AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{|x-5|}{x-5} & x\neq5 \\ 0 & x=5 \end{cases}$$. Answ

Advanced Analysis of an Oscillatory Function

Consider the function $$ f(x)= \begin{cases} \sin(1/x), & x\ne0 \\ 0, & x=0 \end{cases} $$.

Algebraic Simplification and Limit Evaluation of a Log-Exponential Function

Consider the function $$z(x)=\ln\left(\frac{e^{3*x}+e^{2*x}}{e^{3*x}-e^{2*x}}\right)$$ for $$x \neq

Analysis of One-Sided Limits and Jump Discontinuity

Consider the piecewise function $$ f(x)=\begin{cases}3x+1,& x<2,\\ x^2,& x \ge 2.\end{cases} $$ Ans

Analyzing End Behavior and Asymptotes

Consider the function $$f(x)= \frac{5x - 7}{\sqrt{x^2 + 1}}$$. Answer the following:

Application of the Squeeze Theorem

Consider the function defined by $$h(x)=\begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if }

Continuity Analysis of a Piecewise Function

Consider the function defined by $$ f(x)=\begin{cases}2x+1, & x<1,\\ x^2, & 1\le x\le 3,\\ 7-x, & x

Continuity of Constant Functions

Consider the constant function $$f(x)=7$$ for all x. Answer the following parts.

Determining Horizontal Asymptotes of a Log-Exponential Function

Examine the function $$s(x)=\frac{e^{x}+\ln(x+1)}{x}$$, which is defined for $$x > 0$$. Determine th

Determining Parameters for a Continuous Log-Exponential Function

Suppose a function is defined by $$ v(x)=\begin{cases} \frac{\ln(e^{p*x}+x)-q*x}{x} & \text{if } x \

Direct Evaluation of Polynomial Limits

Let $$ f(x)=x^3-5*x+2 $$.

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Estimating Derivatives Using Limit Definitions from Data

The position of an object (in meters) is recorded at various times (in seconds) in the table below.

Exponential Function Limits

Consider the function $$f(x) = \frac{e^x - 1}{x}$$ for $$x \neq 0$$, with the definition $$f(0) = 1$

Exponential Limit Parameter Determination

Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,

Factoring a Cubic Expression for Limit Evaluation

Consider the function $$f(x)= \frac{x^3 - 8}{x - 2}$$ for x \(\neq\) 2. Answer the following parts.

Graph Analysis of Discontinuities

A graph of a function f(x) shows a jump discontinuity at x = 1 and a removable discontinuity (a hole

Graph-Based Analysis of Discontinuity

Examine the graph of a function that appears to be defined by $$f(x)= 3x - 2$$ for all $$x \neq 2$$,

Horizontal Asymptote and End Behavior

Consider the rational function $$f(x)=\frac{3*x^2 + x - 5}{6*x^2 - 4*x + 7}$$. Answer the following

Implicit Differentiation and Tangent Slopes

Consider the circle defined by $$x^2 + y^2 = 25$$. Answer the following:

Intermediate Value Theorem Application

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

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 with Square Root and Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sqrt{4*x+8}-4}{x-2} & x\neq2 \\ 1 & x=2 \end{cases

Limits Involving Absolute Value Functions

Consider the function $$ f(x)=\frac{|x-3|}{x-3} $$.

Limits Involving Composition and Square Roots

Consider the function $$ f(x)=\sqrt{x+4}-2 $$.

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

Oscillatory Function and the Squeeze Theorem

Consider the function $$f(x)=x*\sin(1/x)$$ for x ≠ 0, with f(0)=0.

Real-world Application: Economic Model of Inventory Growth

A company monitors its inventory \(I(t)\) (in units) over time (in months) using the rate function $

Redefining a Function for Continuity

A function is defined by $$f(x) = \frac{x^2 - 1}{x - 1}$$ for $$x \neq 1$$, while $$f(1)$$ is left u

Removable Discontinuity in a Rational Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-16}{x-4} & x\neq4 \\ 3*x+1 & x=4 \end{cases}$$.

Squeeze Theorem Application with Trigonometric Functions

Let $$f(x)= x^2 \sin(1/x)$$ for $$x\neq0$$, and define $$f(0)=0$$.

