- Calculus 1
- Apply Rolle's Theorem
- Find the derivatives of inverse trigonometric functions
- Use logarithmic differentiation
- Solve optimization problems
- Apply first and second derivative tests
- Find extreme values and critical points
- Apply implicit differentiation and solve for related rates
- Find the definition of the derivative
- Calculate limits
- Find tangent line equations
- Evaluate logarithmic functions
- Evaluate exponential functions
- Evaluate trigonometric functions
- Combine functions
- Transform functions
- Set up and solve function word problems
- Sketch functions
- Determine properties of functions
- General Chemistry 1
- Solve problems relating to the Born-Haber Cycle
- Solve problems relating to Coulomb's Law and properties of ionic compounds
- Solve problems relating to vapor pressure and phase diagrams
- Assess problems relating to intermolecular forces
- Apply the Molecular Orbital Theory
- Apply Lewis, VSEPR, and Valence Bond Theory simultaneously
- Assess dipole moment, polarity, bond order, bond length, and bond energies
- Apply the VSEPR Theory
- Assess electronegativity differences and polarity of bonds
- Draw Lewis structures with exceptions to the octet rule
- Draw Lewis structures
- Solve problems relating to the periodic table and its trends
- Solve problems relating to electron configuration
- Solve problems relating to quantum numbers
- Solve problems relating to quantum mechanics
- Solve problems relating to atomic spectra of hydrogen
- Solve problems relating to electromagnetic radiation
- Solve problems relating to Hess's Law and standard enthalpies of formation
- Solve problems relating to enthalpy and internal energy
- Solve problems relating to pressure-volume work
- Solve problems relating to calorimetry
- Solve problems relating to quantity of heat
- Solve problems relating to real gases
- Solve problems relating to Graham's Law
- Solve problems relating to kinetic-molecular theory
- Solve problems relating to collecting a gas over water
- Solve problems relating to Dalton's Law of Partial Pressures
- Solve problems relating to the ideal gas law
- Solve problems relating to simple gas laws
- Use a manometer to measure gas pressure
- Write redox equations
- Write net ionic equations
- Solve problems relating to limiting reactant
- Solve problems relating to stoichiometry
- Name compounds
- Assign oxidation states
- Solve problems relating to empirical and molecular formulas
- Solve problems relating to Avogadro's number, moles, and mass
- Solve problems relating to isotopes
- Identify the number of protons, neutrons, and electrons in atoms and ions
- Apply laws to solve problems
- Classify matter
- Report measurement
- Calculus 2
- Solve problems relating to Taylor and Maclaurin series
- Solve problems relating to power series
- Solve problems relating to convergence and divergence
- Solve problems relating to the Alternating Series Estimation Theorem
- Solve problems relating to conditional convergence
- Solve problems relating to absolute convergence
- Solve problems relating to comparison tests
- Solve problems relating to the P-Series Test
- Solve problems relating to the Integral Test
- Solve problems relating to the Nth Term Test
- Find the telescopic sum
- Solve problems relating to geometric series
- Solve problems relating to partial sums
- Solve problems relating to sequences
- Evaluate improper integrals by cases
- Apply integration techniques
- Use integration to find the work done
- Find the volume of a solid of revolution
- Solve problems relating to arc length
- Use numerical integration
- Analyze phase planes
- Solve problems relating to polar coordinates
- Apply Euler's method
- Solve problems relating to first-order differential equations
- Solve problems relating to polar coordinate equations
- Find the area under the curve
- Solve problems relating to parametric curves
- Sketch slope fields
- Solve problems relating to area in polar coordinates
- Solve problems relating to intersection of polar curves
- Solve problems relating to logistic models
- AP Calculus AB
- Solve derivatives of trigonometric and natural logarithmic functions
- Connect differentiability and continuity: determine when derivatives do and do not exist
- Solve volume with disc method: revolving around the x- or y-axis
- Define the derivative of a function and use derivative notations
- Estimate limit values from graphs and tables
- Solve Riemann sums, summation notation, and definite integral notation
- Sketch graphs of functions and their derivatives
- Apply the Chain Rule
- Apply derivative rules: power, constant, sum, difference, and constant multiple
- Work with the Intermediate Value Theorem
