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35 Cards in this Set
- Front
- Back
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cos^2(x) =
sin^2(x) = tan^2(x) = |
1 - sin^2(x)
1 - cos^2(x) sec^2(x) - 1 |
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sec^2(x) =
csc^2(x) = cot^2(x) = |
tan^2(x) + 1
cot^2(x) + 1 csc^2(x) - 1 |
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List the half angle identities
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sin^2(x) = (1 - cos(2x))/2
cos^2(x) = (1 + cos(2x))/2 |
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What do we do for ∫cos^3(x) or ∫sin^3(x)?
What do we do for ∫cos^2(x) or ∫sin^2(x)? |
Split up using sin^2(x) + cos^2(x) = 1.
Use half angle indentities. |
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What can you do for something like ∫sqrt(4-x^2) dx?
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Right triangle integration
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When doing integration of 2 polynomials ∫(Poly.1/Poly2), what has to be true?
What do you do if it's not. |
The largest power of the denominator has to be larger than the largest of the numarator.
You use long division. |
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What is ∫1/(x^2 + a^2)dx?
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(1/a)*tan^-1(x/a) + C
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When do you use partial fractions?
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When you can factor the denominator.
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Write out the formula for Simpson's rule.
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Sn = [deta(x)/3]*[f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)....
+4f(x(n-1))+f(xn)] delta(x) = (b-a)/n |
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Negative error coorlinate to approximations ____ the true value.
Positive error coorlinate to approximations ____ the true value. |
greater than the true value
less than the true value |
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The trap rule gives and error of approximately 1. What is the error for the middlesum?
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0.5.
Mn have about 1/2 the error of Tn. |
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The n in a trap rule or a midsum doubles. What does the error decrease by?
n increases by a factor of p. What does the error decrease by? |
It decreases by a factor of 4.
It decreases by a factor of p^2. |
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What's the error function for the trap rule?
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The absolute value of the error |Et|=
|∫f(x)dx - Tn| = (K(b-a)^3)/(12n^2) K is the maximum of |f"(x)| on [a,b]. |
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What's the error function for the midsum rule?
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The absolute value of the error |Et|=
|∫f(x)dx - Tn| = (K(b-a)^3)/(24n^2) K is the maximum of |f"(x)| on [a,b]. |
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What's the error function for Simpson's Rule?
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The absolute value of the error |Es|=
|∫f(x)dx - Sn| = (K(b-a)^5)/(180n^4) K is the maximum of |f''''(x)| on [a,b]. |
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In the rightsum formula, do you start with f(x0) or f(x1)?
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Start with f(x1)
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In the leftsum formula, do you start with f(x0) or f(x1)?
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Start with f(x0)
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If f(x) is increasing, is Rn an overestimate or an underestimate?
What is Ln? |
Rn is an overestimate.
Ln is an underestimate. |
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If f(x) is decreasing, is Rn an underestimate or an underestimate?
What is Ln? |
Rn is an underestimate.
Ln is an overestimate. |
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Write the trapsum formula.
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Tn =
(Δx/2)*[f(x1)+2f(x2)+2f(x3)...+ 2f(x(n-1))+f(xn)] |
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When does Tn give an overestimate?
When does Tn give an underestimate? |
When f(x) is concave up.
When f(x) is concave down. |
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When does Mn give an overestimate?
When does Mn give an underestimate? |
When f(x) is concave down.
When f(x) is concave up. |
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What's the midpoint formula?
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((b-a)/n)*(the midpoints)
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What's the leftsum formula?
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((b-a)/n)*[(f(x1)+f(x2)+f(x3)+...
+f(x(n-1)] |
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What's the rightsum formula?
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((b-a)/n)*[(f(x1)+f(x2)+f(x3)+...
+f(x(n-1)+f(xn)] |
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tan^-1(0)=
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0
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When dealing with improper integrals, is the - sign used for apprximations on the left or on the right?
And the plus sign? |
The - sign is for approximations on coming from the left.
The + sign is for approximations on coming from the right. |
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sin^2(x) oscillates between what 2 numbers?
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0 and 1
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What should we take out of ∫1/sqrt(x^3+x)
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Take out x^3. In the denominator, the larger exponents matter less.
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Give the formula for force
What are the units of force? |
ma (mass*acceleration or m*9.8 m/s)
kg*m/s^2 or Newtons Also expressed in lbs. |
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Give the formula for work.
What are its units? |
Work = Force x Distance
Expressed in N*m, kg*m^2/s^2, or Joules Also expressed as lbs.*ft. |
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If we have a 3 lb. book, and we want to lift it 2 feet off the ground.
What's the force in this problem? What's the work done in this problem? |
Force = 3 lbs.
Work = 6 ft.lbs. |
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If we have a 2 kg book, and we want to lift it 3 meters.
What's the force? What's the work done in this problem? |
Force = 19.6 Newtons
Work = 58.8 Joules. |
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Why is ∫1/(4x-2)dx from 0 to 1 improper?
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Because 0.5 is located between 0 and 1, and f(0.5) is an assymptote.
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What has to be true of L'Hopital's Rule?
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g'(x) (the denominator) can't = 0.
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