Squeeze Theorem for an Oscillatory Function

Define the function $$f(x)= x \cos(\frac{1}{x})$$ for x ≠ 0, and let f(0)= 0.

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

Approximating Derivatives Using Secant Lines

For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line

Approximating Tangent Line Slopes

A curve is given by the function $$f(x)= \ln(x) + e^{-x}$$, modeling a physical measurement obtained

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

Comparative Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the data in the following table: | x | f(x) | |---|------| |

Critical Points of a Log-Quotient Function

Let $$f(x)= \frac{\ln(x)}{x}$$ for $$x > 0$$. This function is used to analyze efficiency in logarit

Derivative of an Exponential Decay Function

Consider the function $$f(t)=e^{-0.5*t}$$, which may represent the decay of a substance over time. A

Difference Quotient for a Cubic Function

Let \(f(x)=x^3\). Using the difference quotient, answer the following parts.

Differentiating a Product of Linear Functions

Let $$f(x) = (2*x^2 + 3*x)\,(x - 4)$$. Use the product rule to find $$f'(x)$$.

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

Economic Cost Function Analysis

A company’s production cost is modeled by $$C(x)= 0.02*x^3 - 0.5*x^2 + 4*x + 100$$, where $$x$$ repr

Economic Model: Revenue and Rate of Change

The revenue for a product is given by $$R(x)= \frac{x^2 + 10*x}{x+2}$$, where $$x$$ is in hundreds o

Electricity Consumption with Renewable Generation

A household has solar panels that generate power at a rate of $$f(t)=50*\sin\left(\frac{\pi*t}{12}\r

Finding the Derivative Using First Principles

Consider the function $$f(x)= 5*x^3 - 4*x + 7$$. Use the definition of the derivative to find the de

Highway Traffic Flow Analysis

Vehicles enter a highway ramp at a rate given by $$f(t)=60+4*t$$ (vehicles/min) and exit the highway

Implicit Differentiation in Demand Analysis

Consider the implicitly defined demand function $$x^2 + x*y + y^2 = 100$$, where x represents the pr

Instantaneous Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

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: Cubic with Linear Term

Consider the function $$f(x)=x^3+x$$ defined for all real numbers.

Inverse Function Analysis: Hyperbolic-Type Function

Consider the function $$f(x)=\sqrt{x^2+1}$$ defined for $$x\geq0$$.

Inverse Function Analysis: Rational Function

Consider the function $$f(x)=\frac{2*x+1}{x+3}$$ defined for all x except $$x=-3$$.

Inverse Function Analysis: Square Root Function

Consider the function $$f(x)=\sqrt{4*x+1}$$ defined for $$x \geq -\frac{1}{4}$$.

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

Marginal Cost from Exponential Cost Function

A company’s cost function is given by $$C(x)= 500*e^{0.05*x} + 200$$, where $$x$$ represents the num

Marginal Profit Calculation

A company’s profit (in thousands of dollars) is given by $$P(x)= -2*x^2 + 50*x - 100$$, where $$x$$

Mountain Stream Flow Adjustment

A mountain stream receives additional water from snowmelt at a rate of $$f(t)=4*t$$ (cubic feet/seco

Optimizing Car Speed: Rate of Change Analysis

A car’s speed in km/h is modeled by the function $$s(t)=50+2*t^2-0.1*t^3$$ for $$0 \leq t \leq 10$$

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

Quotient Rule Application

Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone

Rates of Change from Experimental Data

A chemical experiment yielded the following measurements of a substance's concentration (in molarity

River Crossover: Inflow vs. Damming

A river receives water from two tributaries at rates $$f_1(t)=7+0.5*t$$ and $$f_2(t)=9-0.2*t$$ (lite

Secant and Tangent Lines

Consider the function $$f(x)= x^2$$. Use graphical and algebraic methods to examine the behavior of

Secant Slopes Limit Interpretation

For a function $$f(x)$$, the secant slopes over the interval from $$x$$ to $$x+h$$ are given by the

Secant vs. Tangent Rate Comparison

For the function $$f(x)=x^2$$, we analyze the relationship between the secant and tangent approximat

Tangent Line and Differentiability

Let $$h(x)=\frac{1}{x+2}$$, modeling the concentration of a substance in a chemical solution over ti

Tangent Line Equation for an Exponential Function

Consider the function $$f(x)= e^{x}$$ and its graph.

