Calculus I Key Concepts: Practice Problems & Videos

The complete list of videos for Calculus I and II can be found on our Video Calculus page.

The following help materials contain practice problems, worked-out solutions, and video lessons for the topics below. The videos referenced in the help materials are from the Video Calculus series, created by Selwyn Hollis. A similar but larger set of help materials for Calculus I and II can be found at our AP Calculus Help by Topic page.

I. Limits and Continuity

1. Calculating limits intuitively.
2. The definition of a limit.
3. The arithmetic of limits; limits of sums, differences, products and quotients.
4. Trigonometric limits.
5. Continuity, including the Intermediate and Extreme Value Theorems.

II. Differentiation

1. Definition of the derivative; calculating derivatives using the definition; interpreting the derivative as the slope of the tangent line.
2. Differentiation formulas; the power, product, reciprocal, and quotient rules.
3. The chain rule.
4. Differentiating trigonometric functions.
5. Higher Order Derivatives.
6. Implicit differentiation.
7. Rates of change per unit time; related rates.
8. Velocity and Acceleration.
9. Differentials and Newton's method.

III. Applications of the Derivative

1. Rolle's Theorem and the Mean Value Theorem.
2. Critical numbers and the first derivative test.
3. Concavity and the second derivative test.
4. Curve sketching.
5. Extreme values on intervals.
6. Optimization problems.

IV. Integration

1. The definite integral; Riemann sums, area, and properties of the definite integral.
2. The fundamental theorem of calculus.
3. Anti-differentiation and indefinite integrals.
4. Integration by substitution.
5. Mean Value Theorem for integrals; average value.

V. Applications of the Definite Integral

1. Area; area under a curve, area between two curves.
2. Volume; volumes by slicing.
3. Volumes of solids of revolutions; disks and washers, cylindrical shells.
4. Center of mass of a rod and centroid of a planar region.