)$$ for x ≠ 0.
Taylor Series Expansion for $$\arctan(x)$$
Consider the function $$f(x)=\arctan(x)$$ and its Taylor series about $$x=0$$.
Temperature Change Analysis
The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi
Application of Derivative to Relative Rates in Related Variables
Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^
Car Motion: Velocity and Acceleration
A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $
Cost Optimization in Production: Derivative Application
A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu
Derivatives of Inverse Functions
Let $$f(x)=\ln(x)$$ with inverse function $$f^{-1}(x)=e^x$$. Answer the following parts.
Differentiation in Exponential Growth Models
A population is modeled by $$P(t)=P_0e^{r*t}$$ with the initial population $$P_0=500$$ and growth ra
Error Bound Analysis for Cos(x) Approximations in Physical Experiments
In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze
Finding the Derivative of a Logarithmic Function
Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:
Graphical Estimation of Tangent Slopes
Using the provided graph of a function g(t), analyze its rate of change at various points.
Growth Rate of a Bacterial Colony
The radius of a bacterial colony is modeled by $$r(t)= \sqrt{4*t+1}$$, where t (in hours) represents
Heat Transfer in a Rod: Modeling and Differentiation
The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me
Implicit Differentiation and Tangent Line Slope
Consider the curve defined by $$x^2 + x*y + y^2 = 7$$. Answer the following:
Implicit Differentiation on a Circle
Consider the circle defined by $$x^2 + y^2 = 25$$.
Instantaneous Velocity from a Displacement Function
A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)
Irrigation Reservoir Analysis
An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters
Linearization and Tangent Approximations
Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$
Maclaurin Series for e^x Approximation
Consider the function $$f(x)=e^x$$, which models many growth processes in nature. Use its Maclaurin
Optimization and Tangent Lines
A rectangular garden is to be constructed along a river with 100 meters of fencing available for thr
Parametric Analysis of a Curve
A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,
Particle Motion in the Plane
A particle moves in the plane with its position given by $$x(t)=t^2-4*t+1$$ and $$y(t)=3*t-2.5$$, wh
Particle Motion on a Straight Line: Average and Instantaneous Rates
A particle moving along a straight line has its position given by $$s(t)=t^3 - 6*t^2 + 9*t + 4$$ for
Pollutant Levels in a Lake
A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre
Radioactive Decay and Derivative
A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is
Savings Account Growth: From Discrete Deposits to Continuous Derivatives
An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con
Secant and Tangent Slope Analysis
Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:
Taylor Expansion of a Polynomial Function Centered at x = 1
Given the polynomial function $$f(x)=3+2*x- x^2+4*x^3$$, analyze its Taylor series expansion centere
Traffic Flow Analysis
A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve
Vector Function and Motion Analysis
A particle moves according to the vector function $$\vec{r}(t)=\langle 2*\cos(t), 2*\sin(t)\rangle$$
Analyzing an Implicit Function with Mixed Variables
Consider the curve defined by $$x^3 + x*y + y^3 = 3$$. Analyze the derivative of the curve at a give
Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment
In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)
Chain Rule with Trigonometric Composite Function
Examine the function $$ h(x)= \sin((2*x^2+1)^2) $$.
Coffee Cooling Dynamics using Inverse Function Differentiation
A cup of coffee cools according to the model $$T=100-a\,\ln(t+1)$$, where $$T$$ is the temperature i
Derivative of an Inverse Function with a Quadratic
Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of
Differentiation of an Inverse Function
Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a
Differentiation of an Inverse Trigonometric Form
Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.
Differentiation of Composite Inverse Trigonometric Function involving a Rational Function
Differentiate the function $$f(x)= \arccos\left(\frac{3*x}{1+x^2}\right)$$ with respect to $$x$$ and
Enzyme Kinetics in a Biochemical Reaction
In an enzymatic reaction, the substrate concentration $$S(t)$$ and the product concentration $$P(t)$
Financial Flow Analysis: Investment Rates
An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$
Higher-Order Derivatives via Implicit Differentiation
Consider the implicit relation $$x^2 + x*y + y^2 = 7$$.
