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Section 1

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(Exponential Growth) If y is a differentiable function satisfying the differential equation, then...

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Calculus Math

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Date created

Mar 1, 2020

Cards (36)

Section 1

(36 cards)

(Exponential Growth) If y is a differentiable function satisfying the differential equation, then...

Front

Back

On Picking u's and v's

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Back

Length of a Curve (Calculus)

Front

Back

Exponential Growth Formula

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Back

State the Order to Carry out Series Tests

Front

1) Test for Divergence 2) By Type: Geometric, Telescoping Series, or P-series 3) Integral Test (f(n) is Positive, Continuous, Decreasing) 4) Comparison Test or Limit Comparison Test (If positive and "ACTS" like another known series) 5) Alternating Series Test (Use this before hand, if alternating terms and does not fit telescoping series) 6) Ratio Test (for factorials and nth powers) 7) Root Test (for nth powers)

Back

Surface area of shell

Front

Surface area of shell = 2π(radius) (height of shell) =2π ∫ x*f(x) dx

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Integral Test for Series

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Back

Total Population (As an Integral)

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Back

Divergence Test

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Back

Infinite Sums of Polynomials Test

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P-Series p > 1

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Reduction formula (sin(x))

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Rotating About a Vertical Line (x = c)

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To rotate about a vertical line x = c, express R-outer and R-inner as functions of y and integrate along the y axis.

Back

Infinite Geometric Convergence

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Definition of Work (Integral); Variable force

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Comparison Test (Integrals)

Front

B.F. converges ∴ S.F converges S.F diverges ∴ B.F diverges

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1/(1+e^x)

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Series Comparison Test

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Comparison of Product Rule and Integration

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Series Balloon Analogy (Direct Comparison Test)

Front

A good analogy for the Direct Comparison Test, as in Figure 5, is one balloon containing the terms for an, inside a bigger balloon containing the terms for bn. As we blow air into the balloons in amounts corresponding to the subsequent terms an and bn, the balloon containing the bn terms will always be bigger than the balloon containing the an terms since an ≤ bn. If the bigger balloon does not contain enough air to make it pop, then the smaller balloon does not pop either. Thus, if the bigger series converges, so does the smaller series. On the other hand, if the smaller balloon contains enough air to make it pop, then the bigger balloon must also pop, implying that if the smaller series diverges, the bigger series diverges as well. But in the cases that the bigger balloon pops or the smaller balloon does not pop, it says nothing about the remaining balloon

Back

(Exponential Growth) Doubling Time

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Integration by Parts Formula

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Ratio Test

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Laminar Flow Rate (As an Integral)

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PV of an Income Stream

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Work =

Front

Work = Force * Distance

Back

Reduction formula (cos(x))

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Back

Alternating Series Test

Front

Use f'(x) < 0 or an+1 < an to show decreasing

Back

Total Mass (As an Integral)

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Rotating About a Horizontal Line (y = c)

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Back

Limit Comparison Test Tips

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Back

P- series for [0,P] U [P,1]

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Derivative of the Inverse

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Back

(Exponential Growth) Half Life

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Back

Limit Comparison Test

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d/dx bⁿ

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