Power Delivered to Accelerating Blocks

Pendulum Speed at Lowest Point

Power Equation Variable Definition

Average Power and Kinetic Energy

Kinetic Energy and Speed Relationship

Power Output on an Incline

Kinetic Energy and Non-Conservative Work

Kinetic Energy and Reference Frames

Work Done by a Perpendicular Force

Potential Energy of Charged Particles

Work Done By Centripetal Force

Power Work and Time Relationship

Nature of Translational Kinetic Energy

Engine Power and Efficiency

Conservative Forces and Potential Energy

Work and Gravitational Potential Energy

Pendulum Energy Conservation

Potential Energy of a Multi-Object System

System Definition for Energy Conservation

Energy Dissipation in a Pendulum

Electric Potential Energy and Scalar Values

Work by Nonconservative Forces

Energy in a Perfectly Inelastic Collision

System Choice for Energy Conservation

Conservation Of Energy System Boundary

Energy Conservation in a Pendulum

Calculating Average Power

Average Power with Kinetic Friction

Work-Energy Theorem Definition

Ratio of Power Output

Energy Transformation in an Inelastic Collision

Pendulum Speed and Conservation of Energy

Work Done By An Angled Force

Average Power During Acceleration

Pendulum Speed at Lowest Point

System Definition for Energy Conservation

Calculating Speed From Kinetic Energy

Conservative Force and Potential Energy

Calculating Average Power

Elastic Potential Energy in a Spring

Definition of Work

Kinetic Energy and Speed Ratio

Work Done by a Constant Force

Comparing Mechanical Power Output

Kinetic Energy and Velocity Relationship

Roller Coaster Energy Transformation

Power Work and Time Relationship

Definition of Mechanical Work

Energy Transformation Due to Friction

Energy Conversion in a Pendulum

Work Done By Gravitational Force

Using L’Hospital’s Rule, evaluate the limit $$\lim_{x\to1}\frac{\sqrt{5*x^2+2*x+1}-\sqrt{8}}{x-1}.$$ Hint: Differentiate the numerator and the denominator with respect to x.

Linear Approximation of a Logarithmic Function

Acceleration From Velocity

Interpreting the Derivative of a Rate

Find the derivative of the function $$f(x)=e^{2*x}$$ at x = 1.

Rate of Change of an Exponential Model

Particle Motion and Increasing Speed

Solving a Radical Equation

Given the water volume in a pool is described by $$V(t)=8*t^2-32*t+4$$ (in gallons), find the rate of change of the volume (in gallons per hour) at t = 2 hours.

Related Rates Shadow Problem

Position Velocity and Acceleration Relationships

Properties of Linear Approximation

Which of the following is an example of using related rates to determine the change in the radius of a circular pool when its area increases at a known rate? (Here, the derivative of the area with respect to time is connected to the derivative of the radius.)

L'Hôpital's Rule Procedure

Conical Tank Related Rates

Finding Velocity from a Position Function

Related Rates of an Expanding Cube

Particle Velocity from Position Function

Interpreting the Derivative as a Rate of Change

Conical Tank Related Rates

Interpreting the Derivative as a Rate

Position Function From Acceleration

Definite Integral As Total Change

Particle Velocity from Position Function

Interpreting the Derivative in Context

Related Rates of a Semicircle

Velocity from Position Function

First Time Particle Velocity Is Zero

Particle Motion Relative Maximum

Contextual Rate Of Change

Identifying and Classifying Critical Points

Limit Definition of the Derivative

Rate of Change of a Rectangle's Area

Evaluating a Function Defined by an Integral

Interpreting the Derivative of a Rate

Particle Acceleration from Velocity

Slope of a Normal Line to an Integral Function

Evaluate the limit $$\lim_{x\to0} \frac{\sin(2*x)}{x}$$ using L'Hôpital's Rule.