Using the Quotient Rule for a Rational Function

Let $$f(x) = \frac{3*x+5}{x-2}$$. Differentiate $$f(x)$$ using the quotient rule.

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 Motion in the Plane using Implicit Differentiation

A particle moves in the xy-plane along a path defined implicitly by $$x^2+x*y+y^2=7$$. Determine the

Chain and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Chain Rule Basics

Consider the function $$f(x) = \sqrt{3*x^2 + 2}$$. Answer the following:

Chain Rule in an Implicitly Defined Function

Consider the equation $$\tan(x+y)=x^2-y^2$$. Answer the following:

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Chain Rule with Nested Trigonometric Functions

Consider the function $$f(x)= \sin(\cos(3*x))$$. This function involves nested trigonometric functio

Composite and Inverse Differentiation in Production Analysis

A factory’s production output is modeled by the composite function $$Q(x)= f(g(x))$$, where $$g(x)=

Composite Function Differentiation Involving Product and Chain Rules

Consider the function $$F(x)= (x^2 + 1)^3 * \ln(2*x+5)$$.

Composite Function Involving Exponential and Cosine

Consider the function $$f(x)= e^(\cos(x^2))$$. Address the following:

Composite Function: Engineering Stress-Strain Model

In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \

Differentiation of Nested Composite Logarithmic-Trigonometric Function

Consider the function $$f(x)=\ln(\sin(3x^2+2))$$.

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Implicit Differentiation in a Circle

Consider the circle $$x^2 + y^2 = 25$$. Answer the following parts.

Implicit Differentiation in a Financial Model

An implicit relationship between revenue $$R$$ (in thousands of dollars) and price $$p$$ (in dollars

Implicit Differentiation in an Economic Model

In an economic model, the relationship between the quantity supplied $$x$$ and the market price $$y$

Implicit Differentiation in Logarithmic Functions

Consider the equation $$\ln(x)+\ln(y)=1$$. Answer the following:

Implicit Differentiation Involving Trigonometric Functions

For the relation $$\sin(x) + \cos(y) = 1$$, consider the curve defined implicitly.

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve the following:

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x} + y = x + e^{y}$$ which relates $$x$$ and $$y$$ via exponential functi

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Implicitly Defined Inverse Relation

Consider the relation $$y + \ln(y)= x.$$ Answer the following:

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 Model

Let $$f(x) = e^{2*x} + x$$, and let g be its inverse function. Answer the following parts.

Logarithmic Differentiation of a Composite Function

For the function $$y= (x^2+1)^(\tan(x))$$, use logarithmic differentiation to address the following

Pendulum Angular Displacement Analysis

A pendulum's angular displacement is modeled by the function $$\theta(t)=\sin(2t^2+3)$$, where t is

Second Derivative via Implicit Differentiation

Given the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$, find the second derivative $$\frac{d^2y}{dx^2}$

Analysis of Experimental Data

The graph below shows the displacement of an object moving in a straight line. Analyze the object's

Car Deceleration

A car moves along a straight road with a velocity function given by $$v(t)=20-4*t$$ (m/s) for $$0 \l

Critical Points and Concavity Analysis

Consider the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ modeling the position of an

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Differentiability of a Piecewise Function

Consider the piecewise function $$ f(x)=\begin{cases} x^2, & x \leq 2 \\ 4x-4, & x>2 \end{cases} $$

Estimating Instantaneous Rates from Discrete Data

In a laboratory experiment, the concentration of a chemical (in molarity, M) is recorded over time (

Evaluating Indeterminate Limits via L'Hospital's Rule

Scientists are studying the limit of the function $$L(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to

FRQ 9: Production Efficiency Analysis

A factory’s production efficiency is modeled by the relation $$L^2 + L*Q + Q^2 = 1500$$, where L rep

FRQ 18: Chemical Reaction Concentration Changes

During a chemical reaction, the concentrations of reactants A and B are related by $$[A]^2 + 3*[A]*[

