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Absolute Value Function Limit Analysis
Consider the function $$f(x)=\frac{|x-3|}{x-3}$$. Answer the following:
Absolute Value Function Limits
Examine the function $$f(x)=\frac{|x-2|}{x-2}$$.
Analysis of a Jump Discontinuity
Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.
Analysis of a Piecewise Function with Multiple Definitions
Consider the function $$h(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x<3, \\ 2*x-1 & \text{if
Application of the Squeeze Theorem with Trigonometric Functions
Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior
Applying the Squeeze Theorem
Let $$f(x)=x^2\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$. Use the Squeeze Theorem to evaluat
Compound Interest and Loan Repayment
A simplified model for a loan repayment assumes that a borrower owes $$10,000$$ dollars and the rema
Continuity Across Piecewise‐Defined Functions with Mixed Components
Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\
Evaluating a Limit Involving a Radical and Trigonometric Component
Consider the function $$f(x)= \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$$. Answer the following:
Finding a Parameter in a Limit Involving Logs and Exponentials
Consider the function $$ f(x)= \frac{\ln(1+kx) - (e^x - 1)}{x^2}, $$ for $$x \neq 0$$. Assume that $
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
Infinite Limits and Vertical Asymptotes
Let $$g(x)=\frac{1}{(x-2)^2}$$. Answer the following:
Intermediate Value Theorem Application
Suppose $$f(x)$$ is a continuous function on the interval $$[1, 5]$$ with $$f(1) = -2$$ and $$f(5) =
Investigating a Function with a Removable Discontinuity
Consider the function $$f(x)=\frac{x^2-9}{x-3}$$ for x \neq 3. Answer the following:
Jump Discontinuity Analysis using Table Data
A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer
Limit Definition of the Derivative for a Polynomial Function
Let $$f(x)=3*x^2-2*x+1$$. Use the limit definition of the derivative to find $$f'(2)$$.
Limits Involving Absolute Value Functions
Consider the function $$f(x)=\frac{|x-3|}{x-3}$$. Answer the following:
Limits Involving Exponential Functions
Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.
Limits Involving Trigonometric Functions
Consider the function $$q(x)=\frac{1-\cos(2*x)}{x^2}$$.
Limits Involving Trigonometric Functions and the Squeeze Theorem
Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E
Limits via Improper Integration Representation
Consider the function defined by the integral $$f(x)= \int_{1}^{x} \frac{1}{t^2} dt$$ for x > 1. Add
One-Sided Limits for a Piecewise Inflow
In a pipeline system, the inflow rate is modeled by the piecewise function $$R_{in}(t)= \begin{case
One-Sided Limits in a Piecewise Function
Consider the function $$f(x)=\begin{cases} \sqrt{x+4}, & x < 5, \\ 3*x-7, & x \ge 5. \end{cases}$$ A
Oscillatory Behavior and Limits
Consider the function $$f(x)=x\sin(1/x)$$ for x \neq 0, with f(0) defined to be 0. Use the following
Piecewise Function Continuity and Differentiability
Consider the piecewise function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\
Removable Discontinuity and Limit Evaluation
Consider the function $$f(x)=\frac{x^2-9}{x-3}$$ for $$x \neq 3$$. Answer the following: (a) Evaluat
Telecommunications Signal Strength
A telecommunications tower emits a signal whose strength decreases by $$20\%$$ for every additional
Temperature Change Analysis
The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi
Analysis of a Quadratic Function
Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer
Analyzing Car Speed from a Distance-Time Table
A car's position (in meters) is recorded at various times (in seconds) as shown in the table. Use th
Car Motion: Velocity and Acceleration
A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $
Continuous Compound Interest Analysis
For an investment, the amount at time $$t$$ (in years) is modeled by $$A(t)=P*e^{r*t}$$, where $$P$$
Cost Minimization in Packaging
A company's packaging box has a cost function given by $$C(x)=0.05*x^2 - 4*x + 200$$, where $$x$$ is
Derivative from First Principles
Let $$f(x)=\sqrt{x}$$. Use the limit definition of the derivative to find $$f'(x)$$.
Difference Quotient and the Derivative Definition
Let $$g(x)=2*x^2-3*x+5$$. Use the difference quotient method to explore the rate of change of this f
Differentiation from First Principles
Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.