First Time Particle Velocity Is Zero

Related Rates of an Expanding Sphere

Particle Acceleration from Velocity

Ordering Rates of Change

Instantaneous Rate of Change of Temperature

Interval of Negative Velocity

Average Rate of Change of a Function

Tangent Line for an Accumulation Function

Determine the derivative of the function $$f(x)=\sin(x)*e^{x}$$ at x = 0.

A rocket’s altitude is given by $$h(t)= 5*t^2+10*t$$ (in meters) for time t in seconds. Compute the instantaneous acceleration (the second derivative) at $$t=3$$ seconds.

Acceleration from a Position Function

Interpreting a Derivative Value

Particle Motion and Total Distance

Interpreting The Derivative In Context

Particle Velocity from Position Function

Maximum Speed on a Closed Interval

Rate of Change for a Sliding Ladder

Related Rates Shadow Problem

Conditions for an Inflection Point

Tangent Line to a Function Defined by an Integral

Acceleration as the Derivative of Velocity

Units of a Rate of Change

Interpreting First and Second Derivatives in Context

Acceleration from Position Function

Balloon Volume Rate of Change

Related Rates of a Circle's Area

Derivative as Marginal Revenue

Interpretation of the Derivative of Velocity

Contextual Interpretation Of The Derivative

Tangent Line to an Integral Function

Interpreting A Derivative In Context

Interpreting a Derivative in Context

Related Rates of a Sand Pile

Interpretation Of The Derivative In Context

Properties of L'Hopital's Rule

Related Rates Problem-Solving Process

For the function $$f(x)=\ln(x^2+1)$$ as shown in the graph, determine the slope of the tangent line (i.e. the derivative) at x = 0.

Consider the function $$f(x)=\sqrt{x+4}+\frac{1}{x+2}$$ defined for $$x>-4$$ with $$x\neq -2$$. Its derivative is given by $$f'(x)=\frac{1}{2*\sqrt{x+4}}-\frac{1}{(x+2)^2}.$$ Solve for the value of x for which $$f'(x)=0$$.

Based on the provided graph of the velocity function $$v(t)= 2*t+1$$, what is the constant acceleration of the object?

Instantaneous Rate of Change of Revenue

Related Rates of a Cone

Average Rate of Change of a Function

Particle Motion and Zero Velocity

Related Rates of a Circle's Area

A bacteria population is modeled by $$P(t)=100*e^{0.5*t}$$, where t is measured in hours. Compute the rate at which the population is growing (in bacteria per hour) at t = 3 hours.

Kinematic Functions and Derivatives

Interpreting the Derivative of a Rate

Local Linear Approximation

Finding the Constant of Integration

Rate of Change of a Volume Function

Derivative as a Rate of Change

Chain Rule Application Example

Particle Motion at a Specific Time

L'Hôpital's Rule for a Limit at Infinity

A projectile’s height is modeled by $$h(t)=-5*t^2+20*t+3$$. Solve the equation $$-5*t^2+20*t+3=3$$ to determine the nonzero time t (in seconds) when the projectile reaches its initial height.

Spherical Balloon Related Rates

Ordering Linear Approximations

A particle’s position is given by $$x(t)=t^3-6*t^2+9*t+5$$ with velocity $$x'(t)=3*t^2-12*t+9$$. Solve the equation $$3*t^2-12*t+9=3$$ to find the time t (with t > 2) at which the particle’s velocity is $$3\,m/s$$.

Approximate Solution to an Equation

Related Rates Angle Of Elevation

Related Rates Sliding Ladder Problem

Average Rate of Change of a Function

Particle Motion Initial Value Problem

Rate of Change of Distance Between Ships

Related Rates of a Cube's Diagonal

Average Rate of Change of a Function

Interpreting the Derivative of a Rate

Mean Value Theorem Application

Tangent Line for an Integral Function

Derivative of a Sum of Functions

Rate of Change of Water in a Tank

Chain Rule with a Cosine Function

Particle Acceleration from Velocity

Local Linear Approximation of a Square Root

Derivative of a Quadratic Function

Spherical Balloon Related Rates

Instantaneous Rate of Change of Position

Particle Acceleration From Velocity

Minimum Acceleration of a Particle

Particle Motion and Zero Velocity

Linear Approximation of a Function

Related Rates for a Sphere's Volume

The temperature of a cup of coffee is modeled by $$T(t)=70+50*e^{-0.1*t}$$, where t (in minutes) represents the time since the coffee was poured. Determine the rate of change of the temperature (in °F/min) at t = 5 minutes.

