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Analyzing a Velocity Function with Nested Discontinuities
A particle’s velocity along a line is given by $$v(t)= \frac{(t-1)(t+3)}{(t-1)*\ln(t+2)}$$ for $$t>0
Analyzing Process Data for Continuity
A manufacturing process produces items whose lengths (in mm) are recorded as a function f(x) of time
Application of the Intermediate Value Theorem
Let the function $$f(x)= x^3 - 4*x - 1$$ be continuous on the interval $$[0, 3]$$. Answer the follow
Applying the Squeeze Theorem with Trigonometric Function
Consider the function $$ f(x)= x^2 \sin(1/x) $$ for $$x\ne0$$, with $$f(0)=0$$. Use the Squeeze Theo
Arithmetic Sequence in Temperature Data and Continuity Correction
A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ
Continuity of a Composite Function
Let $$g(x) = \sqrt{x+3}$$ and $$h(x) = x^2 - 4$$. Define the composite function $$f(x) = g(h(x))$$.
Continuity of Composite Functions
Let $$f(x)=x+2$$ for all x, and let $$g(x)=\begin{cases} \sqrt{x}, & x \geq 0 \\ \text{undefined},
Ensuring Continuity for a Piecewise-Defined Function
Consider the piecewise function $$p(x)= \begin{cases} ax + 3 & \text{if } x < 2, \\ x^2 + b & \text{
Evaluating a Compound Limit Involving Rational and Trigonometric Functions
Consider the function $$f(x)= \frac{\sin(x) + x^2}{x}$$. Answer the following:
Exponential Limit Parameter Determination
Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,
Factorization and Limit Evaluation
Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e
Factorization and Removable Discontinuity
Consider the function $$f(x) = \frac{x^2 - 9}{x - 3}$$ for $$x \neq 3$$. Answer the following:
Graph Analysis: Identify Limits and Discontinuities
A graph of a function f(x) is provided in the stimulus. The graph shows a removable discontinuity at
Graph Reading: Left and Right Limits
A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det
Graph-Based Analysis of Discontinuity
Examine the graph of a function that appears to be defined by $$f(x)= 3x - 2$$ for all $$x \neq 2$$,
Graphical Estimation of a Limit
The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re
Identifying Discontinuities in a Rational Function
Consider the function $$f(x)=\frac{x^2 - 9}{x - 3}$$ defined for $$x \neq 3$$. Answer the following
Implicit Differentiation and Tangent Slopes
Consider the circle defined by $$x^2 + y^2 = 25$$. Answer the following:
Intermediate Value Theorem in Particle Motion
Consider a particle with position function $$s(t)= t^3 - 7*t+3$$. According to the Intermediate Valu
Limit and Integration in Non-Polynomial Particle Motion
A particle moves along a line with velocity defined by $$v(t)= \frac{e^{2*t}-e^{4}}{t-2}$$ for \(t \
Limit Evaluation using Conjugate Multiplication
Consider the function $$f(x)= \frac{\sqrt{x+3}-2}{x-1}$$.
Limits and the Squeeze Theorem Application
Consider two scenarios: (1) A function f(x) satisfying $$ -|x| \le f(x) \le |x| $$ for all x near 0,
Limits Involving a Removable Discontinuity
Consider the function $$g(x)= \frac{(x+3)(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the followin
Limits Involving Radical Functions
Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.
Limits Involving Radicals and Algebra
Consider the function $$f(x)= \sqrt{x^2 + x} - x$$. Answer the following parts.
Limits Near Vertical Asymptotes
Consider the function $$f(x) = \frac{1}{x - 4}$$. (a) Determine $$\lim_{x \to 4^-} f(x)$$. (b) Dete
Oscillatory Behavior and Discontinuity
Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans
Rational Function Limits and Removable Discontinuities
Consider the function $$f(x)=\frac{(x+3)(x-2)}{(x+3)(x+5)}$$. Answer the following:
Real-World Application: Temperature Sensor Calibration
A temperature sensor in a lab records temperatures (in °C) according to the function $$f(t)= \frac{t
Removable Discontinuity and Limit Evaluation
Consider the function $$f(x) = \frac{(x + 3) * (x - 2)}{x + 3}$$ for $$x \neq -3$$. Answer the follo
Return on Investment and Asymptotic Behavior
An investor’s portfolio is modeled by the function $$P(t)= \frac{0.02t^2 + 3t + 100}{t + 5}$$, where
Squeeze Theorem with Trigonometric Function
Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:
Table Analysis for Estimating a Limit
The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll
Trigonometric Limit Evaluation
Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.
