Key Takeaways
Lesson Description
Key Takeaways
• Limits describe the behavior of functions near a point and form the foundation of calculus.
• Derivatives measure instantaneous rates of change and slopes of tangent lines.
• Differentiation is used to analyze functions, solve optimization problems, and model real-world change.
• Integration measures accumulation and area under curves.
• The Fundamental Theorem of Calculus connects derivatives and integrals.
• Differential equations model changing systems over time.
• Calculus emphasizes understanding relationships between functions, rates of change, and accumulation rather than memorizing formulas alone.
Must Know Derivatives
d/dx(xⁿ)=nxⁿ⁻¹
d/dx(eˣ)=eˣ
d/dx(lnx)=1/x
d/dx(sinx)=cosx
d/dx(cosx)=−sinx
Chain Rule:
(f(g(x)))'=f'(g(x))g'(x
Must Know Integrals
∫xⁿdx=(xⁿ⁺¹)/(n+1)+C
∫eˣdx=eˣ+C
∫(1/x)dx=ln|x|+C
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫abf(x)dx=F(b)-F(a)
Must Know Vocabulary
Limit
Continuity
Derivative
Tangent Line
Chain Rule
Critical Point
Concavity
Inflection Point
Integral
Antiderivative
Fundamental Theorem of Calculus
Differential Equation
Slope Field
Euler's Method
AP Exam Focus
• Interpret graphs of functions, derivatives, and integrals.
• Explain calculus concepts using mathematical reasoning.
• Connect limits, derivatives, and integrals within a single problem.
• Apply calculus to optimization, motion, accumulation, and modeling problems.
• Justify answers using correct notation, graphs, and calculus principles.