Derivative Value Using the Product Rule

Limit Definition of the Derivative

Power Rule with Rational Exponents

Limit Definition of the Derivative

Derivative of the Cosine Function

Equation of a Tangent Line

Derivative Using The Quotient Rule

Power Rule With Fractional Exponents

Limit Definition of the Derivative

Power Rule For Differentiation

Using the graph of $$f(x)=\frac{x^3-2*x}{x-1}$$, determine the value of the derivative $$f'(2)$$ by applying the quotient rule.

The limit $$\lim_{h\to0}\frac{(x+h)^3-x^3}{h}$$ represents the instantaneous rate of change of the function $$f(x)=x^3.$$ Setting this derivative equal to 12 gives the equation $$3x^2=12.$$ Solve for x.

Derivative of the Natural Logarithm Function

Chain Rule with a Trigonometric Function

Product Rule With A Trigonometric Function

Derivative Using the Product Rule

Average Rate of Change of a Rational Function

Definition of Differentiability

Derivative of an Exponential Function

I. The slope of the tangent line at any point on the graph represents the instantaneous rate of change of the function at that point. II. The slope of the secant line between two points on the graph approximates the average rate of change over that interval. III. The difference quotient used to compute the derivative is applicable only for linear functions. Which of the above statements are true?

Derivative Notation

Power Rule with Radical Functions

Product Rule For Differentiation

Limit Definition of an Exponential Derivative

Difference Quotient Approximation

Marginal Cost from Cost Function

Derivative of a Constant Function

I. The slope of the tangent line to the graph at any given point represents the instantaneous rate of change of the function. II. A secant line through two points on the curve approximates the average rate of change over the interval between those points. III. A steep tangent line always indicates that the function is increasing at that point. Which of the above statements are true?

Limit Definition of a Derivative

Estimate the instantaneous rate of change of $$f(x)=\cos(x)$$ at $$x=\frac{\pi}{3}$$ from the graph.

Average Rate of Change Definition

Evaluating a Derivative with the Power Rule

Finding Critical Points of a Polynomial

Power Rule with Fractional Exponents

The graph above shows f(x) = $$\sqrt{x}$$. Compute the average rate of change on the interval [4, 9] and the instantaneous rate of change at x = 9, then find the difference between these two values.

Limit Definition of the Derivative

Chain Rule Application

Limit Definition of the Derivative

Limit Definition of the Derivative

Quotient Rule for Differentiation

Derivative Using the Quotient Rule

Quotient Rule at a Point

Derivative Using the Quotient Rule

Evaluating a Limit as a Derivative

Rate of Change of a Velocity Function

Average Rate of Change on an Interval

Limit Definition of a Derivative

Average Rate of Change of a Function

Average Rate of Change of a Cosine Function

Find the derivative of $$y=\arccos(3*x^2)$$ with respect to x.

Derivative of an Inverse Sine Function

Derivative of an Inverse Tangent Function

Identifying a Composite Function

Which of the following is the correct derivative of $$y=\sqrt{\tan(2*x)}$$ using the Chain Rule?

Derivative of an Inverse Function

Chain Rule With a Radical Function

Find the derivative of the function $$y=\arctan(3*x)$$ with respect to x.

For the circle defined by $$x^2+y^2=25$$, first use implicit differentiation to find $$\frac{dy}{dx}$$, and then determine the second derivative $$\frac{d^2y}{dx^2}$$ at the point (3,4).