Analyzing Differentiability of an Absolute Value Function
Consider the function $$f(x)= |x-2|$$.
Approximating Derivatives Using Secant Lines
For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line
Car's Position and Velocity
A car’s position is modeled by \(s(t)=t^3 - 6*t^2 + 9*t\), where \(s\) is in meters and \(t\) is in
Concavity and the Second Derivative
Consider the function $$f(x)=x^4-4*x^3+6*x^2$$. Answer the following:
Derivative Applications in Population Growth
A population of cells is modeled by $$P(t)=100*e^{0.2*t}$$, where $$t$$ is in hours. Answer the foll
Derivative of the Square Root Function via Limit Definition
Let $$g(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following parts.
Deriving the Derivative from First Principles for a Reciprocal Square Root Function
Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f
Differentiability of an Absolute Value Function
Consider the function $$f(x)=|x-3|$$, representing the error margin (in centimeters) in a calibratio
Economic Marginal Revenue
A company's revenue function is given by \(R(x)=x*(50-0.5*x)\) dollars, where \(x\) represents the n
Event Ticket Sales Dynamics
For a popular concert, tickets are sold at a rate of $$f(t)=100-3*t$$ (tickets/hour) while cancellat
Finding Derivatives of Composite Functions
Let $$f(x)= (3*x+1)^4$$.
Implicit Differentiation of a Circle
Consider the equation $$x^2 + y^2 = 25$$ representing a circle with radius 5. Answer the following q
Instantaneous Rate of Change of Temperature
The temperature in a room is modeled by $$T(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$, where $$t$$
Instantaneous Rate of Temperature Change in a Coffee Cup
The temperature of a cup of coffee is recorded at several time intervals as shown in the table below
Inverse Function Analysis: Cosine and Linear Combination
Consider the function $$f(x)=\cos(x)+x$$ defined on the interval $$[0,\frac{\pi}{2}]$$.
Inverse Function Analysis: Sum with Reciprocal
Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.
Kinematics and Position Function Analysis
A particle’s position is modeled by $$s(t)=4*t^3-12*t^2+5*t+2$$, where $$s(t)$$ is in meters and $$t
Linking Derivative to Kinematics: the Position Function
A particle's position is given by $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, with $$t$$ in seconds and $$s(t)$$
Particle Motion on a Straight Road
A particle moves along a straight road. Its position at time $$t$$ seconds is given by $$s(t) = t^3
Position Function from a Logarithmic Model
A particle’s position in meters is modeled by $$s(t)= \ln(t+1)$$ for $$t \geq 0$$ seconds.
Proof of Scaling in Derivatives
Let $$f(x)$$ be a differentiable function and let $$k$$ be a constant. Consider $$g(x)= k*f(x)$$. Us
Quotient Rule Application
Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone
Real-World Cooling Process
In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.
Related Rates: Balloon Surface Area Change
A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu
Riemann Sums and Derivative Estimation
A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds
Tangent Lines and Local Linearization
Consider the function $$f(x)=\sqrt{x}$$.
Advanced Composite Function Differentiation in Biological Growth
A biologist models bacterial growth by the function $$P(t)= e^{\sqrt{t+1}}$$, where $$t$$ is time in
Advanced Implicit and Inverse Function Differentiation on Polar Curves
Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,
Analyzing a Function and Its Inverse
Consider the invertible function $$f(x)= \frac{x^3+1}{2}$$.
Chain Rule in Angular Motion
An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$
Chain Rule in Temperature Variation
A metal rod's temperature along its length is given by the function $$T(x)= \cos((4*x+2)^2)$$, where
Chain Rule with Exponential and Trigonometric Functions
A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq
Chain Rule with Logarithmic and Radical Functions
Let $$R(x)=\sqrt{\ln(1+x^2)}$$.
Composite Differentiation of an Inverse Trigonometric Function
Let $$H(x)= \arctan(\sqrt{x+3})$$.