Estimating Temperature Change
A scientist recorded the temperature of a liquid at different times (in minutes) as it was heated. U
Graphical Estimation of Tangent Slopes
Using the provided graph of a function g(t), analyze its rate of change at various points.
Implicit Differentiation in Circular Motion
A particle moves along the circle defined by $$x^2 + y^2 = 25$$. Answer the following parts.
Implicit Differentiation with Exponential and Trigonometric Functions
Consider the curve defined implicitly by $$e^(y) + x*\cos(y) = x^2$$.
Linearization and Tangent Approximations
Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$
Market Price Rate Analysis
The market price of a product (in dollars) has been recorded over several days. Use the table below
Parametric Analysis of a Curve
A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,
Population Growth Rate
A population is modeled by $$P(t)=\frac{3*t^2 + 2}{t+1}$$, where $$t$$ is measured in years. Analyze
Population Model Rate Analysis
A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base
Position Recovery from a Velocity Function
A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for
Product Rule Application in Kinematics
A particle’s distance along a path is given by $$s(t)= t*e^(2*t)$$, where $$t$$ is in seconds. Answe
Related Rates: Changing Shadow Length
A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts
Related Rates: Expanding Balloon
A spherical balloon is being inflated so that its volume $$V$$ (in m³) and radius $$r$$ (in m) satis
Related Rates: Sweeping Spotlight
A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit
Tangent Line Approximation
Consider the function $$f(x)=\cos(x)$$. Answer the following:
Tangent Line Estimation from Experimental Graph Data
A function $$f(x)$$ is represented by the following graph of experimental data approximating $$f(x)=
Taylor Series of ln(x) Centered at x = 1
A researcher studies the natural logarithm function $$f(x)=\ln(x)$$ by constructing its Taylor serie
Trigonometric Function Differentiation
Consider the function $$f(x)=\sin(x)+\cos(x)$$. Answer the following:
Urban Population Flow
A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t
Velocity and Acceleration Analysis
A particle moving along a straight line has a velocity function given by $$v(t)=2*t^2 - 8*t + 3$$ (i
Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment
In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)
Chemical Mixing: Implicit Relationships and Composite Rates
In a chemical mix tank, the solute amount (in grams) and the concentration (in mg/L) are related by
Design Optimization for a Cylindrical Can
A manufacturer wants to design a cylindrical can that holds a fixed volume of $$V = 1000$$ cm³. The
Differentiation in a Logistic Population Model
The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}
Differentiation Involving an Inverse Function and Logarithms
Let $$f(x)= x^2+ 3*x+ 2$$, and let y be defined by $$y= \ln(f^{-1}(e^{x}))$$, where $$f^{-1}$$ is th
Differentiation of Composite Exponential and Trigonometric Functions
Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe
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 Implicit Differentiation in a Nonlinear Model
Assume that \(x\) and \(y\) are related by the nonlinear equation $$e^{x*y} + x - \ln(y) = 5$$ with
Implicit Differentiation in a Hyperbola-like Equation
Consider the equation $$ x*y = 3*x - 4*y + 12 $$.
Implicit Differentiation in a Radical Equation
The relationship between $$x$$ and $$y$$ is given by $$\sqrt{x} + \sqrt{y} = 6$$.
Implicit Differentiation in an Elliptical Orbit
An orbit of a satellite is modeled by the ellipse $$4*x^2 + 9*y^2 = 36$$. At the point $$\left(1, \f
Implicit Differentiation Involving Exponential Functions
Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.
Implicit Differentiation of a Circle
The equation of a circle is given by $$x^2+y^2=25$$. Answer the following parts:
Implicit Differentiation of a Product and Composite Function
Consider the equation $$x^2*\sin(y)+e^{y}=x$$, which defines y implicitly as a function of x. Answer
Implicit Differentiation with Trigonometric Components
Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti
Implicit Differentiation: Circle and Tangent Line
The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva
Inverse Function Derivative for the Natural Logarithm
Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal
Inverse Function Derivative in a Cubic Function
Let $$f(x)= x^3+ 2*x - 1$$, a one-to-one differentiable function. Its inverse function is denoted as
Inverse Function Derivatives in a Sensor Model
An instrument outputs a reading defined by $$f(x)= x^3 + 2$$, where $$x$$ represents the voltage inp
Inverse Function Differentiation for a Trigonometric-Polynomial Function
Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with
Inverse Function Differentiation in Economics
In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan
Polar and Composite Differentiation: Arc Slope for a Polar Curve
Consider the polar curve $$r(\theta)=2+\cos(\theta)$$. Answer the following parts:
Population Dynamics in a Fishery
A lake is being stocked with fish as part of a conservation program. The number of fish added per da
Second Derivative via Chain Rule
Let $$h(x)=(e^{2*x}+1)^4$$. Answer the following parts.
