AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

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

Analyzing a Piecewise Defined Function Near a Boundary

Consider the function $$g(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2, \\ 2*x+1 & \text{if

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

Asymptotic Behavior of a Water Flow Function

In a reservoir, the net water flow rate is modeled by the rational function $$R(t)=\frac{6\,t^2+5\,t

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

Continuity Analysis of a Piecewise Function

Consider the piecewise function $$f(x) = \begin{cases} x^2+1 & \text{if } x < 2, \\ 3*x-3 & \text{if

Continuity Analysis of an Integral Function

Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Exponential Function Limits at Infinity

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

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

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

Intermediate Value Theorem in Engineering Context

In a structural analysis, the stress on a beam is modeled by a continuous function $$S(x)$$ on the i

Investigating Limits at Infinity and Asymptotic Behavior

Given the rational function $$f(x)=\frac{5*x^2-3*x+2}{2*x^2+x-1}$$, answer the following: (a) Evalua

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Limits at Infinity for High-Degree Rational Functions

Consider the function $$r(x)=\frac{3*x^4-2*x^2+1}{x^4+5*x^2-4}.$$ Answer the following parts.

Limits from Table and Graph

A function $$g(x)$$ is studied in a lab experiment. The following table gives sample values of $$g(x

Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$

Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

One-Sided Infinite Limits in Rational Functions

Consider the function $$f(x)= \frac{1}{(x-2)^2}$$.

One-Sided Limits and Jump Discontinuities

Consider the piecewise function $$j(x)=\begin{cases}x+2 & \text{if } x< 3,\\ 5-x & \text{if } x\ge 3

Piecewise Function Continuity and Differentiability

Consider the piecewise function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\

Rate of Change in a Chemical Reaction (Implicit Differentiation)

In a chemical reaction the concentration C (in M) of a reactant is related to time t (in minutes) by

Seasonal Temperature Curve Analysis

A graph represents the average daily temperature (in $$^\circ C$$) as a function of the day of the y

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

Using Power Series to Estimate a Trigonometric Function

The power series for $$\sin(x)$$ is $$Q(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n+1}}{(2*n+1)!}.$$

Bacterial Culture Growth: Discrete to Continuous Analysis

In a controlled laboratory, a bacterial culture doubles every hour. The discrete model after n hours

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

Composite Function and Chain Rule Application

Consider the function $$h(x)=\sin(2*x^2+3)$$. Using the chain rule, answer the following parts:

Derivative from the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Differentiation and Linear Approximation for Error Estimation

Let $$f(x) = \ln(x)*x^2$$. Use differentiation and linear approximation to estimate changes in the f

Economic Model Rate Analysis

A company models its cost variations with respect to price $$p$$ using the function $$C(p)=e^{-p}+\l

Efficiency Ratio Rate Change Using the Quotient Rule

An efficiency ratio is modeled by $$E(x) = \frac{x^2+2}{3*x-1}$$, where x represents an input variab

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Estimating Instantaneous Acceleration from Velocity Data

An object's velocity (in m/s) is recorded over time as shown in the table below. Use the data to ana

Interpreting Derivative Notation in a Real-World Experiment

A reservoir's water level (in meters) is measured at different times (in minutes) as shown in the ta

Piecewise Function and Discontinuity Analysis

Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2 \\ 3 & x = 2 \en

Polar Coordinates and Tangent Lines

Consider the polar curve $$r(\theta)=1+\cos(\theta)$$. Answer the following:

Population Growth Rate

A population is modeled by $$P(t)=\frac{3*t^2 + 2}{t+1}$$, where $$t$$ is measured in years. Analyze

Profit Rate Analysis in Economics

A firm’s profit function is given by $$\Pi(x)=-x^2+10*x-20$$, where $$x$$ (in hundreds) represents t

Related Rates: Draining Conical Tank

Water is draining from an inverted conical tank with a height of 6 m and a top radius of 3 m. The vo