Composite Function and Tangent Line
Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without
Composite Function Chain Reaction
A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos
Composite Function from an Implicit Equation
Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, which defines $$y$$ as an implicit function
Composite Function: Engineering Stress-Strain Model
In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \
Composite, Implicit, and Inverse Combined Challenge
Consider a dynamic system defined by the equation $$\sin(y)+\sqrt{x+y}=x$$, which implicitly defines
Differentiation of Nested Exponential Functions
Let $$F(x)=e^{\sin(x^2)}$$.
Implicit Curve Analysis: Horizontal Tangents
Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:
Implicit Differentiation in a Circle
Consider the circle $$x^2 + y^2 = 25$$. Answer the following parts.
Implicit Differentiation of a Logarithmic-Exponential Equation
Consider the equation $$\ln(x+y) + e^{x*y} = 7$$, which implicitly defines $$y$$ as a function of $$
Implicit Differentiation of a Trigonometric Composite Function
Consider the curve defined implicitly by $$\sin(y) + y^2 = x$$.
Implicit Differentiation on a Circle
Consider the circle defined by $$x^2+y^2=25$$.
Implicit Differentiation with Chain and Product Rules
Consider the curve defined implicitly by $$e^{xy} + x^2y = 10$$. Assume that the point $$(1,2)$$ lie
Implicit Differentiation with Product Rule
Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici
Inverse Function Derivative and Recovery
Let $$f(x)=x^3+x$$, which is one-to-one on a suitable interval. Answer the following parts.
Inverse Function Differentiation
Let $$f(x)=x^3+x+1$$, a one-to-one function, and let $$g$$ be the inverse of $$f$$. Use inverse func
Inverse Function Differentiation in an Exponential Context
Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:
Inverse Trigonometric Function Differentiation
Consider the function $$y=\arctan(2*x)$$. Answer the following:
Manufacturing Optimization via Implicit Differentiation
A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre
Multilayer Composite Function Differentiation
Let $$y=\cos(\sqrt{5*x+3})$$. Answer the following:
Population Dynamics via Composite Functions
A biological population is modeled by $$P(t)= \ln\left(20*e^(0.1*t^2)+ 5\right)$$, where t is measur
Airplane Altitude Change
An airplane's altitude (in meters) as a function of time is modeled by $$A(t)= 500*t - 4.9*t^2 + 100
Analysis of a Piecewise Function with Discontinuities
Consider the function $$f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2 \\ x+1 & \text{if } x\
Bacterial Growth Analysis
The number of bacteria in a culture is given by $$P(t)=500e^{0.2*t}$$, where $$t$$ is measured in ho
Complex Piecewise Function Analysis
Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &
Cooling Coffee: Exponential Decay Model
A cup of coffee cools according to $$T(t) = 70 + 50e^{-0.1t}$$, where $$T(t)$$ (in °F) is the temper
Economic Cost Function Linearization
A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $
Elasticity of Demand Analysis
A product’s demand function is given by $$Q(p) = 150 - 10p + p^2$$, where $$p$$ is the price, and $$
Error Estimation in Pendulum Period
The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length
Estimating Small Changes using Differentials
In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame
Expanding Balloon: Related Rates with a Sphere
A spherical balloon is being inflated so that its volume increases at a constant rate of $$dV/dt = 1
FRQ 1: Vessel Cross‐Section Analysis
A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^
FRQ 2: Balloon Inflation Analysis
A spherical balloon is being inflated. Its volume is given by $$V = \frac{4}{3}\pi r^3$$, and the ra
FRQ 4: Revenue and Cost Implicit Relationship
A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat
FRQ 12: Airplane Climbing Dynamics
An airplane’s altitude is modeled by the equation $$y = 0.1*x^2$$, where x (in km) is the horizontal
L'Hôpital's Rule in Chemical Kinetics
In a chemical kinetics experiment, the reaction rate is modeled by the function $$f(x)=\frac{\ln(1+3
Limit Evaluation Using L'Hôpital's Rule
Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 4x^2 + 1}{7x^3 + 2x - 6}$$.
Linearization and Differential Approximations
Let $$f(x)=x^4$$. Use linearization to approximate $$f(3.98)$$ near $$x=4$$.
Linearization and Differentials
Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.