Growth Rate Estimation in a Biological Experiment

In a biological experiment, the mass $$M(t)$$ (in grams) of a bacteria colony is recorded over time

Inflating Balloon Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

Linear Approximation for Function Values

Consider the function $$f(x) = x^3$$. Use linearization at $$x = 4$$ to approximate the value of $$f

Linear Approximations: Estimating Function Values

Let $$f(x)=x^4$$. Use linear approximation to estimate $$f(3.98)$$. Answer the following:

Linearization in Medicine Dosage

A drug’s concentration in the bloodstream is modeled by $$C(t)=\frac{5}{1+e^{-t}}$$, where $$t$$ is

Linearization of a Nonlinear Function

Suppose $$f(x)=\ln(x)$$. Use linearization about $$x=4$$ to approximate $$\ln(4.1)$$. Answer the fol

Minimizing Material in Packaging Design

A company wants to design a closed cylindrical can that holds 1000 mL of liquid. The surface area of

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Optimization: Minimizing Material for a Box

A company wants to design an open-top box with a square base that holds 32 cubic meters. Let the bas

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

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Rate of Change in a Population Model

A population model is given by $$P(t)=30e^{0.02t}$$, where $$P(t)$$ is the population in thousands a

Region Area and Volume by Rotation

Consider the region R bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ from $$x=0$$ to $$x

Related Rates in a Spherical Balloon

A spherical balloon is being inflated, and its volume $$V$$ (in cubic inches) is related to its radi

Related Rates in Expanding Circular Oil Spill

An oil spill forms a circular patch. Its area is given by $$A= \pi*r^2$$. If the area is increasing

Revenue Function and Marginal Revenue Analysis

A company's revenue is modeled by $$R(x)= -0.5*x^3 + 20*x^2 + 15*x$$, where $$x$$ represents the num

Studying a Bouncing Ball Model

A bouncing ball reaches a maximum height after each bounce modeled by $$h(n)= 100*(0.8)^n$$, where n

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x) = \sin(x)$$ defined on the interval $$[0, \pi]$$. Answer the following:

Chemical Reactor Temperature Optimization

In a chemical reactor, the temperature is controlled by the rate of coolant inflow. The coolant infl

Continuous Compound Interest

An investment account is governed by the formula $$A(t)= A_0 * e^{r*t}$$, where $$r$$ is the continu

Cooling of a Cup of Coffee

A cup of coffee cools according to the model $$T(t)= T_{room}+(T_{initial}-T_{room})e^{-kt}$$ with $

Cost Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Determining Absolute and Relative Extrema

Analyze the function $$f(x)= \frac{x}{1+x^2}$$ on the interval $$[-2,2]$$.

Evaluating Pollution Concentration Changes

A study recorded the concentration of a pollutant (in ppm) in a river over time (in hours). Use the

Hydroelectric Dam Efficiency

A hydroelectric dam experiences water inflow and outflow that affect its efficiency. The inflow is g

Increase and Decrease Analysis of a Polynomial Function

Consider the function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ on the interval $$[-1, 3]$$.

Inverse Analysis of a Composite Function

Consider the function $$f(x)=e^(x)+x$$. Although its inverse cannot be written in closed form, answe

Inverse Analysis of a Linear Function

Consider the function $$f(x)=3*x+2$$. Analyze its inverse function by answering all parts below.

Inverse Analysis of a Quadratic Function (Restricted Domain)

Consider the function $$f(x)=x^2-4*x+7$$ defined on the restricted domain $$[2, \infty)$$. Analyze t

Inverse Analysis of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$. Perform the following analysis regarding its inver

Inverse Analysis: Logarithmic Ratio Function in Financial Context

Consider the function $$f(x)=\ln\left(\frac{x+4}{x+1}\right)$$ with domain $$x > -1$$. This function

Investment with Continuous Compounding and Variable Rates

An investment grows continuously with a variable rate given by $$r(t)= 0.05+0.01e^{-0.5*t}$$. Its va

Oil Spill Cleanup

In an oil spill scenario, oil continues to enter an affected area while cleanup efforts remove oil.