Sediment Accumulation in a Dam

Sediment enters a dam reservoir at a rate of $$S_{in}(t)=5\ln(t+1)$$ kg/hour, while sediment is remo

Tangent and Normal Lines to a Curve

Given the function $$p(x)=\ln(x)$$ defined for $$x > 0$$, analyze its rate of change at a specific p

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

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

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

Widget Production Rate

A widget manufacturing plant produces widgets according to the function $$P(t)=4*t^2 - 3*t + 10$$ wh

Chain Rule for a Multi-layered Composite Function

Let $$f(x)= \sqrt{\ln((3*x+2)^5)}$$. Answer the following:

Chain Rule in a Nested Composite Function

Consider the function $$f(x)= \sin\left(\ln((2*x+1)^3)\right)$$. Answer the following parts:

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

Composite Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Continuity and Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Differentiation of an Inverse Exponential Function

Let $$f(x)=e^{2*x}-7$$, and let g denote its inverse function. Answer the following parts.

Differentiation of an Inverse Trigonometric Function

Define $$h(x)= \arctan(\sqrt{x})$$. Answer the following:

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Implicit Differentiation and Inverse Challenges

Consider the implicit relation $$x^2+ x*y+ y^2=10$$ near the point (2,2).

Implicit Differentiation in a Chemical Reaction

In a chemical process, the concentrations of two reactants, $$x$$ and $$y$$, satisfy the relation $$

Implicit Differentiation in an Economic Model

A company’s production is modeled by the implicit relationship $$x*y^2 + \ln(x+y) = 10$$, where $$x$

Implicit Differentiation in Geometric Optics

A parabolic mirror used in a geometric optics experiment is described by the implicit equation $$x^2

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 of a Product Equation

Consider the equation $$ x*y + x + y = 10 $$.

Implicit Differentiation with Trigonometric Equation

Consider the curve defined implicitly by $$\sin(x*y) + x^2 = y^3$$. Answer the following parts:

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

Inverse Function Differentiation in a Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Inverse Function Differentiation in Economics

A product’s demand is modeled by a one-to-one differentiable function $$Q = f(P)$$, where $$P$$ is t

Inverse Function Differentiation in Economics

In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan

Inverse of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases} x^2 & x \ge 0 \\ -x & x < 0 \end{cases}$$. Anal

Inverse Trigonometric Differentiation

Consider the function $$y= \arctan(\sqrt{x+2})$$.

Inverse Trigonometric Functions in Navigation

A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Physics Lab: Logarithmic Chain Rule in a Kinetics Experiment

In a kinetics experiment, the reactant concentration is modeled by $$C(t)=\ln(3*e^{2*t}+4)$$, where

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Second Derivative via Implicit Differentiation

Given the relation $$x^2 + x*y + y^2 = 7$$, answer the following:

Taylor/Maclaurin Polynomial Approximation for a Logarithmic Function

Let $$f(x) = \ln(1+3*x)$$. Develop a second-degree Maclaurin polynomial, determine its radius of con

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

Analyzing Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

Analyzing Motion on a Curved Path

A particle moves along a path defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$t \in [0,2\pi]$

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

Application of L’Hospital’s Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 2x + 1}{3x^3 + 7}$$ using L’Hospital’s Rule.

Approximating Function Values Using Differentials

Let $$f(x)=\sqrt{x}$$. Use linearization near $$x=25$$ to approximate $$\sqrt{25.5}$$.

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

Biology: Logistic Population Growth Analysis

A population is modeled by the logistic function $$P(t)= \frac{100}{1+ 9e^{-0.5*t}}$$, where $$t$$ i

Cycloid Tangent Line

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

Economics: Cost Function and Marginal Analysis

A company's cost function is given by $$C(x)= 0.5*x^3 - 4*x^2 + 10*x + 100$$, where $$x$$ represents

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, which represents a track. A runn

Inflating Balloon: Radius and Surface Area

A spherical balloon is being inflated such that its volume increases at a constant rate of 12 cm³/s.

Inflating Balloon: Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of 10 in³/s.