Marginal Analysis in Economics
A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the
Open-top Box Optimization
A manufacturer wants to design an open‐top rectangular box with a square base that has a fixed volum
Optimization in Packaging
An open-top box with a square base is to be constructed so that its volume is fixed at $$1000\;cm^3$
Particle Motion Analysis
A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$, where $$t$$
Particle Motion with Changing Acceleration
A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²), w
Projectile Motion Analysis
The height of a projectile is modeled by the function $$h(t) = -4.9t^2 + 20t + 2$$, where $$t$$ is i
Projectile Motion with Velocity Components
A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica
Related Rates in a Conical Tank
Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.
Shadow Length Problem
A person 1.80 m tall walks away from a 3.0 m tall lamppost at a rate of 1.2 m/s. Let $$x$$ be the di
Temperature Change Analysis
The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i
Vehicle Deceleration Analysis
A car's position function is given by $$s(t)= 3*t^3 - 12*t^2 + 5*t + 7$$, where $$s(t)$$ is measured
Vehicle Position and Acceleration
A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where
Analyzing a Supply and Demand Model Using Derivatives
A product's price as a function of the number of units produced is given by $$P(q)= 50 - 3*q + 0.5*q
Application of Rolle's Theorem for a Quadratic Function
Let $$f(x)= x^2 - 4$$ be defined on the interval $$[-2,2]$$. In this problem, you will verify the co
Application of the Extreme Value Theorem
Consider the function $$f(x) = \sqrt{x} + (8-x)$$ defined on the interval $$[0, 8]$$. Answer the fol
Area Bounded by $$\sin(x)$$ and $$\cos(x)$$
Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$
Chemical Mixing in a Tank
A 200-liter tank initially contains pure water. A salt solution with a concentration of 0.5 kg/L flo
Composite Function with Piecewise Exponential and Logarithmic Parts
Consider the function $$ f(x) = \begin{cases} e^{x}-1, & x < 2, \\ \ln(x+1), & x \ge 2. \end{cases}
Concavity Analysis of a Trigonometric Function
For the function $$f(x)= \sin(x) - \frac{1}{2}\cos(x)$$ defined on the interval $$[0,2\pi]$$, analyz
Derivative and Concavity of f(x)= e^(x) - ln(x)
Consider the function $$f(x)= e^{x}-\ln(x)$$ for $$x>0$$. Answer the following:
Designing an Enclosure along a River
A farmer wants to build a rectangular enclosure adjacent to a river, using the river as one side of
Discontinuity and Derivative in a Hybrid Piecewise Function
Consider the function $$ f(x) = \begin{cases} x^2, & x \le 1, \\ 3x - 2, & x > 1. \end{cases} $$ A
Extrema in a Cost Function
A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the
FRQ 8: Mean Value Theorem and Non-Differentiability
Examine the function $$f(x)=|x|$$ on the interval [ -1, 1 ].
FRQ 10: First Derivative Test for a Cubic Profit Function
A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the
FRQ 12: Optimization in Manufacturing: Minimizing Cost
A company’s cost function is given by $$C(x)= 0.5*x^2 - 10*x + 125$$ (in dollars), where $$x$$ repre
FRQ 20: Profit Analysis Combining MVT and Optimization
A company’s profit function is given by $$P(x)= -2*x^3 + 18*x^2 - 48*x + 40$$, where $$x$$ (in thous
Increasing/Decreasing Behavior in a Financial Model
A financial analyst models the performance of an investment with the function $$f(x)= \ln(x) - \frac
Instantaneous Velocity Analysis via the Mean Value Theorem
A particle moves along a straight line with its displacement given by $$s(t)= t^3 - 6*t^2 + 9*t + 3$
Inverse Analysis of a Piecewise Function
Consider the piecewise function $$f(x)=\begin{cases}2*x+1 & x\le 0,\\ 3*x+1 & x>0\end{cases}$$. Ans
Jump Discontinuity in a Piecewise Linear Function
Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end
Mean Value Theorem for a Cubic Function
Consider the function $$f(x)= x^3 - 2*x^2 + x$$ on the closed interval $$[0,2]$$. In this problem, y
Minimizing Average Cost in Production