Optimizing a Cylindrical Water Tank

A cylindrical water tank without a top is to be built with a fixed surface area of 100 m². Let $$r$$

Piecewise Function and the Mean Value Theorem

Consider the piecewise function $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x - 1 & \text{

Polynomial Rational Discontinuity Investigation

Consider the function $$ g(x) = \begin{cases} \frac{x^3 - 8}{x - 2}, & x \neq 2, \\ 5, & x = 2. \en

Relative Extrema of a Rational Function

Examine the function $$f(x)= \frac{x+1}{x^2+1}$$ and determine its relative extrema using derivative

Reservoir Evaporation and Rainfall

A reservoir gains water through rainfall and loses water by evaporation. Rainfall occurs at a rate g

Revenue Optimization in Economics

A company's revenue is modeled by the function $$R(x)= x*e^{-0.1*x}$$, where $$x$$ (in thousands) re

Temperature Regulation in a Greenhouse

A greenhouse is regulated by an inflow of warm air and an outflow of cooler air. The inflow temperat

Traffic Intersection Flow Analysis

At a busy urban intersection, traffic flow is modeled by an inflow rate $$I(t)=30+5*t$$ and an outfl

Accumulation Function and Its Derivative

Define the function $$F(x)= \int_0^x \Big(e^{t} - 1\Big)\,dt$$. Answer the following parts related t

Approximating the Area with Riemann Sums

Consider the linear function $$f(x) = 2*x + 1$$ on the interval $$[1,5]$$. Use Riemann sums to appro

Area Between Curves

An engineering design problem requires finding the area of the region enclosed by the curves $$y = x

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

Average Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Composite Functions and Accumulation

Let the accumulation function be defined by $$F(x)=\int_{2}^{x} \sqrt{t+1}\,dt.$$ Answer the followi

Computing a Definite Integral Using the Fundamental Theorem of Calculus

Let the function be defined as $$f(x) = 2*x$$. Use the Fundamental Theorem of Calculus to evaluate t

Definite Integral and the Fundamental Theorem of Calculus

Consider the function $$f(x)= 3*x^2 - 2*x + 1$$ defined on the interval $$[1,4]$$. Use the Fundament

Displacement from a Velocity Function

A particle moves along a straight line with velocity function $$v(t)=3*t^2 - 4*t + 2$$ (in m/s). Det

Estimating Accumulated Water Inflow Using Riemann Sums

A water tank fills at varying rates. The table below shows the inflow rate in liters per second at d

Estimating Distance Traveled Using the Trapezoidal Rule from Speed Data

During a car journey, the speed (in km/hr) is recorded at regular intervals. The table below shows s

Evaluating an Integral with a Piecewise Function

Consider the function defined by $$f(x)=\begin{cases} x^2 & \text{if } x<2,\\ 4*x-4 & \text{if } x

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

FRQ2: Inverse Analysis of an Antiderivative Function

Consider the function $$ G(x)=\int_{0}^{x} (t^2+1)\,dt $$ for all real x. Answer the following parts

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

FRQ19: Inverse Analysis with a Fractional Integrand

Let $$ M(x)=\int_{2}^{x} \frac{t}{t+2}\,dt $$. Answer the following parts.

General Antiderivatives and the Constant of Integration

Given the function $$f(x)= 4*x^3$$, address the following questions about antiderivatives.

Growth of Investment with Regular Contributions and Withdrawals

An investment account receives contributions at a rate of $$C(t)= 100e^{0.05t}$$ dollars per year an

Integration Using U-Substitution

Consider the function $$g(x)= (x-3)^4$$ defined on the interval $$[3,7]$$.