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 in a Circle
Consider the circle defined by $$ x^2+y^2=49 $$.
Implicit Differentiation in a Cost-Production Model
In an economic model, the relationship between the production level $$x$$ (in units) and the average
Implicit Differentiation in Economic Equilibrium
In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand
Implicit Differentiation: Circle and Tangent Line
The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva
Inverse Function Derivative in an Exponential Model
Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p
Inverse Function Differentiation for a Quadratic Function
Let $$ f(x)= (x+1)^2 $$ with the domain $$ x\ge -1 $$. Consider its inverse function $$ f^{-1}(y) $$
Inverse Function Differentiation for a Trigonometric-Polynomial Function
Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with
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 Function Differentiation with Composite Trigonometric Functions
Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$
Multi-step Differentiation of a Composite Logarithmic Function
Consider the function $$F(x)= \sqrt{\ln\left(\frac{1+e^{2*x}}{1-e^{2*x}}\right)}$$, defined for valu
Rocket Fuel Consumption Analysis
A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)
Second Derivative of an Implicit Function
The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:
Tangent Line for a Parametric Curve
A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.
Trigonometric Composite Inverse Function Analysis
Consider the function $$f(x)=\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{
Analysis of Particle Motion
A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest
Analyzing Rate of Approach in a Pursuit Problem
Two cars are traveling on perpendicular roads. Car A is moving east at 60 km/h and is 3 km from the
Approximating Function Values Using Differentials
Let $$f(x)=\sqrt{x}$$. Use linearization near $$x=25$$ to approximate $$\sqrt{25.5}$$.
Differentiation and Concavity for a Non-Motion Problem: Water Filling a Tank
The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an
Estimating the Rate of Change from Reservoir Data
A reservoir's water level h (in meters) was recorded at different times, as shown in the table below
Exponential Function Inversion
Consider the function $$f(x)=e^{2*x}+3$$ which models the growth of a certain variable. Analyze the
Forensic Gas Leakage Analysis
A gas tank under investigation shows leakage at a rate of $$O(t)=3t$$ (liters per minute) while it i
GDP Growth Analysis
A country's GDP (in billions of dollars) is modeled by the function $$G(t)=200e^{0.04*t}$$, where t
Interpreting the Derivative in Straight Line Motion
A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a
Linear Account Growth in Finance
The amount in a savings account during a promotional period is given by the linear function $$A(t)=1
Linearization Approximation
Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans
Motion on a Straight Line with a Logarithmic Position Function
A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),
Parametric Curve Motion
A particle’s trajectory is given by the parametric equations $$x(t)=t^2-1$$ and $$y(t)=2*t+3$$ for $
Pool Water Volume Change
The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea
Projectile Motion with Exponential Term
A projectile's height is given by $$h(t)=50t-5t^2+e^{-0.5t}$$, where h is measured in meters and t i
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 and Marginal Analysis
A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit
River Flow Diversion
At a river junction, water flows in at a rate of $$I(t)=30+5t$$ (cubic feet per second) and exits at
Road Trip Distance Analysis
During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is
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 $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$
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
Temperature Change of Coffee: Exponential Cooling
The temperature of a cup of coffee is modeled by the function $$x(t)= 70 + 50e^{-0.1*t}$$, where $$t
Analysis of an Exponential-Linear Function
Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:
Application of Rolle's Theorem to a Trigonometric Function
Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:
Application of the Extreme Value Theorem in Economics
A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold
Arc Length of a Parametric Circular Arc
A curve is defined parametrically by $$x(t) = 2*\cos(t)$$ and $$y(t) = 2*\sin(t)$$, where t varies f
Car Depreciation Analysis
A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de
Composite Functions and Derivatives
Let $$h(x)=f(g(x))$$ where $$f(u)=u^2+3$$ and $$g(x)=\sin(x)$$. Analyze the composite function on th
Concavity Analysis in a Revenue Model
A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -
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.