L'Hôpital's Rule in Inverse Function Context

Consider the function $$f(x)=x+e^{-x}$$. Although its inverse cannot be expressed in closed form, an

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Logarithmic Function Series Analysis

The function $$L(x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n}$$ represents $$\ln(x)$$ centere

Logistic Population Model Inversion

Consider the logistic population model given by $$f(t)=\frac{50}{1+e^{-0.3*(t-5)}}$$. This function

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

Parametric Motion in the Plane

A particle moves in the plane according to the parametric equations $$x(t)=t^2-2*t$$ and $$y(t)=3*t-

Particle Motion Analysis Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Particle Motion with Changing Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=8-3*t$$ (in m/s²). Given

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Population Growth Rate Analysis

A population grows exponentially according to $$P(t)=1200e^{0.15t}$$, where t is measured in months.

Rational Function Inversion

Consider the rational function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze its inverse.

Reactant Flow in a Chemical Reactor

In a chemical reactor, a reactant is introduced at a rate $$I(t)=15\sin(\frac{t}{2})$$ (grams per mi

Related Rates in Conical Tank Draining

Water is draining from a conical tank. The volume of water is given by $$V=\frac{1}{3}\pi*r^2*h$$, a

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Road Trip Distance Analysis

During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is

Series Approximation in an Exponential Population Model

A population is modeled by $$P(t)= 1000 \times \sum_{n=0}^{\infty} \frac{(0.05t)^n}{n!}$$, which is

Series Approximation in Population Dynamics

A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans

Series Approximation with Center Shift

Consider the function $$f(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (3x-1)^n}{n+1}$$. Answer the followin

Series Differentiation in Heat Transfer Analysis

A heat transfer rate is modeled by $$H(t)= \sum_{n=0}^{\infty} \frac{(-1)^n (0.5t)^{2*n}}{(2*n)!}$$,

Series-Based Analysis of Experimental Data

An experiment models a measurement function as $$g(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x/4)^n}{n+1

Temperature Change of Cooling Coffee

The temperature of a cup of coffee is modeled by $$T(t)=70+50*e^{-0.1*t}$$ (in °F), where $$t$$ is t

Vertical Projectile Motion

An object is thrown vertically upward with an initial velocity of 20 m/s and experiences a constant

Analysis of a Function with Oscillatory Behavior

Consider the function $$f(x)=x+\sin(x)$$. Answer the following parts:

Analysis of a Logarithmic Function

Consider the function $$q(x)=\ln(x)-\frac{1}{2}*x$$ defined on the interval [1,8]. Answer the follow

Analysis of a Rational Function

Consider the function $$f(x)= \frac{x^2+4}{x+1}$$ defined for $$x\neq -1$$. Analyze its behavior.

Analysis of an Exponential-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Application of the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Area and Volume of Region Bounded by Exponential and Linear Functions

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+2$$. The region enclosed by these curves will be

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Concavity & Inflection Points for a Rational Polynomial Function

Examine the function $$f(x)= \frac{x}{x^2+1}$$ to determine its concavity and identify any inflectio

Convergence and Differentiation of a Series with Polynomial Coefficients

The function $$P(x)=\sum_{n=0}^\infty \frac{n^2 * (x-1)^n}{3^n}$$ is used to model stress in a mater

Derivative Analysis of a Rational Function

Consider the function $$s(x)=\frac{x}{x^2+1}$$. 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

Extreme Value Analysis

Consider the function $$f(x) = x^3 - 3*x^2 + 4$$ on the closed interval $$[0,3]$$. Use the Extreme V

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

Ink Drop Diffusion and Intensity Loss

When a drop of ink is placed in water, it spreads out in concentric rings. The intensity of the ink

Inverse Analysis for a Logarithmic Function

Let $$f(x)= \ln(2*x+5)$$ for $$x > -2.5$$. Answer the following parts.