- Connect limits at infinity and horizontal asymptotes
- Connect infinite limits and vertical asymptotes
- Confirm continuity over an interval
- Define continuity at a point
- Use the first and second derivative test to determine relative (local) extrema
- Find particular solutions using initial conditions and separation of variables
- Find the area between curves expressed as functions of y
- Integrate using substitution
- Solve exponential models with differential equations
- Find general solutions using separation of variables
- Reason using slope fields
- Sketch slope fields
- Verify solutions for differential equations
- Model situations with differential equations
- Solve volume with washer method: revolve around other axes
- Solve volume with washer method: revolve around the x- or y-axis
- Solve volume with disc method: revolving around other axes
- Solve volumes with cross sections: triangles and semicircles
- Solve volumes with cross sections: squares and rectangles
- Find the area between curves that intersect at more than two points
- Find the area between curves expressed as functions of x
- Use accumulation functions and definite integrals in applied contexts
- Connect position, velocity, and acceleration of functions using integrals
- Find the average value of a function on an interval
- Select techniques for antidifferentiation
- Integrate functions using long division and complete the square
- Find antiderivatives and definite integrals: basic rules and notation - Part 2
- Find antiderivatives and definite integrals: basic rules and notation - Part 1
- Find antiderivatives and indefinite integrals: basic rules and notation
- Apply properties of definite integrals
- Interpret the behavior of accumulation functions involving area
- Solve the fundamental theorem of calculus and accumulation functions
- Approximate areas with Riemann sums
- Determine concavity of functions over their domains
- Use the candidates test to determine absolute (global) extrema
- Determine intervals on which a function is increasing or decreasing
- Solve problems relating to Extreme Value Theorem, global vs. local extrema, and critical points
- Approximate values of a function using local linearity and linearization
- Straight-line motion: connect position, velocity, and acceleration
- Interpret the meaning of the derivative in context
- Calculate higher-order derivatives
- Differentiate inverse functions
- Use implicit differentiation
- Apply the Product Rule and Quotient Rule
- Define average and instantaneous rates of change at a point
- Connect multiple representations of limits
- Determine limits by the Squeeze Theorem
- Determine limits by algebraic manipulation
- Determine limits by direct substitution
- Determine limits by limit properties
- AP Calculus BC
- Apply the Comparison Tests for convergence
- Apply derivative rules: power, constant, sum, difference, and constant multiple
- Find the area of the region bounded by two polar curves
- Find the area between curves that intersect at more than two points
- Integrate using substitution
- Find antiderivatives and indefinite integrals: basic rules and notation
- Approximate values of a function using local linearity and linearization
- Interpret the meaning of the derivative in context
- Differentiate inverse functions
- Apply the Chain Rule
- Connect position, velocity, and acceleration of functions using integrals
- Interpret the behavior of accumulation functions involving area
- Use the first and second derivative test to determine relative (local) extrema
- Determine concavity of functions over their domains
- Use the candidates test to determine absolute (global) extrema
- Determine intervals on which a function is increasing or decreasing
- Straight-line motion: connect position, velocity, and acceleration
- Use implicit differentiation
- Find the derivatives of tan, cot, sec, and/or csc functions
- Work with the Intermediate Value Theorem
- Find radius and interval of convergence of power series
- Sketch slope fields
- Model situations with differential equations
- [BC only] Solve the arc length of a smooth, planar curve and distance traveled
- Solve volume with washer method: revolve around other axes
- Solve volume with disc method: revolving around other axes
- Solve volumes with cross sections: squares and rectangles
- Find the area between curves expressed as functions of y
- [BC only] Evaluate improper integrals
- Represent functions as power series
- Find Taylor or Maclaurin series for a function
- Apply Lagrange error bound
- Find Taylor polynomial approximations of functions
- Apply alternating series error bound
- Determine absolute or conditional convergence
- Apply the Ratio Test for convergence