Implicit Differentiation

Chain Rule with Power Rule

Differentiate $$f(x)=\ln(\sqrt{1+2*x})$$ with respect to x.

Logarithmic Implicit Differentiation

Differentiate $$y = \arcsin(3*x)$$ with respect to x.

Applying The Chain Rule

Derivative of an Inverse Function

Implicit Differentiation at a Point

Using the graph above of the function $$f(x)$$, rank the following steps for finding the derivative of its inverse function $$f^{-1}(x)$$ at a certain point (from first to last): A) Identify the x-value on the graph that corresponds to the given y-value. B) Compute the derivative $$f'(x)$$ at that x-value. C) Take the reciprocal of $$f'(x)$$ to obtain $$\left(f^{-1}\right)'(y)$$. D) Simplify the resulting expression to obtain the final derivative.

If $$f(x)=\sqrt{2*x+3}$$ is invertible and its inverse is denoted by $$g(x)$$, determine $$g'(5)$$ knowing that if $$f(x_0)=5$$ then $$g'(5)=\frac{1}{f'(x_0)}$$.

Chain Rule With Inverse Tangent

Implicit Differentiation With Logarithms

Chain Rule with a Logarithmic Function

Let $$f(x)=x^3+2*x$$ be one-to-one on a restricted domain. If $$f(1)=3$$, determine $$(f^{-1})'(3)$$.

Chain Rule with Natural Logarithm

For the relation $$\sin(x+y)=x*y$$, use implicit differentiation to find an expression for $$\frac{dy}{dx}$$ in terms of x and y.

Differentiate the function $$y= \sqrt{\ln(3*x+2)}$$ with respect to $$x$$.

Derivative Using Implicit Differentiation

Chain Rule Application Analysis

Solve for the derivative $$\frac{d}{dx}[\arccos(3*x)]$$ using the chain rule.

Implicit Differentiation of a Trigonometric Function

Derivative of an Inverse Function

Find the derivative of $$y=(3*x^2+2*x)^5$$ with respect to $$x$$.

Second Derivative with Implicit Differentiation

Let $$f(x) = (x+1)^2$$ and suppose $$g(x)=f^{-1}(x)$$. Compute $$g'(16)$$.

Suppose a differentiable function f(x) is given by the graph above where $$f(2)=5$$ and $$f'(2)=3$$. Using inverse function differentiation, compute $$(f^{-1})'(5)$$.

Second Derivative with Implicit Differentiation

Applying The Chain Rule

Derivative of an Inverse Function

Implicit Differentiation with an Exponential

Chain Rule with a Power Function

Differentiate $$y = \arccos(2*x)$$ with respect to x.

Implicit Differentiation with an Exponential Function

Differentiate $$y=\arccos(4*x)$$ with respect to $$x$$.

Solve for the derivative of the function $$f(x)= \cos(5*x^3)$$ using the chain rule.

Consider the graph provided representing $$f(x)= (x^3+1)^2$$. All of the following statements regarding its derivative using the chain rule are true except:

