Course Content
A. Understanding Limits Graphically
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A.01. Identify limits from graphs.
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A.02. Determine one-sided limits from graphs.
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A.03. Determine if a limit exists using graphical analysis.
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A.04. Interpret limits in the context of real-world applications.
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A.05. Sketch graphs exhibiting various limit behaviors.
B. Calculating Limits Algebraically
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B.01. Apply basic limit laws (addition, subtraction, multiplication).
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B.02. Apply the division law for limits.
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B.03. Apply power and root laws for limits.
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B.04. Evaluate limits of polynomials and rational functions.
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B.05. Evaluate limits involving factorization and rationalization techniques.
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B.06. Evaluate limits involving absolute value functions.
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B.07. Evaluate limits involving trigonometric functions.
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B.08. Prove limit properties using the epsilon-delta definition (advanced).
C. Infinite Limits and Vertical Asymptotes
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C.01. Identify infinite limits from graphs.
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C.02. Determine limits at vertical asymptotes of rational functions.
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C.03. Analyze the behavior of functions near vertical asymptotes.
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C.04. Connect infinite limits to the concept of vertical asymptotes.
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C.05. Apply L’Hôpital’s rule to evaluate indeterminate forms (introduced later, referenced here).
D. Limits at Infinity and End Behavior
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D.01. Determine end behavior of functions from graphs.
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D.02. Determine end behavior of polynomial and rational functions.
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D.03. Analyze horizontal asymptotes.
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D.04. Relate end behavior to the degrees of polynomials in rational functions.
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D.05. Sketch graphs based on end behavior analysis.
E. Continuity
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E.01. Identify continuous functions graphically.
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E.02. Determine continuity using graphical analysis.
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E.03. Determine one-sided continuity graphically.
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E.04. Identify and analyze points of discontinuity graphically.
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E.05. Determine continuity on an interval graphically.
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E.06. Determine the continuity of a piecewise function at a point.
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E.07. Construct continuous piecewise functions.
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E.08. Apply and prove the Intermediate Value Theorem.
F. Introduction to Derivatives: Rates of Change
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F.01. Calculate average rate of change.
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F.02. Interpret average rate of change in context.
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F.03. Find instantaneous rates of change using limits.
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F.04. Understand velocity as a rate of change.
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F.05. Find values of derivatives using limits.
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F.06. Find the slope of a tangent line using limits.
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F.07. Find equations of tangent lines using limits.
G. Derivative Rules
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G.01. Apply the sum and difference rules.
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G.02. Apply the product rule.
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G.03. Apply the quotient rule.
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G.04. Apply the power rule.
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G.05. Apply the chain rule.
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G.06. Apply the inverse function rule.
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G.07. Derive the power rule from the definition of the derivative (advanced).
H. Calculating Derivatives of Various Functions
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H.01. Find derivatives of polynomials.
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H.02. Find derivatives of rational functions.
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H.03. Find derivatives of sine and cosine functions.
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H.04. Find derivatives of other trigonometric functions.
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H.05. Find derivatives of exponential functions.
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H.06. Find derivatives of logarithmic functions.
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H.07. Find derivatives of inverse trigonometric functions.
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H.08. Find derivatives of radical functions.
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H.09. Find derivatives using the product rule.
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H.10. Find derivatives using the quotient rule.
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H.11. Find derivatives of trigonometric functions using product and quotient rules.
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H.12. Find derivatives using the chain rule.
I. Advanced Derivative Techniques
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I.01. Find derivatives using implicit differentiation.
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I.02. Find tangent lines using implicit differentiation.
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I.03. Find derivatives using logarithmic differentiation.
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I.04. Apply derivative rules to solve more complex problems.
J. Higher-Order Derivatives
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J.01. Find higher-order derivatives of polynomials.
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J.02. Find higher-order derivatives of rational and radical functions.
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J.03. Find second derivatives of trigonometric, exponential, and logarithmic functions.
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J.04. Find higher-order derivatives of exponential and trigonometric functions.
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J.05. Find higher-order derivatives of logarithmic and power functions.
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J.06. Interpret the physical meaning of higher-order derivatives.
K. Applications of Derivatives: Rates of Change
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K.01. Relate position, velocity, speed, and acceleration using derivatives.
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K.02. Solve related rates problems.
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K.03. Analyze motion problems using derivatives.
L. L’Hôpital’s Rule
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L.01. Apply L’Hôpital’s rule to indeterminate forms involving quotients.
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L.02. Apply L’Hôpital’s rule to other indeterminate forms (advanced).
M. Analyzing Functions Using the First Derivative
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M.01. Apply the Mean Value Theorem.
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M.02. Find absolute extrema on a closed interval.
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M.03. Identify the graph of the derivative from the graph of a function.
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M.04. Analyze monotonicity and critical points using the first derivative.
N. Analyzing Functions Using the Second Derivative
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N.01. Identify the graph of the second derivative from the graph of a function.
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N.02. Determine concavity and inflection points using the second derivative.
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N.03. Relate the first and second derivatives to graph sketching.
O. Optimization Problems
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O.01. Solve optimization problems.
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O.02. Formulate and solve real-world optimization problems.
P. Linear Approximation
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P.01. Perform linear approximation.
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P.02. Analyze the error in linear approximation.
Q. Introduction to Integration: Approximating Area
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Q.05. Evaluate definite integrals using graphs.
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Q.06. Apply Riemann sums.
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Q.01. Approximate the area under a curve using left and right endpoints.
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Q.02. Approximate the area under a curve using midpoints.
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Q.03. Approximate the area under a curve using trapezoids.
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Q.04. Understand definite integrals and net area.
R. Fundamental Theorem of Calculus
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R.01. Understand and apply the Fundamental Theorem of Calculus, Part 1.
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R.02. Understand and apply the Fundamental Theorem of Calculus, Part 2.
S. Antiderivatives and Indefinite Integrals
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S.01. Find antiderivatives.
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S.02. Find indefinite integrals using the power rule.
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S.03. Find indefinite integrals involving exponential and logarithmic functions.
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S.04. Find indefinite integrals involving trigonometric functions.
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S.05. Find indefinite integrals: mixed review.
T. Definite Integrals
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T.01. Evaluate definite integrals using the power rule.
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T.02. Evaluate definite integrals involving exponential and logarithmic functions.
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T.03. Evaluate definite integrals involving trigonometric functions.
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T.04. Evaluate definite integrals: mixed review.
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T.05. Apply properties of definite integrals.
U. Applications of Integration
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U.01. Calculate the average value of a function.
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U.02. Solve problems involving area and volume calculations.
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U.03. Apply integration to solve real-world problems.
