Overview
These notes summarize the key concepts and problem-solving strategies taught in Khan Academy’s AP Calculus BC course. Each section includes essential ideas and sample flashcard-style questions to reinforce understanding.
Limits and Continuity
- Limits describe how a function behaves as the input approaches a specific value.
- Flashcard: How do you estimate a limit from a graph?
Answer: Observe the y-value the function approaches as x nears the target value from both sides.
- Continuity at a point requires the function to be defined, the limit to exist, and both to be equal.
- Flashcard: What three conditions must be met for a function to be continuous at a point?
Answer: The function is defined at the point, the limit exists, and the function’s value equals the limit.
- Indeterminate forms (like 0/0) can be resolved using algebraic manipulation, factoring, conjugates, or trigonometric identities.
- Flashcard: How do you resolve a 0/0 indeterminate form?
Answer: Simplify the expression using factoring, conjugates, or trig identities, then re-evaluate the limit.
- The squeeze theorem helps find limits when a function is trapped between two others with known limits.
- Flashcard: When do you use the squeeze theorem?
Answer: When a function is bounded above and below by two functions that share the same limit at a point.
- Discontinuities can be removable, jump, or infinite.
- Flashcard: How do you identify a removable discontinuity?
Answer: If redefining the function at a point would make it continuous.
- The Intermediate Value Theorem guarantees that a continuous function takes on every value between two points.
- Flashcard: What does the Intermediate Value Theorem state?
Answer: If a function is continuous on [a, b], it takes every value between f(a) and f(b).
Differentiation: Definition and Rules
- The derivative measures the instantaneous rate of change and is defined as a limit.
- Flashcard: How is the derivative at a point defined?
Answer: As the limit of the average rate of change as the interval approaches zero.
- The power rule, product rule, and quotient rule are used to differentiate various functions.
- Flashcard: What is the power rule for derivatives?
Answer: d/dx[xⁿ] = n·xⁿ⁻¹.
- Derivatives of trigonometric, exponential, and logarithmic functions are fundamental.
- Flashcard: What is the derivative of sin(x)?
Answer: cos(x).
- Differentiability implies continuity, but a function can be continuous without being differentiable.
- Flashcard: Does continuity guarantee differentiability?
Answer: No; a function can be continuous but not differentiable at a point.
Differentiation: Composite, Implicit, and Inverse Functions
- The chain rule is used to differentiate composite functions.
- Flashcard: How do you differentiate f(g(x))?
Answer: f'(g(x)) · g'(x).
- Implicit differentiation is used when y is not isolated.
- Flashcard: How do you find dy/dx for an equation involving x and y?
Answer: Differentiate both sides with respect to x, treating y as a function of x, and solve for dy/dx.
- Derivatives of inverse and inverse trigonometric functions have specific formulas.
- Flashcard: What is the derivative of arcsin(x)?
Answer: 1/√(1 - x²).
- Second derivatives provide information about concavity and inflection points.
- Flashcard: What does it mean if f''(x) > 0?
Answer: The function is concave up at x.
Contextual Applications of Differentiation
- Derivatives model rates of change in real-world contexts like motion, biology, and economics.
- Flashcard: How do you interpret the derivative in a motion problem?
Answer: It represents velocity (rate of change of position).
- Related rates problems involve finding how two or more quantities change with respect to time.
- Flashcard: What is the first step in a related rates problem?
Answer: Write an equation relating the variables, then differentiate both sides with respect to time.
- Linear approximation uses the tangent line to estimate function values near a point.
- Flashcard: How do you approximate f(x) near x = a?
Answer: Use f(a) + f'(a)(x - a).
- L'Hôpital's Rule is used to evaluate limits of indeterminate forms like 0/0 or ∞/∞.
- Flashcard: When can you apply L'Hôpital's Rule?
Answer: When the limit yields 0/0 or ∞/∞ and the derivatives of numerator and denominator exist.
Applying Derivatives to Analyze Functions
- The Mean Value Theorem connects average and instantaneous rates of change.
- Flashcard: What does the Mean Value Theorem guarantee?
Answer: There is at least one point where the instantaneous rate equals the average rate over [a, b].
- Critical points occur where the derivative is zero or undefined; these help find extrema.
- Flashcard: How do you find local maxima and minima?
Answer: Find where f'(x) = 0 or undefined, then use the first or second derivative test.
- The second derivative reveals concavity and inflection points.
- Flashcard: What is an inflection point?
Answer: A point where the concavity of the function changes.
- Optimization uses derivatives to solve maximum and minimum problems in real-world scenarios.
- Flashcard: What is the general process for optimization?
Answer: Write an equation for the quantity to optimize, find critical points, and test endpoints if needed.
Integration and Accumulation of Change
- Definite integrals calculate net change or area under a curve.
