Course Content
A. Functions
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A.01. Find function values from graphs.
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A.02. Evaluate functions at given inputs.
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A.03. Add, subtract, multiply, and divide functions.
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A.04. Compose functions.
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A.05. Identify inverse functions graphically.
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A.06. Find inverse function values from tables and graphs.
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A.07. Determine inverse functions algebraically.
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A.08. Analyze the domain and range of functions and their inverses.
B. Families of Functions
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B.01. Identify and apply basic function transformations (translations, reflections, dilations).
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B.02. Describe transformations using function notation.
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B.03. Combine multiple transformations on a single function.
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B.04. Graph transformed functions given the parent function and transformations.
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B.05. Analyze how transformations affect key features of a function (e.g., intercepts, asymptotes).
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B.06. Model real-world scenarios using function transformations.
C. Trigonometric Functions
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C.01. Convert between degrees and radians.
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C.02. Identify quadrants and determine the signs of trigonometric functions in each quadrant.
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C.03. Understand and apply reference angles.
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C.04. Find trigonometric ratios using right triangles and the unit circle.
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C.05. Evaluate trigonometric functions using reference angles.
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C.06. Determine inverse trigonometric functions and their properties.
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C.07. Graph sine and cosine functions, identifying key features (amplitude, period, phase shift, vertical shift).
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C.08. Write equations of sine and cosine functions given their graphs or properties.
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C.09. Solve trigonometric equations.
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C.10. Apply trigonometric functions to model periodic phenomena.
D. Exponential and Logarithmic Functions
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D.01. Define and identify exponential and logarithmic functions.
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D.02. Convert between exponential and logarithmic forms.
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D.03. Evaluate logarithms using different bases.
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D.04. Use the properties of logarithms (product, quotient, power rules).
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D.05. Solve exponential and logarithmic equations.
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D.06. Understand and apply the change of base formula.
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D.07. Analyze exponential growth and decay models.
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D.08. Model real-world phenomena using exponential and logarithmic functions.
E. Introduction to Limits
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E.01. Estimate limits graphically.
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E.02. Determine one-sided limits graphically.
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E.03. Determine if a limit exists using graphical analysis.
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E.04. Evaluate limits algebraically using limit laws.
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E.05. Explore limits involving infinity.
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E.06. Understand the concept of continuity.
F. Calculate Limits
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F.01. Apply limit laws (addition, subtraction, multiplication, division, power, root) to evaluate limits.
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F.02. Evaluate limits of polynomial and rational functions.
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F.03. Evaluate limits involving factorization and rationalization techniques.
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F.04. Evaluate limits involving absolute value functions.
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F.05. Evaluate limits involving trigonometric functions.
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F.06. Evaluate limits using L’Hopital’s rule (introduced later).
G. Limits Involving Infinity
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G.01. Find limits at vertical asymptotes graphically.
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G.02. Determine end behavior graphically.
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G.03. Determine end behavior of polynomial and rational functions analytically.
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G.04. Identify horizontal asymptotes.
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G.05. Analyze the behavior of functions as x approaches positive and negative infinity.
H. Rational Functions
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H.01. Determine vertical asymptotes of rational functions.
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H.02. Determine horizontal asymptotes of rational functions.
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H.03. Analyze the behavior of rational functions near asymptotes.
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H.04. Sketch graphs of rational functions.
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H.05. Solve rational equations.
I. Continuity
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I.01. Identify continuous functions graphically.
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I.02. Determine continuity using graphical analysis.
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I.03. Determine one-sided continuity.
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I.04. Identify and analyze points of discontinuity graphically.
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I.05. Determine continuity on an interval.
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I.06. Determine the continuity of piecewise functions.
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I.07. Construct continuous piecewise functions.
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I.08. Apply the Intermediate Value Theorem.
J. Introduction to Derivatives
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J.01. Calculate average rate of change.
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J.02. Understand the concept of instantaneous rate of change.
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J.03. Interpret the derivative as a slope of a tangent line.
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J.04. Calculate derivatives using the limit definition.
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J.05. Relate derivatives to velocity and acceleration.
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J.06. Find equations of tangent lines using derivatives.
K. Derivative Rules
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K.01. Apply the power rule for differentiation.
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K.02. Apply the sum and difference rules.
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K.03. Apply the product rule.
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K.04. Apply the quotient rule.
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K.05. Apply the chain rule.
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K.06. Apply the inverse function rule.
L. Calculate Derivatives
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L.01. Find derivatives of polynomial functions.
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L.02. Find derivatives of rational functions.
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L.03. Find derivatives of trigonometric functions.
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L.04. Find derivatives of exponential functions.
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L.05. Find derivatives of logarithmic functions.
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L.06. Find derivatives of inverse trigonometric functions.
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L.07. Find derivatives of radical functions.
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L.08. Apply multiple derivative rules in combination.
M. Derivative Strategies
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M.01. Apply implicit differentiation.
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M.02. Find tangent lines using implicit differentiation.
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M.03. Apply logarithmic differentiation.
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M.04. Solve related rate problems.
N. Calculate Higher Derivatives
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N.01. Find higher-order derivatives of polynomials.
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N.02. Find higher-order derivatives of rational and radical functions.
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N.03. Find higher-order derivatives of trigonometric, exponential, and logarithmic functions.
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N.04. Identify patterns in higher-order derivatives.
O. Rates of Change
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O.01. Relate position, velocity, speed, and acceleration using derivatives.
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O.02. Solve related rates problems.
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O.03. Apply derivatives to model real-world problems involving rates of change.
P. L’Hôpital’s Rule
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P.01. Apply L’Hôpital’s rule to evaluate indeterminate forms (0/0, ∞/∞).
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P.02. Understand the limitations of L’Hôpital’s rule.
Q. Analyze Functions Using the First Derivative
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Q.01. Apply the Mean Value Theorem.
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Q.02. Find absolute extrema on a closed interval.
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Q.03. Sketch graphs of functions using first derivative information (increasing/decreasing intervals, local extrema).
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Q.04. Analyze the relationship between a function and its first derivative graphically.
R. Analyze Functions Using the Second Derivative
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R.01. Determine concavity using the second derivative.
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R.02. Identify inflection points.
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R.03. Sketch graphs of functions using second derivative information (concavity, inflection points).
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R.04. Analyze the relationship between a function and its second derivative graphically.
S. Optimization
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S.01. Solve optimization problems involving finding maximum and minimum values.
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S.02. Apply optimization techniques to real-world problems.
T. Linear Approximation
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T.01. Use linear approximation to estimate function values.
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T.02. Understand the limitations of linear approximation.
U. Introduction to Integration
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U.01. Approximate area under a curve using left, right, midpoint, and trapezoidal rules.
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U.02. Understand the concept of definite integrals and net area.
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U.03. Evaluate definite integrals using graphs.
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U.04. Apply properties of definite integrals.
V. Fundamental Theorem of Calculus
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V.01. Understand and apply the Fundamental Theorem of Calculus, Part 1.
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V.02. Understand and apply the Fundamental Theorem of Calculus, Part 2.
W. Antiderivatives and Indefinite Integrals
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W.01. Find antiderivatives.
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W.02. Find indefinite integrals using the power rule.
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W.03. Find indefinite integrals involving exponential and logarithmic functions.
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W.04. Find indefinite integrals involving trigonometric functions.
X. Definite Integrals
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X.01. Evaluate definite integrals using the power rule.
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X.02. Evaluate definite integrals involving exponential and logarithmic functions.
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X.03. Evaluate definite integrals involving trigonometric functions.
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X.04. Apply the properties of definite integrals to simplify calculations.
Y. Applications of Integration
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Y.01. Calculate the average value of a function over an interval.
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Y.02. Solve problems involving area and accumulation.
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Y.03. Apply integration to solve real-world problems.