Tangent Line to an Ellipse
Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan
Analysis of a Piecewise Function with Discontinuities
Consider the function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} &\text{if } x \neq 2 \\ 3 &\text{if }
Analyzing a Motion Graph
A car's velocity over time is modeled by the piecewise function given in the graph. For $$0 \le t <
Arc Length Calculation
Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.
Business Profit Optimization
A firm's profit is modeled by $$P(x)= -4*x^2 + 240*x - 1000$$, where $$x$$ (in hundreds) represents
Compound Interest Rate Change
An investment grows according to $$A(t)=5000e^{0.07t}$$, where t is measured in years. Answer the fo
Conical Tank Filling - Related Rates
A conical water tank has its volume given by $$V= \frac{1}{3}\pi*r^2*h$$, where \(r\) is the radius
Cubic Curve Linearization
Consider the curve defined implicitly by $$x^3 + y^3 - 3*x*y = 0$$.
Differentiating a Product: f(x)=x sin(x)
Let \(f(x)=x\sin(x)\). Analyze the behavior of \(f(x)\) near \(x=0\).
Draining Hemispherical Tank
A hemispherical tank of radius $$5$$ m is draining. The volume of water in the tank is given by $$V
Ellipse Tangent Line Analysis
Consider the ellipse given by the implicit equation $$9*x^2 + 4*y^2 = 36$$. Answer the following par
Exponential Relation
Consider the equation $$e^{x*y} = x + y$$.
Financial Model Inversion
Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex
Graphical Analysis of an Inverse Function
Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the
Inflating Spherical Balloon: A Related Rates Problem
A spherical balloon is being inflated so that its volume increases at a constant rate of $$12\; in^3
Infrared Sensor Distance Analysis
An infrared sensor measures the distance to a moving target using the function $$d(t)=50*e^{-0.2*t}+
Inversion in a Light Intensity Decay Model
A laboratory experiment records the decay of light intensity over time, modeled by $$f(t)=80*e^{-0.2
Inversion of an Absolute Value Function
Consider the function $$f(x)=|x-3|+2$$ with the domain restricted to $$x\ge3$$. Analyze its inverse.
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
Linear Account Growth in Finance
The amount in a savings account during a promotional period is given by the linear function $$A(t)=1
Motion along a Curved Path
A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo
Optimizing Area of a Rectangular Field
A farmer has 100 meters of fencing to enclose three sides of a rectangular field (the fourth side be
Polar Coordinates: Arc Length of a Spiral
Consider the polar curve defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0 \le \theta \le \pi$$.
Population Growth Analysis
A certain bacterial population in a lab grows according to the model $$P(t)=100\cdot e^{0.03*t}$$, w
Population Growth Differential
Consider an implicit relationship between a population $$N$$ and time $$t$$ given by $$\ln(N) + t =
Related Rates: Pool Water Level
Water is being drained from a circular pool. The surface area of the pool is given by $$A=\pi*r^2$$.
Security Camera Angle Change
A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the
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
Vector Function: Particle Motion in the Plane
A particle moves in the plane with a position vector given by $$\mathbf{r}(t)=\langle t, t^2 \rangle
Water Filtration Plant Analysis
A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an
Analysis of a Function with Oscillatory Behavior
Consider the function $$f(x)=x+\sin(x)$$. Answer the following parts:
Analysis of a Quartic Function as a Perfect Power
Consider the function $$f(x)=x^4-4*x^3+6*x^2-4*x+1$$. Answer the following parts:
Analysis of an Exponential Function
Consider the function $$f(x)=e^{-x}*(x^2)$$. Answer the following parts:
Analysis of Critical Points for Increasing/Decreasing Intervals
Consider the function $$ f(x)=x^3-6x^2+9x+2. $$ Answer the following parts:
Application of Rolle's Theorem
Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following
Bacterial Culture with Periodic Removal
A laboratory experiment involves a bacterial culture that, at the beginning of an hour, has 200 bact
Concavity and Inflection Points
Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:
Determining Absolute Extrema for a Trigonometric-Polynomial Function
Consider the function $$f(x)= x+\cos(x)$$ defined on the closed interval $$[0, 2\pi]$$. Determine th
Differentiability of a Piecewise Function
Consider the piecewise function $$r(x)=\begin{cases} x^2, & x \le 2 \\ 4*x-4, & x > 2 \end{cases}$$.