A company’s cost function is given by $$C(x)= 0.5*x^3 - 6*x^2 + 20*x + 100$$, where $$x$$ represents
Monotonicity and Inverse Function Analysis
Consider the function $$f(x)= x + e^{-x}$$ defined for all real numbers. Investigate its monotonicit
Motion Analysis with Acceleration Function
A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). G
Optimization in Production with Exponential Cost Function
A manufacturer’s cost function is modeled by $$C(x)= 200 + 50*x + 100*e^{-0.1*x}$$ where $$x$$ repre
Optimizing an Open-Top Box from a Metal Sheet
A rectangular sheet of metal with dimensions 24 cm by 18 cm is used to create an open-top box by cut
Pharmacokinetics: Drug Concentration Decay
A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe
Projectile Motion and Derivatives
A projectile is launched so that its height is given by $$h(t) = -4.9*t^2 + 20*t + 1$$, where $$t$$
Rational Function Optimization
Consider the rational function $$f(x)= \frac{x^2 + 1}{x - 1}$$ defined on the interval $$[2,6]$$. An
Related Rates in an Evaporating Reservoir
A reservoir’s volume decreases due to evaporation according to $$V(t)= V_0*e^{-a*t}$$, where $$t$$ i
Reservoir Sediment Accumulation
A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim
Water Reservoir Net Change
A water reservoir receives water from a river at a rate $$R_{in}(t)=5+0.5*t$$ and discharges water a
Accumulated Chemical Concentration
A scientist observes that the rate of change of chemical concentration in a solution is given by $$r
Accumulated Water Volume in a Tank
A water tank is being filled at a rate given by $$R(t) = 4*t$$ (in cubic meters per minute) for $$0
Accumulation and Inflection Points
Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ Answer the following parts:
Analyzing Work Done by a Variable Force
An object is acted on by a force given by $$F(x)= 3*x^2 - x + 2$$ (in newtons), where $$x$$ is the d
Area Between Curves
Consider the curves defined by $$f(x)=x^2$$ and $$g(x)=2*x$$. The region enclosed by these curves is
Area Under a Parabola
Consider the quadratic function $$f(x)=x^2 - 4*x + 3$$. Analyze the function on the interval $$[1,4]
Area Under a Piecewise Function
Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for
Bacterial Growth Modeling with Antibiotic Administration
A bacterial culture is subject to both growth and treatment simultaneously. The bacterial growth rat
Chemical Accumulation in a Reactor
A chemical reactor has a net accumulation rate given by $$R(t)=5*\cos(t) + 2$$ (in kg/hour), where $
Comparing Riemann Sum and the Fundamental Theorem
Let $$f(x)=3*x^2$$ on the interval $$[1,4]$$.
Convergence of Riemann Sum Estimations
Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,4]$$. Answer the following questions re
Cooling of a Liquid Mixture
In a tank, the cooling rate is given by $$C(t)=20e^{-0.3t}$$ J/min while an external heater adds a c
Economic Accumulation of Revenue
The marginal revenue (MR) for a company is given by $$MR(x)=50*e^{-0.1*x}$$ (in dollars per item), w
Elevation Profile Analysis on a Hike
A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy
Evaluating a Definite Integral Using U-Substitution
Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.
Exploring the Fundamental Theorem of Calculus
Let the function $$F(x) = \int_{1}^{x} \frac{1}{t^2+1}\,dt$$ represent an accumulation function. Ans
Finding the Area of a Parabolic Arch
An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans
FRQ5: Inverse Analysis of a Non‐Elementary Integral Function
Consider the function $$ P(x)=\int_{0}^{x} e^{t^2}\,dt $$ for x ≥ 0. Answer the following parts.
FRQ6: Inverse Analysis of a Displacement Function
Let $$ S(t)=\int_{0}^{t} (6-2*u)\,du $$ for t in [0, 3], representing displacement in meters. Answer
FRQ8: Inverse Analysis of a Piecewise-Defined Accumulation Function
Let $$ R(x)=\begin{cases} \int_{1}^{x} t\,dt, & 1 \le x \le 3 \\ \int_{1}^{x} (2*t-1)\,dt, & x > 3 \
FRQ16: Inverse Analysis of an Integral Function via U-Substitution
Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.