- Apply the Alternating Series Test for convergence
- Apply the Harmonic Series and P-Series
- Apply the Integral Test for convergence
- Apply the Nth Term Test for divergence
- Work with geometric series
- Define convergent and divergent infinite series
- Find the area of a polar region or area bounded by a single polar curve
- Define polar coordinates and differentiate in polar form
- Solve motion problems using parametric and vector-valued functions
- Integrate vector-valued functions
- Define and differentiate vector-valued functions
- Find arc lengths of curves given by parametric equations
- Find second derivatives of parametric equations
- Define and differentiate parametric equations
- [BC only] Solve logistic models with differential equations
- Solve exponential models with differential equations
- Find particular solutions using initial conditions and separation of variables
- Find general solutions using separation of variables
- [BC only] Approximate solutions using Euler's method
- Reason using slope fields
- Verify solutions for differential equations
- Solve volume with washer method: revolve around the x- or y-axis
- Solve volume with disc method: revolving around the x- or y-axis
- Solve volumes with cross sections: triangles and semicircles
- Find the area between curves expressed as functions of x
- Use accumulation functions and definite integrals in applied contexts
- Find the average value of a function on an interval
- Select techniques for antidifferentiation
- [BC only] Integrate using integration by parts
- Integrate functions using long division and complete the square
- Find antiderivatives and definite integrals: basic rules and notation - Part 2
- Find antiderivatives and definite integrals: basic rules and notation - Part 1
- Apply properties of definite integrals
- Solve the fundamental theorem of calculus and accumulation functions
- Solve Riemann sums, summation notation, and definite integral notation
- Approximate areas with Riemann sums
- Sketch graphs of functions and their derivatives
- Solve problems relating to Extreme Value Theorem, global vs. local extrema, and critical points
- Calculate higher-order derivatives
- Apply the Product Rule and Quotient Rule
- Solve derivatives of cos x, sin x, e^x, and ln x
- Connect differentiability and continuity: determine when derivatives do and do not exist
- Define the derivative of a function and use derivative notations
- Define average and instantaneous rates of change at a point
- Connect limits at infinity and horizontal asymptotes
- Connect infinite limits and vertical asymptotes
- Confirm continuity over an interval
- Define continuity at a point
- Connect multiple representations of limits
- Determine limits by the Squeeze Theorem
- Determine limits by algebraic manipulation
- Determine limits by direct substitution
- Determine limits by limit properties
- Estimate limit values from graphs and tables
- General Chemistry 2
- Solve problems relating to solubility and Ksp
- Solve problems relating to the equilibrium constant expression
- Relate equilibrium concentrations of reactants and products
- Solve problems relating to the effect of temperature on reaction rates
- Solve problems relating to the rate law
- Solve problems involving equilibrium calculations
- Solve problems involving Ecell as a function of concentrations
- Solve problems involving strong acids, strong bases, weak acids, and weak bases
- Solve problems involving acids, bases, and conjugate acid-base pairs
- Solve problems relating to standard Gibbs energy and K
- Solve problems involving redox reactions
- Use Le Chatelier's Principle, Q, and K
- Use molecular structure to predict acid-base behavior and demonstrate the common-ion effect
- Solve problems involving titrations
- Solve problems relating to entropy
- Measure reaction rates
- Solve problems relating to reaction mechanisms
- Solve problems relating to precipitation
- Solve problems involving electrolysis
- Solve problems relating to standard Gibbs energy
- Solve problems relating to reactions of various orders
- Solve problems involving buffer solutions
- Write equilibrium constant expressions for reactions involving pure solids or liquids
- Solve problems relating to polyprotic acids and ions as acids and bases
- Solve problems relating to zero-, first-, and second-order reactions
- AP Chemistry
- Solve problems involving galvanic (voltaic) and electrolytic cells
- Solve problems involving solubility equilibria
- Solve problems involving buffers and titrations
- Solve problems involving second-order reactions
- Solve problems involving nomenclature (molecular compounds)
- Solve problems involving ionic and net ionic equations
