MATH 1261 Review Sheet - Georgia College & State University



MATH 1261 Calculus I

Final Exam Review Sheet

Definitions that you should know

• function

• ε−δ definition of limit

continuity of a function f at a point

derivative (limit definition)

antiderivative

definite integral

Riemann Sum

average value of a function f on [a,b]

Theorems that you should be able to state

Squeeze Theorem

IVT (Intermediate Value Theorem)

EVT (Extreme Value Theorem)

MVT (Mean Value Theorem (for derivatives))

FTC (Fundamental Theorem of Calculus (Parts I/II))

Mean Value Theorem for Integrals

theorem relating derivatives and continuity

theorem relating antiderivatives of the same function

theorem relating sign of derivative to increasing/decreasing/constant

Other Topics

Limit Laws

• ε−δ proofs of limit (primarily linear functions)

Infinite Limits

Limits at infinity

Interpretation of the Derivative (slope of the tangent line, (instantaneous) rate of change)

Derivative Rules (Sum, Difference, Constant, Constant Multiple, Product, Quotient, Chain)

Derivatives of power functions, trigonometric functions

Higher Order Derivatives (First derivative, Second derivative, Third derivative, nth derivative)

Implicit Differentiaton

Graphing using derivatives

extrema (local (relative) and absolute (global))

critical numbers (first order (CNs), second order (CN2s))

inflection points

increasing/decreasing/constant (first derivative)

concave up/ concave down (second derivative)

intercepts (horizontal/vertical)

asymptotes (horizontal (HA), vertical (VA), oblique (OA))

Applications of the derivative

Differentials and Linear Approximation

Related Rates

Optimization Problems

Newton's Method

Antiderivatives, solving simple differential equations (with/without initial conditions)

Interpretations of the definite integral (in terms of area, net change in a function)

Properties of Integrals

Integration by substitution ("reversing" the chain rule for derivatives)

Proving Part II of the FTC (Fundamental Theorem of Calculus)

Applications of definite integral

Area between curves (VCS, HCS)

Volumes of Solids of Revolution (disks, washers, shells)

Average Value

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