Logistically Modeled Accumulation in Biology

A biologist is studying the growth of a bacterial culture. The rate at which new bacteria accumulate

Net Change vs Total Accumulation in a Velocity Function

A particle moves with velocity $$v(t)=5-t^2$$ (in m/s) for t in [0,4]. Answer the following:

Population Change in a Wildlife Reserve

In a wildlife reserve, animals immigrate at a rate of $$I(t)= 10\cos(t) + 20$$ per month, while emig

Population Growth in a Bacterial Culture

A bacterial culture grows at a rate given by $$r(t)=0.5*e^{-0.3*t}$$ (in thousands of bacteria per h

Rainfall Accumulation Analysis

The rainfall intensity at a location is modeled by the function $$i(t) = 0.5*t$$ (inches per hour) f

Reservoir Accumulation Problem

A reservoir is filled at a rate given by $$R(t)=\frac{8}{1+e^{-0.5*t}}$$ cubic meters per minute, wh

Riemann Sum Approximation from a Table

The table below gives values of a function $$f(x)$$ at selected points: | x | 0 | 2 | 4 | 6 | 8 | |

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Temperature Change in a Chemical Reaction

During an exothermic chemical reaction, the temperature (in °C) is recorded over a 15-minute period.

Temperature Cooling: An Initial Value Problem

An object cools according to the differential equation $$T'(t)=-0.2\,(T(t)-20)$$, where $$T(t)$$ is

Total Distance from Velocity Data

A car’s velocity, in meters per second, is recorded over time as given in the table below: | Time (

Total Distance Traveled from Velocity Data

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for t in [0

Total Fuel Used Over a Trip

A car consumes fuel at a rate modeled by $$r(t) = 0.2*t + 1.5$$ (in gallons per hour) during a long

Trapezoidal Rule in Estimating Accumulated Change

A rising balloon has its height measured at various times. A portion of the recorded data is given i

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 Direction Fields for $$dy/dx = y-1$$

Consider the differential equation $$dy/dx = y - 1$$. A slope field for this equation is provided. A

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

Bacterial Culture with Antibiotic Treatment

A bacterial culture grows at a rate proportional to its size, but an antibiotic is administered cont

Bernoulli Differential Equation

Solve the Bernoulli differential equation $$\frac{dy}{dx}-\frac{1}{x}y=-x*y^2$$ for $$x>0$$ with the

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

Chemical Reactor Temperature Profile

In a chemical reactor, the temperature $$T$$ (in °C) is recorded over time (in minutes) as shown. Th

Differential Equation in Business Profit

A company's profit $$P(t)$$ changes over time according to $$\frac{dP}{dt} = 100\,e^{-0.5t} - 3P$$.

Direction Fields and Integrating Factor

Consider the differential equation $$\frac{dy}{dx}=\frac{2*x}{1+y^2}$$ with the initial condition $$

Direction Fields for an Autonomous Equation

Consider the differential equation $$\frac{dy}{dx}=y^2-9$$. Analyze the behavior of its solutions.

Ecosystem Nutrient Cycle

In a forest ecosystem, nitrogen is deposited from the atmosphere at a rate of $$2$$ kg/ha/year while

Exponential Growth and Doubling Time

A bacterial culture grows according to the differential equation $$\frac{dy}{dt} = k * y$$ where $$y

Implicit Differential Equation and Asymptotic Analysis

Consider the differential equation $$\frac{dy}{dx}= \frac{y(1-y)}{x}$$ for $$x > 0$$ with the initia

Implicit Differentiation of a Circle

Consider the circle defined by $$x^2+ y^2= 25$$. Answer the following:

Implicit Differentiation of a Transcendental Equation

Consider the equation $$e^{x*y} + y^3= x$$. Answer the following:

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

Insulin Concentration Dynamics

The concentration $$I$$ (in μU/mL) of insulin in the blood follows the model $$\frac{dI}{dt}=-k(I-I_

Linear Differential Equation and Integrating Factor

Consider the differential equation $$\frac{dy}{dx} = y - x$$. Use the method of integrating factor t

Logistic Growth Model for Population Dynamics

A population $$P$$ is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\fr

Logistic Population Growth

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\left(1

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

Mixing Problem with Variable Inflow Concentration

A tank initially contains 50 L of water with 5 kg of dissolved salt. A solution enters the tank at a

Mixing Problem with Variable Volume

A tank initially contains 200 liters of solution with 10 kg of solute. A solution with concentration

Motion Under Gravity with Air Resistance

An object is falling vertically under the influence of gravity and air resistance. Its velocity $$v(

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

Newton's Law of Cooling with Temperature Data

A thermometer records the temperature of an object cooling in a room. The object's temperature $$T(t

Nonlinear Cooling of a Metal Rod

A thin metal rod cools in an environment at $$15^\circ C$$ according to the differential equation $$

Oil Spill Cleanup Dynamics

To mitigate an oil spill, a cleanup system is employed that reduces the volume of oil in contaminate

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda N$$,

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-kN$$. If the

Radioactive Decay and Half-Life

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda N$$.