FRQ20: Inverse Analysis of a Function with a Piecewise Continuous Integrand
Consider the function $$ I(x)= \begin{cases} \int_{0}^{x}\cos(t)\,dt, & 0 \le x \le \pi/2 \\ \int_{0
Function Transformations and Their Integrals
Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze
Medication Concentration and Absorption Rate
A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.
Medication Infusion in Bloodstream
A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m
Motion Analysis from Velocity Data
A particle moves along a straight line with the following velocity data (in m/s) recorded at specifi
Net Change in Population Growth
A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$
Net Change in Salaries: An Accumulation Function
A company models its annual bonus savings with the rate function $$B'(t)= 500*(1+\sin(t))$$ dollars
Particle Motion with Variable Acceleration
A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). T
Rainfall and Evaporation in a Greenhouse
In a greenhouse, rainfall is modeled by $$R(t)= 8\cos(t)+10$$ mm/hr, while evaporation occurs at a c
Related Rates: Expanding Balloon
A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d
Riemann Sum Approximation of f(x) = 4 - x^2
Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the
Seismic Data Analysis: Ground Acceleration
A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table
Ski Lift Passengers: Boarding and Alighting Rates
On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r
Tabular Riemann Sums for Electricity Consumption
A household's daily electricity consumption (in kWh) over 5 consecutive days is recorded in the tabl
Trigonometric Integral with U-Substitution
Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(t)\tan(t)\,dt$$.
Trigonometric Integration via U-Substitution
Evaluate the integral $$I=\int_{0}^{\frac{\pi}{4}} \tan(x)*\sec^2(x)\,dx.$$ Answer the following par
U-Substitution in a Trigonometric Integral
Evaluate the integral $$\int \sin(2*x) * \cos(2*x)\,dx$$ using u-substitution.
Water Flow in a Tank
Water flows into a tank at a rate given by $$R(t)=3*t+2$$ (in liters per minute) for $$0 \le t \le 6
Analyzing Slope Fields for $$dy/dx=x\sin(y)$$
Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid
Bernoulli Differential Equation via Substitution
Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ
Boat Crossing a River with Current
A boat is attempting to cross a river flowing at a constant speed of $$2$$ m/s. The boat is directed
Carbon Dating and Radioactive Decay
Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia
Chemical Reaction Rate
The concentration $$y$$ (in moles per liter) of a reactant in a chemical reaction is modeled by the
Cooling with a Time-Dependent Coefficient
A substance cools according to $$\frac{dT}{dt} = -k(t)(T-25)$$ where the cooling coefficient is give
Differential Equation in Business Profit
A company's profit $$P(t)$$ changes over time according to $$\frac{dP}{dt} = 100\,e^{-0.5t} - 3P$$.
Falling Object with Air Resistance
A falling object experiences air resistance proportional to the square of its velocity. Its velocity
Implicit Solution for $$\frac{dy}{dx}=\frac{x+2}{y+1}$$
Solve the differential equation $$\frac{dy}{dx} = \frac{x+2}{y+1}$$ with the initial condition $$y(0
Investment Growth with Continuous Contributions
An investment account grows continuously with an annual interest rate of 5% while continuous deposit
Investment Growth with Continuous Deposits
An investment account accrues interest continuously at an annual rate of 0.05 and receives continuou
Linear Differential Equation using Integrating Factor
Solve the linear differential equation $$\frac{dy}{dx} + 2y = x$$ with the initial condition $$y(0)=
Logistic Growth Model
A population is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\frac{P}{
Mixing Problem in a Salt Solution Tank
A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o
Mixing Problem in a Tank
A tank initially contains 200 L of water with 10 kg of dissolved salt. Brine with a salt concentrati
Mixing Problem with Evaporation and Drainage
A tank initially contains 200 L of water with 20 kg of pollutant. Water enters the tank at 2 L/min w
Newton's Law of Cooling
An object is heated to $$100^\circ C$$ and left in a room at $$20^\circ C$$. According to Newton's l
Population Dynamics with Harvesting
A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula
Radioactive Decay
A radioactive substance decays according to $$\frac{dy}{dt} = -0.05\,y$$ with an initial mass of $$y
Radioactive Decay
A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda N$$,
Radioactive Decay and Half-Life
A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda N$$.
