Pre-Calculus/Calculus: Calculus Concepts

For a basic understanding of calculus, you should be comfortable with the following concepts:

The cartesian plane and functions

• The real number line
• The cartesian plane and the distance formula
• Lines in the plane; slope
• Circles
• Graphs of equations
• Functions

Limits

• Limits
• Continuity
• Limits & asymptotes
• Curve sketching

Differentiation

• The derivative as the slope of a curve
• Differentiability and continuity
• The derivative as a rate of change
• Higher order derivatives
• The product and quotient rules
• Position - velocity - acceleration functions
• The chain rule and the general power rule
• Implicit differentiation
• Related rates

Applications of differentiation

• Extrema on an interval
• The mean value theorem
• Increasing and decreasing functions
• The first derivative test
• Concavity & the second derivative test
• Limits at infinity (horizontal asymptotes)
• Curve sketching (including extrema & concavity)
• Optimization problems (max/min problems)
• Newton's method
• Differentials

Integration

• The indefinite integral
• U-substitution and the general power rule
• The fundamental theorem of calculus
• Area and the definite integral
• Trapezoid rule

Applications of integration

• Average value of a function
• Area between two curves
• Volumes of revolution ("discs" & "shells")
• Volumes of solids with known cross section

Exponential and logarithmic functions

• Exponential functions
• Differentiation and integration of exponentials
• Inverse functions
• Logarithmic functions
• Natural logarithmic differentiation
• Natural logarithmic integration
• Differential equations
• Growth and decay
• L'Hoptial's rule

Trigonometric and inverse trigonometric functions

• Trigonometric graphs and limits
• Derivatives of trigonometric functions
• Integrals of trigonometric functions
• Inverse trigonometric functions and differentiation
• Inverse trigonometric functions and integration

Integration techniques

• Review of integration formulas
• Integration by parts