Chain Rule with Powers and Roots

Implicit Differentiation With Sine

Implicit Differentiation at a Point

Chain Rule With Trigonometric Functions

Center of Mass of a Rod-Mass System

Free-Body Diagram Components

Forces on a Projectile in Flight

Force Interactions in Rocket Propulsion

Centripetal Acceleration and Speed

Properties of Kinetic Friction

Orbital Period and Planetary Properties

Energy Conservation in a Spring-Mass System

Force and Center of Mass Motion

Center of Mass and External Forces

Conservation of Momentum with Two Blocks

Particle Model Approximation

Conditions for Translational Equilibrium

Newton's Third Law and Acceleration

Characteristics of Uniform Circular Motion

Center of Mass of a Particle System

Normal Force in an Accelerating Elevator

Normal Force On An Inclined Plane

Orbital Period and Gravitational Constant

Equivalence Principle and Pendulum Motion

Center Of Mass Of A System Of Masses

Apparent Weight in an Elevator

Acceleration on a Rough Inclined Plane

Tangential Acceleration in Circular Motion

Forces On A Frictionless Banked Curve

Centripetal Acceleration and Speed

Minimum Speed in a Vertical Loop

Validity of the Particle Model

Static Friction and Applied Force

Net Force from Parallel Springs

Apparent Weight in an Elevator

Orbital Period and Stellar Mass

Newton's Third Law of Gravitation

Tension in an Ideal String

Gravitational Field Strength Calculation

Static Equilibrium of Forces

Net Force and Resulting Motion

Newton's Third Law Interaction

Newton's Second Law and Proportionality

Forces on a Frictionless Incline

Contact Force Between Two Blocks

Newton's First Law and Net Force

Newton's Third Law Force Pair

Ideal Spring Force Law

Newton's Third Law and Gravitational Force

Balanced Forces and Static Friction

Forces on a Projectile in Upward Motion

Force Mass and Acceleration Relationship

Tension Force On A Suspended Object

Ideal Spring Force and Displacement

Center of Mass of a Linear System

Force Ratio in Parallel Springs

Free-Body Diagram of a Pendulum

Center of Mass Acceleration

Net Force and Acceleration

Satellite Orbital Speed and Radius

Pendulum Motion in Orbit

Fundamental Cause of Contact Forces

Mass and Gravitational Weight

Properties of Kinetic Friction

Center of Mass and Internal Forces

Free-Body and Vector Component Diagrams

Center of Mass With Removed Section

Center of Mass of a Non-Uniform Rod

Tension in a Static Pulley System

Comparing Coefficients of Kinetic Friction

Forces From Two Compressed Springs

Energy Transfer in a Damped Pendulum

Hooke's Law and Spring Constant

Tension in a Connected System

Static Friction Threshold

Gravitational Potential Energy and Distance

Direction of Spring Force

Net Acceleration in Non-Uniform Circular Motion

Orbital Radius and Period

Origin of Tension Force

Forces and Systems in Rocket Motion

Center Of Mass Of A Non-Uniform Rod

Newton's Third Law and Vehicle Acceleration

Center of Mass of a Two-Mass System

Shift in Center of Mass

Factors of Kinetic Friction

Gravitational Force and Mass Proportionality

Orbital Period and Radius Relationship

Ideal Speed on a Banked Curve

Forces at a Distance in AP Physics 1

Gravitational Force and Planetary Radius

Condition for Constant Velocity

Gravitational Force and Planetary Mass Ratio

Calculating the Period of Oscillation

Apparent Weight During Free Fall

Center of Mass of a Rod and Point Mass

Center of Mass of a Two-Object System

Center of Mass of a Particle System

Force Opposing Surface Motion

Newtons Second Law Mass and Acceleration

Kinetic Friction and Contact Area

Net Force and Acceleration Comparison

Gravitational Field Strength and Distance

Center of Mass of a Rod-Mass System

Time-Independent Kinematic Equation

Graphical Representation of Vector Magnitude

Average Velocity from Position and Time

Acceleration From Velocity Components

Acceleration and Velocity Relationship

Projectile Motion Time of Flight

Relative Direction of Motion

Scaling Force Vectors

Resultant Displacement Magnitude

One-Dimensional Vector Operations

Relative Velocity of a Dropped Object

Kinematic Equation Variables

Vector Component Signs and Magnitudes

Calculating Average Acceleration

Definition of Acceleration

Projectile Motion and Gravity

Calculating Average Acceleration

Motion Diagrams From Position-Time Graphs

Calculating Average Acceleration

Time to Reach Maximum Height

Net Force From Collinear Forces

Definition of Average Velocity

Constant Velocity Position Vector

One-Dimensional Vector Addition

Vector Quantity Definition

Velocity Vector Angle

Effect of Gravity Approximation on Period

Projectile Motion Velocity Vector Angle

Displacement From an Acceleration Function

Resultant Velocity of a Boat