Expanding Oil Spill - Related Rates
A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius
Exponential Growth and Logarithmic Transformation
A bacteria population is modeled by $$P(t)= A*e^{k*t}$$, where $$t$$ is measured in hours, A is the
Extreme Value Theorem for a Piecewise Function
Let $$h(x)$$ be defined on $$[-2,4]$$ as $$ h(x)= \begin{cases} -x^2+4 & \text{if } x \le 1, \\ 2x-
Implicit Differentiation in Economic Context
Consider the curve defined implicitly by $$x*y + y^2 = 12$$, representing an economic relationship b
Logarithmic Function Derivative Analysis
Consider the function $$f(x)= \ln(x^2+1)$$. Answer the following questions about its behavior.
Logarithmic-Exponential Function Analysis
Consider the function $$f(x)= e^(x) + x$$ defined for all real numbers. Answer the following questio
MVT Application: Rate of Temperature Change
The temperature in a room is modeled by $$T(t)= -2*t^2+12*t+5$$, where $$t$$ is in hours. Analyze th
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
Related Rates: Changing Shadow Length
A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp
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
Ski Resort Snow Accumulation and Melting
At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due
Accumulated Change Prediction
A population grows continuously at a rate proportional to its size. Specifically, the growth rate is
Accumulated Displacement from a Piecewise Velocity Function
A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\
Accumulated Displacement from Acceleration
A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini
Accumulation Function from a Rate Function
The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for
Approximating an Exponential Integral via Riemann Sums
Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below
Area Between the Curves f(x)=x² and g(x)=2x+3
Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete
Area Estimation Using Riemann Sums for $$f(x)=x^2$$
Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef
Average Value of an Exponential Function
For the function $$f(x)= x*e^{-x}$$, determine the average value on the interval $$[0,2]$$. Answer t
Bacteria Population Accumulation
A bacteria culture grows at a rate given by $$r(t)=0.5*t^2-t+3$$ (in thousand bacteria per hour) for
Bacterial Growth Accumulation
The instantaneous growth rate of a bacterial culture is modeled by $$r(t)= 0.3*t$$ million cells per
Chemical Reactor Concentration
In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu
Comparing Riemann Sum Approximations for an Increasing Function
A function f(x) is given in the table below: | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x) | 3
Cost Accumulation from Marginal Cost Function
A company’s marginal cost function $$MC(q)$$ (in dollars per unit) for producing $$q$$ units is give
Cross-Sectional Area of a River Using Trapezoidal Rule
The depth $$h(x)$$ (in meters) of a river’s cross-section is measured at various points along a hori
Estimating Area Under a Curve via Riemann Sums
The following table shows values of a function f(t): | t | 0 | 2 | 4 | 6 | 8 | |---|---|---|---|---
Evaluating a Trigonometric Integral
Evaluate the integral $$\int_{0}^{\pi/2} \cos(3*x)\,dx$$.
Heat Energy Accumulation
The rate of heat transfer into a container is given by $$H(t)= 5\sin(t)$$ kJ/min for $$t \in [0,\pi]
Integrated Growth in Economic Modeling
A company experiences revenue growth at an instantaneous rate given by $$r(t)=0.5*t+2$$ (in millions
Integration by Parts: Logarithmic Function
Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f
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
Rainfall Accumulation Over Time
A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l
Riemann Sum from a Table: Plant Growth Data
A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar
Series Representation and Term Operations
Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+
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
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 Accumulation Using Trapezoidal Sum
A reservoir is monitored over time and its water level (in meters) is recorded at various times (in
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
Capacitor Discharge in an RC Circuit
In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio
City Population with Migration
The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe
Coffee Cooling: Differential Equation Application
A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur
Combined Differential Equations and Function Analysis
Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y
Complex Related Rates Problem Involving a Moving Ladder
A 10-meter ladder leans against a vertical wall. The bottom of the ladder slides away from the wall
Compound Interest with Continuous Payment
An investment account grows with a continuous compound interest rate $$r$$ and also receives continu
Cooling of an Object Using Newton's Law of Cooling
An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto
Exponential Growth with Shifted Dependent Variable
The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th
Free-Fall with Air Resistance
An object falling under gravity experiences air resistance proportional to the square of its velocit
FRQ 2: Separable Differential Equation with Initial Condition
Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(
FRQ 6: Radioactive Decay
A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda * N$$
FRQ 8: RC Circuit Analysis
In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$
FRQ 11: Linear Differential Equation via Integrating Factor
Solve the differential equation $$\frac{dy}{dx}+\frac{1}{x}y=\sin(x)$$ with the initial condition $$
FRQ 18: Enzyme Reaction Rates
A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\
Implicit Differential Equations and Slope Fields
Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.
