AP/Honors Calculus Chapter 4 Applications of Derivatives Chapter 4 Applications of Derivatives.
28937876 AP Calculus BC Study Guide
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Calculus Study Guide Rationalization to find limits: Squeeze Theorem: & DNE (oscillates) . Important Limits: Intermediate Value Theorem: If is continuous on [a,b] and k i s any number between (a) and (b), then there is one number c in [a,b] such that (c)=k. Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b) then there exists a number c such that . ( Rolle’s Theorem is the same, where ) Fundamental Theorem of Calculus: If f is continuous on [a,b] and F is the antiderivative of f on [a,b] then Second Fundamental Theorem: Mean Value Theorem for Integrals: If f (x) is continuous on [a,b] then there exists a c value between a and b such that A rectangle with length b-a and height f(c) is called the Average Value of the Function. Trapezoidal Rule: To approximate area under a curve. As the sum approaches Trapezoida l Error: If f has a continuous second derivative on [a,b] then the error E in approximating by the trapezoidal rule is
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Transcript of 28937876 AP Calculus BC Study Guide
8/7/2019 28937876 AP Calculus BC Study Guide
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Calculus Study Guide
Rationalization to find limits:
Squeeze Theorem:
& DNE (oscillates).
Important Limits:
Intermediate Value Theorem:If is continuous on [a,b] and k is any number
between (a) and (b), then there is one number c in [a,b] such that (c)=k.
Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b) then
there exists a number c such that . (Rolle’s Theorem is the same,
where )
Fundamental Theorem of Calculus: If f is continuous on [a,b] and F is theantiderivative of f on [a,b] then
Second Fundamental Theorem:
Mean Value Theorem for Integrals: If f (x) is continuous on [a,b] then there
exists a c value between a and b such that
A rectangle with length b-a and height f(c) is called the Average Value of the Function.
Trapezoidal Rule: To approximate area under a curve.
As the sum approaches
Trapezoidal Error: If f has a continuous second derivative on [a,b] then the
error E in approximating by the trapezoidal rule is
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Arc Length: Derivative must exist. If not, use dy instead of dx .
Trigonometric Substitution: Replace x in with because
Replace x in with because
Replace x in with becauseExample:
Substitute with Picture:
Odd Man Out Rule: If sine and cosine have positive powers and only one of thepowers is odd:Keep one of the odd-powered factors, then convert the rest to the other trigexpression using .
Power Reducing Formula:
Steven or Todd Rule: Use if positive powers of secant and tangent. If secant iseven, use Steven. If tangent is odd, use Todd.
Steven: Save a . Convert remaining secants to tangents.Todd: Save a . Convert remaining tangents to secants.
Integration by Parts:
L’hopital’s Rule: To solve indeterminate limits, take derivative of both numeratorand denominator until you can take a limit by direct substitution.
Inverse Trigonometric Functions- Integration:
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Direct Comparison Test: If has no negative terms: Test for Convergence: converges if there is a convergent series with.
Test for Divergence: diverges if there is a divergent series with .
Limit Comparison Test: If is positive and there is a convergent/divergent series
that is also positive and , then converges/diverges.
Alternating Series: An alternating series converges if:, , and
Root Test: series converges if (If equal to one, test is inconclusive)
Ratio Test: Series converges if:
(Inconclusive if the limit equals 1)
Trigonometric Identities:
Integration of Trigonometric Functions:
Taylor Polynomials:
Lagrange Form of the Remainder:
Important Polynomials:
Polar Functions: Use these to convert to Parametric.Area
Vectors: Magnitude of a velocity vector is speed.