- Solve problems involving single- and double-replacement reactions
- Solve problems involving entropy and Gibbs free energy
- Solve problems involving spectroscopy and the electromagnetic spectrum
- Solve problems involving acid-base equilibria
- Solve problems involving atomic structure and electron configuration
- Solve problems involving gases
- Solve problems relating to Lewis structures and molecular geometry
- Solve problems involving compounds and ions
- Solve problems involving types of bonds
- Solve problems involving internal energy, heat, and work
- Solve problems involving nomenclature (ionic compounds)
- Solve problems involving zero- and first-order reactions
- Solve problems involving intermolecular forces
- Solve problems involving cell potentials under nonstandard conditions
- Solve problems involving the factors that affect chemical equilibrium
- Solve problems involving acids, bases, and pH
- Solve problems involving standard cell potentials
- Solve problems involving Arrhenius equation and reaction mechanisms
- Solve problems involving atoms
- Solve problems involving the equilibrium constant
- Solve problems involving reaction rates and the rate law
- Solve problems involving enthalpy
- Solve problems involving precipitation reactions
- Solve problems electrolytic cells and electrolysis
- Solve problems involving periodic table trends
- Solve problems involving solutions and mixtures
- Solve problems involving redox reactions
- Microeconomics
- Elucidate Application of Demand and Supply
- Understanding Ecotourism
- Identify and Explain Factors affecting Demand and Supply
- Explaining the Democratic System of Government
- Understanding International Externalities
- Understanding the concept of Public Goods
- Understanding economic systems and globalization
- Explaining Imperfect Information
- Understanding Immigration
- Elucidate definitions calculation and graphs of Costs and profit in short run
- Explain the functioning of a Monopoly
- Defining Positive externality
- Finding out Market-Oriented Environmental Tools
- Define and Calculate Different Types of Elasticity
- Explain key concepts in the process of Production in Short Run with key definitions
- Identifying Government encouragement to innovation
- Defining Negative Externality
- Understanding How Businesses Raise Financial Capital
- Explaining the concept of Labor union
- Define and Explain Types of Costs and Profit
- Explain and Analyze Efficiency in Perfect Competitive Markets
- Understanding Insurance and Imperfect Information
- Explaining the Theories of International Trade
- Explain the functioning of a Monopoly, Anti-competitive behaviour and regulations
- Understanding the Labor Markets
- Explain and Analyze Price Controls, Surplus and Efficiency
- Analyze the Factors affecting Consumption Choices
- Explain Changes in Elasticity
- Explain the broad aspects of economic systems
- Explain functioning of Monopolistic Competition and Oligopoly
- Understanding Special Interest Politics
- Define and Explain Perfect Competition - Price, Output and profit
- Evaluating the Government Policies to Reduce Income Inequality
- Define and Analyze the Problems of Scarcity and Choices
- Finding out the Tradeoff between Economic Output and Environment
- Explaining Income Inequality
- Discusssing the Arguments in Support of Restricting Imports
- Understanding Protectionism
- Understanding the reason behind Intra-industry Trade between Similar Economies
- Knowing about Public Economy
- Explaining Households Supply of Financial Capital
- Understanding Poverty
- Decoding Employment Discrimination
- Undertaking Marginal Analysis of pollution reduction
- Analyze Costs in the Long run
- Explain Relationship between Consumer Choices and Utility
- Define and Graph Demand, Supply and Equilibrium
- Analyze Problems of Scarcity and Social Choices
- Macroeconomics
- Exchange rates demand and supply shifts
- Automatic stablizers
- National debt
- International Trade and Capital flows
- Graphs, variables, and economic models
- Equilibrium and Stabilization
- National Income
- Unemployment and related terms
- Aggregate Demand
- Money Supply
- Fiscal policy
- Phillips Curve
- Unemployment: Short run
- Opportunity Cost
- Measures of Inflation
- Solving for Scarcity
- Monetary and Fiscal Policy: Application
- Cost and Benefit of Inflation
- Discretionary fiscal policy
- Unemployment: Long run
- AD-AS Model Equilibrium
- Globalization and Protectionism
- Policies and events affecting an open economy
- Loanable funds
- Exchange Rate polcies
- Foreign Exchange
- Macro economics of open economies
- Velocity and Quantity equation
- Inflation
- Money
- Federal Reserve System
- Interest rates