Logistic Growth Model
A population P grows according to the logistic differential equation $$\frac{dP}{dt} = rP\left(1-\fr
Logistic Population Model
A fish population is modeled by the logistic differential equation $$\frac{dP}{dt}= r*P\left(1-\frac
Mixing in a Chemical Reaction
A solution in a tank is undergoing a chemical reaction described by the differential equation $$\fra
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
Modeling Disease Spread with Differential Equations
In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin
Modeling Exponential Growth
A population follows the differential equation $$\frac{dP}{dt} = k*P$$. Given that the population do
Newton's Law of Cooling
An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie
Particle Motion with Variable Acceleration
A particle moves along a straight line with an acceleration given by $$a(t)=6-4*t$$. At time t = 0,
Population Dynamics in Ecology
Consider the differential equation that models the growth of a fish population in a lake: $$\frac{dP
Population Dynamics with Harvesting
A fish population in a lake is modeled by the logistic equation with harvesting: $$\frac{dP}{dt}=r\,
Radio Signal Strength Decay
A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra
RL Circuit Analysis
An RL circuit is described by the differential equation $$L\frac{di}{dt} + R*i = V$$, where $$L=0.5\
Saltwater Mixing Problem
A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of
Second-Order Differential Equation: Oscillations
Consider the second-order differential equation $$\frac{d^2y}{dx^2}= -9*y$$ with initial conditions
Solution Curve from Slope Field
A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio
Tank Mixing Problem
A tank contains 1000 L of a well‐mixed salt solution. Brine containing 0.5 kg/L of salt flows into t
Verification of Integral Representation of Solutions
Let $$y(x)=\int_0^x e^{-(x-t)} f(t)\,dt$$, where $$f(t)$$ is a continuous function. Answer the follo
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
Water Tank Inflow-Outflow Model
A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters
Accumulated Rainfall
The rainfall intensity in a region is given by $$R(t)=0.2*t^2+1$$ (in cm/hour), where $$t$$ is measu
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
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
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 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: Parabolic and Linear Functions
Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu
Area Between Exponential Curves
Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=e^{-2*x}$$ for $$x\ge0$$. Answer the following:
Area Calculation: Region Under a Parabolic Curve
Let $$f(x)=4-x^2$$. Consider the region bounded by the curve $$f(x)$$ and the x-axis.
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 Value of a Polynomial Function
Consider the function $$f(x)=2*x^2 - 4*x + 3$$ defined on the closed interval $$[0,4]$$. Answer the
Displacement vs. Distance: Analysis of Piecewise Velocity
A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^
Fluid Flow Rate and Total Volume
A river has a flow rate given by $$Q(t)=50+10*\cos(t)$$ (in cubic meters per second) for $$t\in[0,\p
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
Kinematics: Motion with Variable Acceleration
A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has
Particle Position and Distance Traveled
A particle moves along a line with velocity $$v(t)=t^3-6*t^2+9*t$$ (m/s) for $$t\in[0,5]$$. Given th
Pollution Concentration in a Lake
A lake has a pollution concentration modeled by $$C(x) = 16 - x^2$$ (in mg/L), where $$x$$ (in meter
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 $$t
Power Series Representation for ln(x) about x=4
The function $$f(x)=\ln(x)$$ is to be expanded as a power series centered at $$x=4$$. Find this seri
Projectile Motion Analysis
A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera
Projectile Motion under Gravity
An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial
Savings Account with Decreasing Deposits
An individual opens a savings account with an initial deposit of $1000 in the first month. Every sub
Volume by the Washer Method: Between Curves
Consider the region between the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x$$ between their
Volume of a Solid Obtained by Rotation
Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat
Volume of a Solid of Revolution Between Curves
Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x \in [0,4]$$.