Sand Pile Dynamics
Sand is added to a pile at a constant rate of $$15$$ kg/min while some sand is simultaneously lost d
Seasonal Temperature Variation
The temperature $$T(t)$$ in a region is modeled by the differential equation $$\frac{dT}{dt} = -0.2\
Separable Differential Equation involving $$y^{1/3}$$
Consider the differential equation $$\frac{dy}{dx} = y^{1/3}$$ with the initial condition $$y(8)=27$
Slope Field Analysis and Asymptotics
Consider the differential equation $$\frac{dy}{dx}=\frac{x}{1+y^2}$$. Solve the equation and analyze
Slope Field Exploration
Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul
Vehicle Deceleration
A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.
Analysis of a Rational Function's Average Value
Consider the rational function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$[-1,1]$$. Analyz
Area Between a Parabola and a Line
Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll
Area Between a Parabolic Curve and a Line
Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ on the interval $$[0,4]$$. The table below sh
Area Between Cost Functions in a Business Analysis
A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C
Area Between Curves: Revenue and Cost Analysis
A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$
Area Between Parabolic Curves
Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x-x^2$$. Determine the area of the region bounded by t
Average of a Logarithmic Function
Let $$f(x)=\ln(x+2)$$ represent a measured quantity over the interval $$[0,6]$$.
Average Speed from a Velocity Function
A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$
Average Speed from Variable Acceleration
A car accelerates along a straight road with acceleration given by $$a(t)=2*t-1$$ (in m/s²) for $$t\
Average Temperature Analysis
A local weather station recorded the temperature throughout a day using the model $$T(t)=-0.5*t+35$$
Average Value of a Polynomial Function
Consider the function $$f(x)= 3*x^2 - 2*x + 1$$ defined on the interval $$[0,4]$$. Use the concept o
Cell Phone Battery Consumption
A cell phone’s battery life degrades over time such that the effective battery life each month forms
Center of Mass of a Lamina with Variable Density
A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x
Charity Donations Over Time
A charity receives monthly donations that form an arithmetic sequence. The first donation is $$50$$
Comparing Sales Projections
A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w
Designing an Open-Top Box
An open-top box with a square base is to be constructed with a fixed volume of $$5000\,cm^3$$. Let t
Determining Field Area from Intersection of Curves
A farmer's field is bounded by the curves $$y=0.5*x^2$$ and $$y=4*x$$. Find the area of the field wh
Discontinuities in a Piecewise Function
Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 2 & \text{if }
Discounted Cash Flow
A company projects that its annual cash flow will grow according to a geometric sequence. The initia
Economics: Consumer Surplus Calculation
Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons
Hollow Rotated Solid
Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i
Ice Rink Design: Volume and Area
An ice rink is designed with a cross-sectional profile given by $$y=4-x^2$$ (with y=0 as the base).
Implicit Differentiation in an Electrical Circuit
In an electrical circuit, the voltage $$V$$ and current $$I$$ are related by the equation $$V^2 + (3
Inverse for a Quadratic Function
Consider the function $$f(x)=x^2+4$$ defined for $$x \ge 0$$. Analyze its inverse function.
Optimization of Average Production Rate
A manufacturing process has a production rate modeled by the function $$P(t)=50e^{-0.1*t}+20$$ (unit
Particle Motion on a Parametric Path
A particle moves along a path given by the parametric equations $$x(t)= t^2 - t$$ and $$y(t)= 3*t -
Pipeline Installation Cost Analysis
The cost to install a pipeline along a route is given by $$C(x)=100+5*\sin(x)$$ (in dollars per mete
Population Growth and Average Rate
A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in
Rebounding Ball
A ball is dropped from a height of $$16$$ meters. Each time the ball bounces, its maximum height is
Resource Consumption in an Ecosystem
The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per
Solid of Revolution: Water Tank
A water tank is formed by rotating the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and t
Volume Calculation via Cross-Sectional Areas
A solid has cross-sectional areas perpendicular to the x-axis that are circles with radius given by
Volume of a Solid Using the Washer Method
Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev
Volume of a Solid with Square Cross Sections
A solid is formed over the region under the line $$f(x)=4-x$$ from $$x=0$$ to $$x=4$$ in the x-y pla
Work Calculation from an Exponential Force Function
An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen
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