AP Calculus AB · Calculus I
Calculus Volume 1
Limits, derivatives and integrals: the first semester of calculus, matched to AP Calculus AB.
6 chapters · 45 lessons · each one taught on the whiteboard, then practiced
Functions and Graphs
- Use functional notation to evaluate a function.
- Determine the domain and range of a function.
- Draw the graph of a function.
- Find the zeros of a function.
- Recognize a function from a table of values.
- Make new functions from two or more given functions.
- Describe the symmetry properties of a function.
Read 1.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
33 exercises from the book · 17 key terms
- Calculate the slope of a linear function and interpret its meaning.
- Recognize the degree of a polynomial.
- Find the roots of a quadratic polynomial.
- Describe the graphs of basic odd and even polynomial functions.
- Identify a rational function.
- Describe the graphs of power and root functions.
- Explain the difference between algebraic and transcendental functions.
- Graph a piecewise-defined function.
- Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position.
Read 1.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
28 exercises from the book · 17 key terms
- Convert angle measures between degrees and radians.
- Recognize the triangular and circular definitions of the basic trigonometric functions.
- Write the basic trigonometric identities.
- Identify the graphs and periods of the trigonometric functions.
- Describe the shift of a sine or cosine graph from the equation of the function.
Read 1.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
33 exercises from the book · 4 key terms
- Determine the conditions for when a function has an inverse.
- Use the horizontal line test to recognize when a function is one-to-one.
- Find the inverse of a given function.
- Draw the graph of an inverse function.
- Evaluate inverse trigonometric functions.
Read 1.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
27 exercises from the book · 5 key terms
- Identify the form of an exponential function.
- Explain the difference between the graphs of $x^b$ and $b^x.$
- Recognize the significance of the number $e.$
- Identify the form of a logarithmic function.
- Explain the relationship between exponential and logarithmic functions.
- Describe how to calculate a logarithm to a different base.
- Identify the hyperbolic functions, their graphs, and basic identities.
Read 1.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
45 exercises from the book · 7 key terms
Limits
- Describe the tangent problem and how it led to the idea of a derivative.
- Explain how the idea of a limit is involved in solving the tangent problem.
- Recognize a tangent to a curve at a point as the limit of secant lines.
- Identify instantaneous velocity as the limit of average velocity over a small time interval.
- Describe the area problem and how it was solved by the integral.
- Explain how the idea of a limit is involved in solving the area problem.
- Recognize how the ideas of limit, derivative, and integral led to the studies of infinite series and multivariable calculus.
Read 2.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
16 exercises from the book · 8 key terms
- Using correct notation, describe the limit of a function.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Use a graph to estimate the limit of a function or to identify when the limit does not exist.
- Define one-sided limits and provide examples.
- Explain the relationship between one-sided and two-sided limits.
- Using correct notation, describe an infinite limit.
- Define a vertical asymptote.
Read 2.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
31 exercises from the book · 4 key terms
- Recognize the basic limit laws.
- Use the limit laws to evaluate the limit of a function.
- Evaluate the limit of a function by factoring.
- Use the limit laws to evaluate the limit of a polynomial or rational function.
- Evaluate the limit of a function by factoring or by using conjugates.
- Evaluate the limit of a function by using the squeeze theorem.
Read 2.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
27 exercises from the book · 9 key terms
- Explain the three conditions for continuity at a point.
- Describe three kinds of discontinuities.
- Define continuity on an interval.
- State the theorem for limits of composite functions.
- Provide an example of the intermediate value theorem.
Read 2.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
25 exercises from the book · 9 key terms
- Describe the epsilon-delta definition of a limit.
- Apply the epsilon-delta definition to find the limit of a function.
- Describe the epsilon-delta definitions of one-sided limits and infinite limits.
- Use the epsilon-delta definition to prove the limit laws.
Read 2.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
17 exercises from the book · 2 key terms
Derivatives
- Recognize the meaning of the tangent to a curve at a point.
- Calculate the slope of a tangent line.