- Flashcard: What does ∫ₐᵇ f(x) dx represent?
Answer: The net area under f(x) from x = a to x = b.
- Riemann sums approximate integrals; taking the limit gives the exact value.
- Flashcard: How do you set up a Riemann sum?
Answer: Divide the interval into subintervals, sum f(x)·Δx for each.
- The Fundamental Theorem of Calculus links differentiation and integration.
- Flashcard: What does the Fundamental Theorem of Calculus state?
Answer: The derivative of the integral of f(x) is f(x), and the definite integral can be found using antiderivatives.
- Antiderivatives and the reverse power rule are used for indefinite integrals.
- Flashcard: What is the reverse power rule for integration?
Answer: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1.
- Integration techniques include u-substitution, integration by parts, partial fractions, and handling improper integrals.
- Flashcard: When do you use u-substitution?
Answer: When the integrand contains a function and its derivative.
Differential Equations
- Differential equations relate a function to its derivatives.
- Flashcard: What is a differential equation?
Answer: An equation involving a function and its derivatives.
- Slope fields provide a graphical way to visualize solutions.
- Flashcard: What does a slope field show?
Answer: The direction of the solution curve at each point.
- Euler’s Method numerically approximates solutions to differential equations.
- Flashcard: How does Euler’s Method work?
Answer: Use the slope at a point to estimate the next value, repeating over intervals.
- Separation of variables and initial conditions are used to find particular solutions.
- Flashcard: How do you solve a separable differential equation?
Answer: Rearrange to isolate variables, integrate both sides, and solve for the function.
- Exponential and logistic models describe growth and decay in real-world systems.
- Flashcard: What is the general solution for dy/dx = ky?
Answer: y = Ce^(kt).
Applications of Integration
- Integrals are used to find area under curves, between curves, and average values.
- Flashcard: How do you find the area between two curves?
Answer: Integrate the difference between the top and bottom functions over the interval.
- Volumes of solids are found using cross-sections, disk, and washer methods.
- Flashcard: What is the disk method?
Answer: Integrate π[r(x)]² dx to find the volume of a solid of revolution.
- Arc length and motion can be determined by integrating appropriate formulas.
- Flashcard: How do you find the arc length of y = f(x) from a to b?
Answer: ∫ₐᵇ √[1 + (f'(x))²] dx.
Parametric, Polar, and Vector-Valued Functions
- Parametric equations express curves using a parameter (usually t); derivatives and integrals are found with respect to t.
- Flashcard: How do you find dy/dx for parametric equations x(t), y(t)?
Answer: dy/dx = (dy/dt) / (dx/dt).
- Polar coordinates describe curves using radius and angle; derivatives and areas are calculated with polar formulas.
- Flashcard: How do you find the area inside a polar curve r(θ)?
Answer: (1/2) ∫[r(θ)]² dθ over the interval.
- Vector-valued functions model motion in the plane or space; derivatives give velocity and acceleration.
- Flashcard: What does the derivative of a vector-valued function represent?
Answer: The velocity vector.
Infinite Sequences and Series
- Sequences may converge (approach a value) or diverge.
- Flashcard: How do you test if a sequence converges?
Answer: Find the limit as n approaches infinity.
- Infinite series are sums of sequences; convergence is tested using various methods (geometric, nth-term, integral, comparison, alternating, ratio, p-series).
- Flashcard: What is the nth-term test for divergence?
Answer: If the nth term does not approach zero, the series diverges.
- Taylor and Maclaurin series approximate functions using polynomials.
- Flashcard: What is a Taylor series?
Answer: An infinite sum representing a function as a series of its derivatives at a point.
- Power series can be integrated and differentiated term by term within their interval of convergence.
- Flashcard: How do you find the interval of convergence for a power series?
Answer: Use the ratio test to determine where the series converges.
Key Terms & Definitions
- Limit: The value a function approaches as the input nears a point.
- Derivative: The instantaneous rate of change of a function.
- Integral: The accumulation of values, often representing area under a curve.
- Differential Equation: An equation involving derivatives of a function.
- Riemann Sum: An approximation of an integral using finite sums.
- Chain Rule: A rule for differentiating composite functions.
- u-Substitution: A technique for integrating complex expressions.
- Slope Field: A graphical representation of a differential equation’s solutions.
- Convergence: When a sequence or series approaches a finite value.
- Power Series: An infinite sum of terms with varying powers of x.
Action Items / Next Steps
- Practice solving problems using the strategies above, focusing on both conceptual understanding and calculation.
- Use flashcards to test yourself on definitions, theorems, and problem-solving steps.
- Complete quizzes and unit tests to check your mastery of each topic.
- Work through solved AP Calculus BC exam problems for exam preparation.
- Review and master foundational skills before attempting the course challenge.