Radioactive Decay with Production

A radioactive substance decays while being produced at a constant rate, and its mass $$M(t)$$ (in kg

Related Rates: Conical Tank Filling

Water is pumped into a conical tank at a rate of $$3$$ m$^3$/min. The tank has a height of $$4$$ m a

Related Rates: Shadow Length

A 2 m tall lamp post casts a shadow of a 1.8 m tall person who is walking away from the lamp post at

Sand Pile Dynamics

Sand is added to a pile at a constant rate of $$15$$ kg/min while some sand is simultaneously lost d

Soot Particle Deposition

In an environmental study, the thickness $$P$$ (in micrometers) of soot deposited on a surface is me

Water Tank Flow Analysis

A water tank receives an inflow of water at a rate $$Q_{in}(t)=50+10*\sin(t)$$ (liters/min) and an o

Accumulated Rainfall Calculation

During a 12-hour storm, the rainfall rate is modeled by $$r(t)=3+\sin(t)$$ (in mm/hr) for $$0 \le t

Area Between a Function and Its Tangent

A function $$f(x)$$ and its tangent line at $$x=a$$, given by $$L(x)=m*x+b$$, are considered on the

Average Concentration in Medical Dosage

A patient’s bloodstream concentration of a medication is modeled by the function $$C(t)=\frac{100}{1

Average Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

Average Reaction Rate Determination

A chemical reaction’s rate is modeled by the function $$r(t)=k*e^{-t}$$, where $$t$$ is in seconds a

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 Value of a Deposition Rate Function

During a sediment deposition experiment, the deposition rate (in mm/hr) was recorded over a 10-hour

Car Motion: Position, Velocity, and Acceleration

A car moving along a straight eastbound road has an acceleration given by $$a(t)=4-0.5*t$$ (in m/s²)

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

Consumer Surplus Calculation

The demand function for a certain product is given by $$D(p)=100-5*p$$ and the supply function by $$

Economic Analysis of Consumer Surplus

A market demand function is given by $$P(x)=50 - 10*\ln(x+1)$$, where $$x$$ represents quantity dema

Economics: Consumer Surplus Calculation

Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons

Hiking Trail: Position from Velocity

A hiker's velocity is given by $$v(t)=3\cos(t/2)+1$$ (in km/h) for 0 ≤ t ≤ 2π. Assuming the hiker st

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

Motion Under Resistive Force

A particle’s acceleration in a resistive medium is modeled by $$a(t)=\frac{10}{1+t} - 2*e^{-t}$$ (in

Particle Motion Analysis

A particle moving along a straight line has an acceleration given by $$a(t)=6-0.5*t$$ (in m/s²) for

Position Analysis of a Particle with Piecewise Acceleration

A particle moving along a straight line experiences a piecewise constant acceleration given by $$a(

Position, Velocity, and Acceleration Analysis of a Moving Car

A car moving along a straight line experiences an acceleration given by $$a(t)= 2*t - 4$$ (in m/s²).

Projectile Motion: Position, Velocity, and Maximum Height

A projectile is launched vertically upward with an initial velocity of $$20\,m/s$$ from a height of

Retirement Savings Auto-Increase

A person contributes to a retirement fund such that the monthly contributions form an arithmetic seq

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

Total Distance Traveled from a Velocity Profile

A particle’s velocity over the interval $$[0, 6]$$ seconds is given in the table below. Note that th

Viral Video Views

A viral video’s daily views form a geometric sequence. On day 1, the video is viewed 1000 times, and

Volume by the Cylindrical Shells Method

A region bounded by $$y=\ln(x)$$, $$y=0$$, and the vertical line $$x=e$$ is rotated about the y-axis

Volume by the Washer Method: Solid of Revolution

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region i

Water Flow in a River: Average Velocity and Flow Rate

A river has a width of 10 meters. The velocity of the water at a distance $$x$$ (in meters) from one