- Identify the derivative as the limit of a difference quotient.
- Calculate the derivative of a given function at a point.
- Describe the velocity as a rate of change.
- Explain the difference between average velocity and instantaneous velocity.
- Estimate the derivative from a table of values.
Read 3.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
27 exercises from the book · 4 key terms
- Define the derivative function of a given function.
- Graph a derivative function from the graph of a given function.
- State the connection between derivatives and continuity.
- Describe three conditions for when a function does not have a derivative.
- Explain the meaning of a higher-order derivative.
Read 3.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book · 5 key terms
- State the constant, constant multiple, and power rules.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- Extend the power rule to functions with negative exponents.
- Combine the differentiation rules to find the derivative of a polynomial or rational function.
Read 3.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
24 exercises from the book · 7 key terms
- Determine a new value of a quantity from the old value and the amount of change.
- Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
- Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
- Predict the future population from the present value and the population growth rate.
- Use derivatives to calculate marginal cost and revenue in a business situation.
Read 3.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
14 exercises from the book · 8 key terms
- Find the derivatives of the sine and cosine function.
- Find the derivatives of the standard trigonometric functions.
- Calculate the higher-order derivatives of the sine and cosine.
Read 3.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
18 exercises from the book
- State the chain rule for the composition of two functions.
- Apply the chain rule together with the power rule.
- Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
- Recognize the chain rule for a composition of three or more functions.
- Describe the proof of the chain rule.
Read 3.6 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
23 exercises from the book · 1 key terms
- Calculate the derivative of an inverse function.
- Recognize the derivatives of the standard inverse trigonometric functions.
Read 3.7 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
26 exercises from the book
- Find the derivative of a complicated function by using implicit differentiation.
- Use implicit differentiation to determine the equation of a tangent line.
Read 3.8 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
17 exercises from the book · 1 key terms
- Find the derivative of exponential functions.
- Find the derivative of logarithmic functions.
- Use logarithmic differentiation to determine the derivative of a function.
Read 3.9 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
18 exercises from the book · 1 key terms
Applications of Derivatives
- Express changing quantities in terms of derivatives.
- Find relationships among the derivatives in a given problem.
- Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.
Read 4.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
24 exercises from the book · 1 key terms
- Describe the linear approximation to a function at a point.
- Write the linearization of a given function.
- Draw a graph that illustrates the use of differentials to approximate the change in a quantity.
- Calculate the relative error and percentage error in using a differential approximation.
Read 4.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
21 exercises from the book · 7 key terms
- Define absolute extrema.
- Define local extrema.
- Explain how to find the critical points of a function over a closed interval.
- Describe how to use critical points to locate absolute extrema over a closed interval.
Read 4.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book · 10 key terms
- Explain the meaning of Rolle’s theorem.
- Describe the significance of the Mean Value Theorem.
- State three important consequences of the Mean Value Theorem.
Read 4.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
26 exercises from the book · 2 key terms
- Explain how the sign of the first derivative affects the shape of a function’s graph.
- State the first derivative test for critical points.
- Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.
- Explain the concavity test for a function over an open interval.
- Explain the relationship between a function and its first and second derivatives.
- State the second derivative test for local extrema.
Read 4.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
32 exercises from the book · 7 key terms
- Calculate the limit of a function as $x$ increases or decreases without bound.
- Recognize a horizontal asymptote on the graph of a function.
- Estimate the end behavior of a function as $x$ increases or decreases without bound.
- Recognize an oblique asymptote on the graph of a function.
- Analyze a function and its derivatives to draw its graph.
Read 4.6 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book · 5 key terms
- Set up and solve optimization problems in several applied fields.
Read 4.7 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
23 exercises from the book · 1 key terms
- Recognize when to apply L’Hôpital’s rule.
- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Describe the relative growth rates of functions.
Read 4.8 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
28 exercises from the book · 2 key terms
- Describe the steps of Newton’s method.
- Explain what an iterative process means.
- Recognize when Newton’s method does not work.
- Apply iterative processes to various situations.