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Lake Ecosystem Nutrient Dynamics

In a lake, nutrients (phosphorus) enter at a rate given by $$N_{in}(t)=5*\sin(t)+10$$ mg/min and are

Manufacturing Optimization in Production

A company’s profit (in thousands of dollars) from producing x (in thousands of units) is given by $$

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

Minimizing Production Cost

A company’s production cost is modeled by the function $$C(x)=0.5*x^2 - 20*x + 300$$, where $$x$$ re

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

Pharmaceutical Dosage and Metabolism

A patient is administered a medication with an initial dose of 50 mg. Due to metabolism, the amount

Projectile Motion Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Projectile Trajectory: Parametric Analysis

A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

Accumulated Displacement from a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

Accumulation Function Analysis

A function $$A(x) = \int_{0}^{x} (e^{-t} + 2)\,dt$$ represents the accumulated amount of a substance

Analyzing an Invertible Cubic Function

Consider the function $$f(x) = x^3 + 2*x + 1$$ defined for all $$x$$. Answer the following questions

Antiderivatives and the Fundamental Theorem

Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i

Chemical Reaction: Rate of Concentration Change

A chemical reaction features a rate of change of concentration given by $$R(t)= 5*e^{-0.5*t}$$ (in m

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 Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Convergence of an Improper Integral

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{x^{p}}\,dx$$, where $$p$$ is a positive

Determining Antiderivatives and Initial Value Problems

Suppose that $$F(x)$$ is an antiderivative of the function $$f(x)=5*x^4 - 2*x + 3$$, and that it is

Integration of a Piecewise Function

A function representing a rate is defined piecewise by $$f(t)= \begin{cases} 2*t, & 0 \le t \le 3 \

Integration of a Rational Function via Partial Fractions

Evaluate the indefinite integral $$\int \frac{2*x+3}{x^2+x-2}\,dx$$ by using partial fractions.

Integration of a Trigonometric Function by Two Methods

Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me

Integration via U-Substitution for a Composite Function

Evaluate the integral of a composite function and its definite form. In particular, consider the fun

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Radioactive Decay: Accumulated Decay

A radioactive substance decays according to $$m(t)=50 * e^(-0.1*t)$$ (in grams), with time t in hour

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Riemann Sum Approximation of Area

Given the following table of values for the function $$f(x)$$ on the interval $$[0,4]$$, use Riemann

Series Convergence and Integration with Power Series

Consider the power series $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$, which represents $$

Solving for Unknowns using Logarithmic Properties in Integration

Consider the definite integral $$\int_(a)^(b) \frac{3}{x} dx$$ which is given to equal 6, where a is

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

Autonomous ODE: Equilibrium and Stability

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(2-y)*(y+1)$$. Answer the following

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Chemical Reaction and Separable Differential Equation

In a particular chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to t

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Direction Fields and Stability Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=y(1-y)$$. Answer the following parts.

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

Exact Differential Equations

Consider the differential equation $$ (2*x + y) + (x + 3*y)\,\frac{dy}{dx} = 0$$.

FRQ 5: Mixing Problem in a Tank

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine with a salt concen

FRQ 6: Radioactive Decay

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

FRQ 10: Cooling of a Metal Rod

A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th

FRQ 17: Slope Field Analysis and Particular Solution

Consider the differential equation $$\frac{dy}{dx}=x-y$$. Answer the following parts.

Logistic Growth Model

A population P grows according to the logistic differential equation $$\frac{dP}{dt} = rP\left(1-\fr

Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

Newton’s Law of Cooling Application

An object is cooling in a room with ambient temperature $$T_a=20^\circ C$$. Its temperature $$T(t)$$

Predator-Prey Model with Harvesting

Consider a simplified model for the prey population in a predator-prey system that includes constant

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Separable Differential Equation and Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

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: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Area Under a Parametric Curve

Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u

Average Force on a Beam

A beam experiences a varying force along its length given by $$F(x)=20 - 0.5*x$$ (in kN) where $$x$$

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Temperature Analysis

A research team models the ambient temperature in a region over a 24‐hour period with the function $

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

Average Value of a Temperature Function

A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

Center of Mass of a Lamina with Constant Density

A thin lamina occupies the region in the first quadrant bounded by $$y=x^2$$ and $$y=4$$. The densit

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Determining Average Value of a Velocity Function

A runner’s velocity is given by $$v(t)=2*t+3$$ (in m/s) for $$t\in[0,5]$$ seconds.