Concavity and Points of Inflection
Consider the function $$f(x)=x^3 - 6*x^2 + 9*x + 2$$. Analyze the concavity of the function using th
Construction Payment Milestones
A construction project is structured around milestone payments. The first payment is $$10000$$ dolla
Differentiability and Optimization of a Piecewise Function
Consider the piecewise function $$f(x)=\begin{cases} x^2, & x \le 2 \\ 4*x - 4, & x > 2 \end{cases}
Error Estimation in Approximating $$e^x$$
For the function $$f(x)=e^x$$, use the Maclaurin series to approximate $$e^{0.3}$$. Then, determine
Finding Local Extrema for an Exponential-Logarithmic Function
The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:
Function Behavior Analysis
Consider the function \( f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1 \). Answer the following parts:
Garden Design Optimization
A gardener wants to design a rectangular garden adjacent to a river, so that fencing is required for
Implicit Differentiation and Tangent to an Ellipse
Consider the ellipse defined by the equation $$4*x^2 + 9*y^2 = 36$$. Answer the following parts:
Inverse Analysis for a Function with Multiple Transformations
Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.
Mean Value Theorem in River Flow
A river cross‐section’s depth (in meters) is modeled by the function $$f(x) = x^3 - 4*x^2 + 3*x + 5$
Optimization in Particle Routing
A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe
Optimizing Fencing for a Rectangular Garden
A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee
Optimizing Material for a Container
An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.
Parameter Estimation in a Log-Linear Model
In a scientific experiment, the data is modeled by $$P(t)= A\,\ln(t+1) + B\,e^{-t}$$. Given that $$P
Radiocarbon Dating in Artifacts
An archaeological artifact contains a radioactive isotope with an initial concentration of 100 units
Related Rates: Expanding Balloon
A spherical balloon is being inflated so that its volume $$V$$ increases at a constant rate of $$\fr
Relative Motion in Two Dimensions
A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$
Revenue Optimization in Business
A company’s price-demand function is given by $$P(x)= 50 - 0.5*x$$, where $$x$$ is the number of uni
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
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.
Accumulated Displacement from a Piecewise Velocity Function
A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\
Analyzing a Cumulative Distribution Function (CDF)
A chemical reaction has a time-to-completion modeled by the cumulative distribution function $$F(t)=
Approximating Water Volume Using Riemann Sums
A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a
Arc Length of an Architectural Arch
An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len
Area Estimation Using Riemann Sums for $$f(x)=x^2$$
Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef
Continuous Antiderivative for a Piecewise Function
A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,
Cyclist's Distance Accumulation Function
A cyclist’s total distance traveled is modeled by $$D(t)= \int_{0}^{t} (5+\sin(u))\, du + 2$$ kilom
Evaluating an Integral Involving an Exponential Function
Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.
Evaluating an Integral Using U-Substitution
Evaluate the indefinite integral $$\int (x-4)^{10}\,dx$$ using u-substitution.
Finding Area Between Two Curves
Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.
Fundamental Theorem and Total Accumulated Growth
A bacteria culture grows according to the logistic model $$\frac{dN}{dt}=N\left(1-\frac{N}{10000}\r
Improper Integral Convergence
Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.
Integration by U-Substitution and Evaluation of a Definite Integral
Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut
Integration of a Piecewise Function for Total Area
Consider the piecewise function $$f(x)$$ defined by: $$f(x)=3-x$$ for $$0 \le x \le 3$$, and $$f(x)=
Integration Using U-Substitution
Evaluate the definite integral $$\int_{0}^{2} (3*x+1)^{4} dx$$ using u-substitution. Answer the foll
Logistic Growth and Population Integration
A population grows according to the logistic differential equation $$\frac{dP}{dt}=k*P\left(1-\frac
Numerical Approximation: Trapezoidal vs. Simpson’s Rule
The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu
Particle Displacement and Total Distance
A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le
Population Model Inversion and Accumulation
Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d
Rate of Production in a Factory
A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per
Reservoir Water Level
A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$
Revenue Estimation Using the Trapezoidal Rule
A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t
Total Cost from a Marginal Cost Function
A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre
Trapezoidal Approximation for a Curved Function
Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:
Vehicle Motion and Inverse Time Function
A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se
Volume of a Solid by the Shell Method
Consider the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and the vertical line $$x=4$$.
Work Done by an Exponential Force
A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\
Work on a Nonlinear Spring
A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di
Analysis of a Piecewise Function with Potential Discontinuities
Consider the piecewise defined function $$ f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x < 2,
Bacterial Growth with Predation
A bacterial culture grows according to the differential equation $$\frac{dB}{dt}= r*B - P$$, where $
Compound Interest and Investment Growth
An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the
Cooling and Mixing Combined Problem
A container holds 2 L of water initially at 80°C. Cold water at 20°C flows into the container at a r
Differential Equation with Exponential Growth and Logistic Correction
Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a
Drug Concentration in the Bloodstream
A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif
Estimating Total Change from a Rate Table
A car's velocity (in m/s) is recorded at various times according to the table below:
Euler's Method Approximation
Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin
Exact Differential Equation
Examine the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y)\,dy = 0 $$. Determine if the
Exact Differential Equations
Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2+2*x*y)\,dy = 0 $$. Answer the followi
FRQ 12: Bacterial Growth with Limiting Resources
A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where
FRQ 16: Harvesting in a Predator-Prey Model
A prey population $$P(t)$$ in a marine ecosystem is modeled by the differential equation $$\frac{dP}
Logistic Model in Product Adoption
A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.
Logistic Population Growth Model
A population is modeled by the logistic differential equation $$\frac{dP}{dt} = r*P\left(1-\frac{P}{
Modeling Ambient Temperature Change
The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT
Newton's Law of Cooling
A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti
Newton's Law of Cooling
A hot liquid cools in a room maintained at a constant temperature $$T_{room}$$. The temperature $$T(
RC Circuit: Voltage Decay
In an RC circuit, the voltage across a capacitor satisfies $$\frac{dV}{dt} = -\frac{1}{R*C} * V$$. G
Relative Motion with Acceleration
A car starts from rest and its velocity $$v(t)$$ (in m/s) satisfies the differential equation $$\fra
Separable Differential Equation with Absolute Values
Consider the differential equation $$\frac{dy}{dx} = \frac{|x|}{y}$$ with the condition that $$y>0$$
Separable Differential Equation with Initial Condition
Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(
Sketching a Solution Curve from a Slope Field
A slope field for the differential equation $$\frac{dy}{dt}=y(1-y)$$ is provided. Use the slope fiel
Slope Field Analysis for $$\frac{dy}{dx}=x$$
Consider the differential equation $$\frac{dy}{dx}= x$$. A slope field for this differential equatio
Temperature Change with Variable Ambient Temperature
A heated object is cooling in an environment where the ambient temperature changes over time. For $$
Variable Carrying Capacity in Population Dynamics
In a modified logistic model, the carrying capacity of a population is time-dependent and given by $
Viral Spread on Social Media
Let $$V(t)$$ denote the number of viral posts on a social media platform. Posts go viral at a consta
Approximating Functions using Taylor Series
Consider the function $$f(x)= \ln(1+2*x)$$. Use Taylor series methods to approximate and analyze thi
Area Between Curves from Experimental Data
In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres
Area Enclosed by a Cardioid in Polar Coordinates
Consider the polar curve given by $$r(\theta)=1+\cos(\theta)$$.
Area Under a Parametric Curve
Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u
Average Cost Function in Production
A factory’s cost function for producing $$x$$ units is modeled by $$C(x)=0.5*x^2+3*x+100$$, where $$
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 Computation
Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=
Average Value of a Piecewise Function
Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x
Average Value of a Piecewise Function
Consider the piecewise function defined on $$[0,4]$$ by $$ f(x)= \begin{cases} x^2 & \text{for } 0
Average Velocity of a Car
A car's velocity is given by $$v(t)=20-4*\ln(t+1)$$ (in m/min) for $$t$$ in minutes on the interval
Balloon Inflation Related Rates
A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a
Bonus Payout: Geometric Series vs. Integral Approximation
A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5
Center of Mass of a Lamina
A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a
Center of Mass of a Plate
A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a
Cyclist Average Speed Calculation
A cyclist’s velocity is given by $$v(t) = t^2 - 4*t + 6$$ (in m/s) for $$t$$ in the interval $$[0,4]
Displacement vs. Distance: Analysis of Piecewise Velocity
A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^
Inflow vs Outflow: Water Reservoir Capacity
A reservoir receives water with an inflow rate given by $$I(t)=20+5\sin(t)$$ (liters/min) and discha
Moment of Inertia of a Thin Plate
A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$
Optimization and Integration: Maximum Volume
A company designs open-top cylindrical containers to hold $$500$$ liters of liquid. (Recall that $$1
Optimizing the Thickness of a Cooling Plate
The local heat conduction efficiency at a point on a cooling plate is modeled by the function $$A(x)
Polar Coordinates: Area of a Region
A region in the plane is described in polar coordinates by the equation $$r= 2+ \cos(θ)$$. Determine
Projectile Motion under Gravity
An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial
Volume about a Vertical Line using Two Methods
A region in the first quadrant is bounded by $$y=x$$, $$y=0$$, and $$x=2$$. This region is rotated a
Volume by Shell Method: Rotated Parabolic Region
Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y
Volume by the Shell Method
Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is revolved about t
Volume of a Solid via the Disc Method
The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th
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
Volume of an Irregular Tank
A water tank has a varying cross-sectional profile described by $$y(x)=\sqrt{25 - (x-5)^2}$$, for $$
Volume with Equilateral Triangle Cross Sections
The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros
Work Done by a Variable Force
A variable force given by $$F(x)= 2*x + 3$$ (in Newtons) is applied to an object as it moves along a
Work Done by a Variable Force
A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as
Analyzing Oscillatory Motion in Parametric Form
The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$
Arc Length of a Parametric Curve with Logarithms
Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \
Arc Length of a Polar Curve
Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:
Arc Length of a Quarter-Circle
Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l
Area Enclosed by a Polar Curve: Lemniscate
The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.
Comparing Parametric, Polar, and Cartesian Representations
An object moves along a curve described by the parametric equations $$x(t)= \frac{t}{1+t^2}$$ and $$
Conversion Between Polar and Cartesian Coordinates
Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.
Conversion to Cartesian and Analysis of a Parametric Curve
Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.
Converting and Analyzing a Polar Equation
Examine the polar equation $$r=2+3\cos(\theta)$$.
Exploring Polar Curves: Spirals and Loops
Consider the polar curve $$r=θ$$ for $$0 \le θ \le 4\pi$$, which forms a spiral. Analyze the spiral
Integration of Speed in a Parametric Motion
For the parametric curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$,
Intersection of Polar and Parametric Curves
Consider the polar curve given by $$r = 2\cos(θ)$$ and the parametric curve defined by $$x(t)= 1+t^2
Lissajous Figures and Their Properties
A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)
Motion Analysis via a Vector-Valued Function
An object's position is described by the vector function $$\mathbf{r}(t)= \langle e^{-t}, \; \ln(1+t
Motion in a Damped Force Field
A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t)
Oscillatory Behavior in Vector-Valued Functions
Examine the vector-valued function $$\mathbf{r}(t)=\langle \cos(2*t), \sin(3*t), \cos(t)\sin(2*t) \r
Oscillatory Motion in a Vector-Valued Function
Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in
Parametric Curves and Intersection Points
Two curves are defined by $$C_1: x(t)=t^2,\, y(t)=2*t+1$$ and $$C_2: x(s)=4-s^2,\, y(s)=3*s$$. Find
Parametric Intersection and Enclosed Area
Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given
Parametric Motion and Change of Direction
A particle moves along a path defined by the parametric equations $$x(t)=t^3-3t$$ and $$y(t)=2t^2$$
Parametric Representation of an Ellipse
An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$
Polar Plots and Intersection Points in Design
A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th
Projectile Motion via Parametric Equations
A projectile is launched with initial speed $$v_0 = 20\,m/s$$ at an angle of $$45^\circ$$. Its motio
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
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 Functions and Curvature
Let the vector-valued function be $$\vec{r}(t)= \langle t, t^2, t^3 \rangle$$.
Vector-Valued Kinematics
A particle follows a path in space described by the vector-valued function $$r(t) = \langle \cos(t),
Wind Vector Analysis in Navigation
A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)
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