Read 4.9 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book · 2 key terms
- Find the general antiderivative of a given function.
- Explain the terms and notation used for an indefinite integral.
- State the power rule for integrals.
- Use antidifferentiation to solve simple initial-value problems.
Read 4.10 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
31 exercises from the book · 3 key terms
Integration
- Use sigma (summation) notation to calculate sums and powers of integers.
- Use the sum of rectangular areas to approximate the area under a curve.
- Use Riemann sums to approximate area.
Read 5.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
32 exercises from the book · 8 key terms
- State the definition of the definite integral.
- Explain the terms integrand, limits of integration, and variable of integration.
- Explain when a function is integrable.
- Describe the relationship between the definite integral and net area.
- Use geometry and the properties of definite integrals to evaluate them.
- Calculate the average value of a function.
Read 5.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
50 exercises from the book · 8 key terms
- Describe the meaning of the Mean Value Theorem for Integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 1.
- Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 2.
- Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
- Explain the relationship between differentiation and integration.
Read 5.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
35 exercises from the book · 4 key terms
- Apply the basic integration formulas.
- Explain the significance of the net change theorem.
- Use the net change theorem to solve applied problems.
- Apply the integrals of odd and even functions.
Read 5.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
25 exercises from the book · 1 key terms
- Use substitution to evaluate indefinite integrals.
- Use substitution to evaluate definite integrals.
Read 5.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
2 key terms
- Integrate functions involving exponential functions.
- Integrate functions involving logarithmic functions.
Read 5.6 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
38 exercises from the book
- Integrate functions resulting in inverse trigonometric functions
Read 5.7 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
60 exercises from the book
Applications of Integration
- Determine the area of a region between two curves by integrating with respect to the independent variable.
- Find the area of a compound region.
- Determine the area of a region between two curves by integrating with respect to the dependent variable.
Read 6.1 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book
- Determine the volume of a solid by integrating a cross-section (the slicing method).
- Find the volume of a solid of revolution using the disk method.
- Find the volume of a solid of revolution with a cavity using the washer method.
Read 6.2 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
34 exercises from the book · 5 key terms
- Calculate the volume of a solid of revolution by using the method of cylindrical shells.
- Compare the different methods for calculating a volume of revolution.
Read 6.3 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
25 exercises from the book · 1 key terms
- Determine the length of a curve, $y = f (x),$ between two points.
- Determine the length of a curve, $x = g (y),$ between two points.
- Find the surface area of a solid of revolution.
Read 6.4 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
30 exercises from the book · 3 key terms
- Determine the mass of a one-dimensional object from its linear density function.
- Determine the mass of a two-dimensional circular object from its radial density function.
- Calculate the work done by a variable force acting along a line.
- Calculate the work done in pumping a liquid from one height to another.
- Find the hydrostatic force against a submerged vertical plate.
Read 6.5 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
19 exercises from the book · 4 key terms
- Find the center of mass of objects distributed along a line.
- Locate the center of mass of a thin plate.
- Use symmetry to help locate the centroid of a thin plate.
- Apply the theorem of Pappus for volume.
Read 6.6 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
21 exercises from the book · 6 key terms
- Write the definition of the natural logarithm as an integral.
- Recognize the derivative of the natural logarithm.
- Integrate functions involving the natural logarithmic function.
- Define the number $e$ through an integral.
- Recognize the derivative and integral of the exponential function.
- Prove properties of logarithms and exponential functions using integrals.
- Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.
Read 6.7 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
24 exercises from the book
- Use the exponential growth model in applications, including population growth and compound interest.
- Explain the concept of doubling time.
- Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling.
- Explain the concept of half-life.
Read 6.8 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
15 exercises from the book · 4 key terms
- Apply the formulas for derivatives and integrals of the hyperbolic functions.
- Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.
- Describe the common applied conditions of a catenary curve.
Read 6.9 in the textbook →Learn
Whiteboard lesson · about 15 min
PracticePractice
27 exercises from the book · 1 key terms