- Discretionary fiscal policy
- Types of fiscal policy
- Taxation
- Government spending
- Indexing
- Causes of inflation
- Understanding Inflation
- Computation of unemployment rate
- Types of unemployment
- Aggregate Supply
- IS-LM Model
- Basic Demand-Supply
- GDP
- Consumer Price Index
- Mechanics 1
- Understanding the aspects of acceleration due to gravity
- Understanding the concept of linear momentum
- Understanding the concept of angular motion
- Understanding the concept of center of mass and using them to solve problems
- Understanding the concept of energy
- Understanding the concept of collisions
- Understanding the concept of conservation of momentum
- Understanding phenomena based on gravity occurring in everyday life
- Graphing and using energy diagrams
- Applying the law of energy conservation
- Understanding potential energy
- Understanding work done by a variable force
- Understanding how work and kinetic energy are related
- Understanding the concept of power
- Understanding the concept of work
- Understanding the concepts of energy and kinetic energy
- Solving problems based on gravity
- Understanding the basics of motion and acceleration
- Understanding the basics of gravity
- Understanding frictional forces
- Understanding projectile motion
- Solving problems using Newton's Laws of Motion
- Understanding kinematic graphs
- Solving problems based on the equations of motion
- Understanding the concept of rotational motion
- Electricity and Magnetism
- Understanding the force on a moving charge in a magnetic field
- Understanding the combinations of capacitors and the energy stored in a capacitor
- Applying Gauss’s law
- Understanding magnetic properties of matter
- Understanding Series-Parallel circuit
- Understanding the magnetic effect of an electric current
- Understanding magnetic dipole moment
- Understanding Ohm's Law & Current electricity
- Understanding DC Parallel circuit
- Understanding dynamic or motional electromagnetic induction
- Understanding Krichoff's Law
- Understanding capacitors
- Understanding Gauss’s law
- Electric potential and electric potential energy
- Electric field and force on charges
- Electric charge and application of Coulomb’s law
- Solving problems based on the applications of electromagnetic induction
- Understanding energy stored in an inductor
- Solving problems based on inductance
- Solving problems by applying the concept of induced EMF
- Understanding the production of induced EMF
- Understanding Power and Power Factor in an AC Circuit
- Understanding RLC circuit
- Understanding capacitive reactance
- Understanding inductive reactance
- Understanding AC waveform
- Understanding DC Series circuit
Mathematics skill
Apply Rolle's Theorem
With this skill you will learn
- Estimate the value/s of c that satisfies Rolle's Theorem using the graph of the function
- Find all numbers c that satisfies Rolle's Theorem
- Sketch the graph of a function on a given interval and showing that the function does not satisfy Rolle's Theorem
- Show that the equation has at most one root, two roots, or three real roots in a given interval
Apply Rolle's Theorem Questions
Questions that need this skill to solve them
be continuous and differentiable at 
. Find the value of 
using Rolle’s theorem.OC user
in Mathematics·
13 Jul 2020
is defined as below. Over the interval 
there is no point where the derivative equals zero. Why doesn't Rolle's Theorem apply to this situation?
, 
OC user
in Mathematics·
8 Jan 2021
in the interval 
.OC user
in Mathematics·
8 Jan 2021
shown below.
OC user
in Mathematics·
8 Jan 2021
shown below.
OC user
in Mathematics·
8 Jan 2021
in the interval 
. If it can, find the values of 
.OC user
in Mathematics·
13 Jul 2020
in the interval OC user
in Mathematics·
13 Jul 2020
in the interval 
. If it can, find the values of OC user
in Mathematics·
8 Jan 2021
. Find all values in the interval 
such that 
.OC user
in Mathematics·
8 Jan 2021
, what are the things we need to prove that 
has at least one root in the interval 
using Rolle’s Theorem?OC user
in Mathematics·
8 Jan 2021
, what are the things we need to prove that 
using Rolle’s Theorem?OC user
in Mathematics·
8 Jan 2021
, what are the things we need to prove that 
using Rolle’s Theorem?OC user
in Mathematics·
8 Jan 2021
shown below.
OC user
in Mathematics·
8 Jan 2021
shown below.
OC user
in Mathematics·
8 Jan 2021
be continuous and differentiable at 
. Find the value of Learn this skill
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Develop a comprehensive understanding of fundamental mathematics concepts and applications in a classroom of students motivating each other.
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