Electrical Charge Distribution

A wire of length 3 meters carries a charge distribution given by $$\rho(x)=\frac{5}{1+x^2}$$ (in cou

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

Particle Motion with Velocity Reversal

A particle moves along a straight line with an acceleration given by $$a(t)=12-6*t$$ (in m/s²) for $

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²), initial velocity

Projectile Maximum Height

A ball is thrown upward with an acceleration of $$a(t)=-9.8$$ m/s², an initial velocity of $$v(0)=20

Rainfall Accumulation Analysis

The rainfall rate (in cm/hour) at a location is modeled by $$r(t)=0.5+0.1*\sin(t)$$ for $$0 \le t \l

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 Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of a Hollow Cylinder Using the Washer Method

A manufacturer designs a hollow cylindrical container. The outer surface is modeled by $$y=10-\sqrt{

Volume of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Volume of a Solid by the Washer Method

The region bounded by $$y=x^2$$ and $$y=4$$ is rotated about the x-axis, forming a solid with a hole

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

Work Done by a Variable Force

A variable force is applied along a frictionless surface and is given by $$F(x)=6-0.5*x$$ (in Newton

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Analyzing a Clock's Second Hand with Polar Coordinates

A clock's second hand rotates uniformly, and its tip traces a circle of radius 12 cm. Its position i

Analyzing the Concavity of a Parametric Curve

A curve is defined by $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$.

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Area Enclosed by a Polar Curve

Consider the polar curve given by $$r = 2*\sin(\theta)$$.

Circular Motion in Vector-Valued Form

A particle moves along a circle of radius 5 with its position given by $$ r(t)=\langle 5*\cos(t),\;

Combined Motion Analysis

A particle’s path is described by the parametric equations $$x(t)= \ln(1+ t^2)$$ and $$y(t)= \sqrt{t

Comparative Particle Motion

Two particles follow the paths given by: Particle A: $$x_A(t)=t^2,\, y_A(t)=2*t$$ and Particle B: $$

Comparing Parametric, Polar, and Cartesian Representations

An object moves along a curve described by the parametric equations $$x(t)= \frac{t}{1+t^2}$$ and $$

Conversion from Polar to Cartesian Coordinates

The polar equation $$r(\theta)=4*\cos(\theta)$$ represents a curve.

Curvature of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t^2$$ for \(t>0\).

Helical Particle Motion

A particle travels along a helical path described by $$\vec{r}(t)= \langle \cos(t),\; \sin(t),\; t \

Intersection of Two Parametric Curves

Two curves are represented parametrically as follows: Curve A is given by $$x(t)=t^2, \; y(t)=2*t+1$

Intersection Points of Polar Curves

Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:

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*

Modeling Circular Motion with Vector-Valued Functions

An object moves along a circle of radius $$3$$ with its position given by $$\mathbf{r}(t)=\langle 3\

Motion in a Damped Force Field

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t)

Multi-Step Problem Involving Polar Integration and Conversion

Consider the polar curves $$r_1(\theta)= 2\cos(\theta)$$ and $$r_2(\theta)=1$$.

Numerical Integration Techniques for a Parametric Curve

A curve is defined by the parametric equations $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t

Oscillatory Motion in a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

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

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

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 Parametric Equations

A projectile is launched with initial speed $$v_0 = 20\,m/s$$ at an angle of $$45^\circ$$. Its motio

Projectile Motion: Rocket Launch Tracking

A rocket is launched with its horizontal position given by $$x(t)=100*t$$ (in meters) and its vertic

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Vector-Valued Functions: Velocity and Acceleration

A particle's position is given by the vector-valued function $$\vec{r}(t) = \langle \sin(t), \ln(t+1

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 Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa