Overview
This is a summary of the topics covered in Khan Academy's Calculus 1 course, including limits, derivatives, integrals, differential equations, and their applications.
Limits and Continuity
- Understand the concept and definition of a limit and one-sided limits.
- Estimate limits from graphs and tables.
- Use direct substitution, factoring, conjugates, and trigonometric identities to find limits.
- Recognize and classify discontinuities; understand continuity at a point and over intervals.
- Apply the Squeeze Theorem and Intermediate Value Theorem.
- Analyze infinite limits and limits at infinity.
Derivatives: Definition and Basic Rules
- Relate secant lines to average rate of change, and tangent lines to instantaneous rate of change.
- Define the derivative as a limit and estimate derivatives graphically/numerically.
- Apply the power rule and derivative rules for sums, differences, and constants.
- Differentiate polynomials, trigonometric, exponential, and logarithmic functions.
- Use product and quotient rules.
- Understand differentiability at a point.
Derivatives: Chain Rule and Advanced Topics
- Identify and differentiate composite functions using the chain rule.
- Apply implicit differentiation and find derivatives of inverse (including inverse trig) functions.
- Differentiate using multiple rules and strategies.
- Compute second derivatives and use logarithmic differentiation.
Applications of Derivatives
- Interpret the meaning of the derivative in real-world contexts.
- Solve motion problems; analyze rates of change in various applications.
- Tackle related rates problems and apply implicit differentiation.
- Use local linearity for approximation and apply L'HΓ΄pital's rule to evaluate limits.
Analyzing Functions
- Use the Mean Value and Extreme Value Theorems.
- Identify and classify critical points, intervals of increase/decrease, and extrema.
- Analyze concavity and inflection points using first and second derivatives.
- Solve optimization problems.
Integrals
- Understand accumulation of change and define the definite integral as a limit of Riemann sums.
- Use left/right, midpoint, and trapezoidal approximations for area under curves.
- Apply the Fundamental Theorem of Calculus.
- Calculate antiderivatives and use the reverse power rule.
- Integrate polynomials, exponential, logarithmic, and trigonometric functions.
- Integrate using substitution, long division, and trigonometric identities.
Differential Equations
- Write and solve simple differential equations; verify solutions.
- Sketch and interpret slope fields.
- Solve separable differential equations and analyze exponential models.
- Find particular solutions with initial conditions.
Applications of Integrals
- Compute the average value of a function.
- Solve problems involving motion using integrals.
- Calculate the area between curves and volumes of solids using cross-sections, disc, and washer methods.
Key Terms & Definitions
- Limit β the value a function approaches as the input approaches a certain point.
- Continuity β a function is continuous if there are no breaks, jumps, or holes at a point or interval.
- Derivative β the instantaneous rate of change of a function with respect to its variable.
- Integral β a mathematical tool to accumulate quantities, often representing area under a curve.
- Riemann sum β an approximation of an integral using a sum of rectangle areas.
- Chain Rule β a formula for computing the derivative of a composite function.
- Implicit Differentiation β finding derivatives when functions are defined implicitly.
- Differential Equation β an equation involving derivatives of an unknown function.
Action Items / Next Steps
- Complete practice problems and quizzes in each unit.
- Review key theorems and formulas before the unit tests.
- Attempt the course challenge to test your overall mastery.