Volume of a Solid Rotated about y = -1
The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$
Volume with Square Cross Sections
The region in the $$xy$$-plane is bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. A solid is formed
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
Work Done in Pumping Water from a Parabolic Tank
A water tank has a parabolic cross-section described by $$y=x^2$$ (with y in meters, x in meters). T
Analyzing a Cycloid
A cycloid is defined by the parametric equations $$x(t)= r*(t - \sin(t))$$ and $$y(t)= r*(1 - \cos(t
Analyzing a Looping Parametric Curve
The curve is defined by the equations $$x(t)=t^3-3t$$ and $$y(t)=t^2$$ for \(-2\le t\le 2\). Due to
Analyzing Oscillatory Motion in Parametric Form
The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$
Area between Two Polar Curves
Given the polar curves $$R(\theta)=3$$ and $$r(\theta)=2$$ for $$0 \le \theta \le 2\pi$$, find the a
Area Between Two Polar Curves
Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:
Area Between Two Polar Curves
Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of
Conversion of Polar to Cartesian Coordinates
Consider the polar curve $$ r=4*\cos(\theta) $$. Analyze its Cartesian equivalent and some of its pr
Curvature and Oscillation in Vector-Valued Functions
Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t
Exponential Decay in Vector-Valued Functions
A particle moves in the plane with its position given by the vector-valued function $$\vec{r}(t)=\la
Finding the Slope of a Tangent to a Parametric Curve
Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.
Implicit Differentiation with Implicitly Defined Function
Consider the equation $$x^2+xy+y^2=7$$, which defines $$y$$ implicitly as a function of $$x$$.
Integrating a Vector-Valued Function
A particle has a velocity given by $$\vec{v}(t)= \langle e^t, \cos(t) \rangle$$. Its initial positio
Intersection of Parametric Curves
Two curves are given by the parametric equations $$x_1(t)=t^2,\; y_1(t)=t^3$$ and $$x_2(s)=1-s^2,\;
Kinematics in Polar Coordinates
A particle’s position in polar coordinates is given by $$r(t)= \frac{5*t}{1+t}$$ and $$\theta(t)= \f
Kinematics on a Circular Path
A particle moves along a circle given by the parametric equations $$x(t)= 5*\cos(t)$$ and $$y(t)= 5*
Motion of a Particle in the Plane
A particle moves in the plane with parametric equations $$x(t)=t^2-4*t$$ and $$y(t)=2*t^3-6*t^2$$ fo
Optimization on a Parametric Curve
A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.
Parameter Elimination in Logarithmic and Quadratic Relationships
Given the parametric equations $$x(t)= \ln(t)$$ and $$y(t)= t^2 - 4*t + 3$$ for $$t > 0$$, eliminate
Parametric Curves and Concavity
Consider the parametric equations $$x(t)= \sin(t)$$ and $$y(t)= \cos(2*t)$$ for $$t \in [0, 2\pi]$$.
Parametric Equations and Intersection Points
Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(
Parametric Intersection and Tangency
Two curves are given in parametric form by: Curve 1: $$x_1(t)=t^2,\, y_1(t)=2t$$; Curve 2: $$x_2(s
Parametric Intersection of Curves
Consider the curves $$C_1: x(t)=\cos(t),\, y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$ and $$C_2: x(s)=1
Particle Motion in the Plane
A particle moves in the plane with parametric equations $$x(t)= 3\cos(t)$$ and $$y(t)= 3\sin(t)$$ fo
Particle Trajectory in Parametric Motion
A particle moves along a curve with parametric equations $$x(t)= t^2 - 4*t$$ and $$y(t)= t^3 - 3*t$$
Polar Spiral: Area and Arc Length
Consider the polar spiral defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0\le\theta\le 2\pi$$. An
Projectile Motion via Vector-Valued Functions
A projectile is launched from the origin with an initial velocity given by \(\mathbf{v}(0)=\langle 5
Symmetry and Area in Polar Coordinates
Consider the polar curve given by $$r=2\cos(\theta)$$. Answer the following:
Vector-Valued Integrals in Motion
A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$
Velocity and Acceleration of a Particle
A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}
Wind Vector Analysis in Navigation
A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)
Work Done by a Force along a Path
A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra
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