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35 Cards in this Set

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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
sec^2(x) =
csc^2(x) =
cot^2(x) =
tan^2(x) + 1
cot^2(x) + 1
csc^2(x) - 1
List the half angle identities
sin^2(x) = (1 - cos(2x))/2
cos^2(x) = (1 + cos(2x))/2
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.
What can you do for something like ∫sqrt(4-x^2) dx?
Right triangle integration
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.
What is ∫1/(x^2 + a^2)dx?
(1/a)*tan^-1(x/a) + C
When do you use partial fractions?
When you can factor the denominator.
Write out the formula for Simpson's rule.
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
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
The trap rule gives and error of approximately 1. What is the error for the middlesum?
0.5.

Mn have about 1/2 the error of Tn.
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.
What's the error function for the trap rule?
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].
What's the error function for the midsum rule?
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].
What's the error function for Simpson's Rule?
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].
In the rightsum formula, do you start with f(x0) or f(x1)?
Start with f(x1)
In the leftsum formula, do you start with f(x0) or f(x1)?
Start with f(x0)
If f(x) is increasing, is Rn an overestimate or an underestimate?

What is Ln?
Rn is an overestimate.

Ln is an underestimate.
If f(x) is decreasing, is Rn an underestimate or an underestimate?

What is Ln?
Rn is an underestimate.

Ln is an overestimate.
Write the trapsum formula.
Tn =
(Δx/2)*[f(x1)+2f(x2)+2f(x3)...+
2f(x(n-1))+f(xn)]
When does Tn give an overestimate?

When does Tn give an underestimate?
When f(x) is concave up.

When f(x) is concave down.
When does Mn give an overestimate?

When does Mn give an underestimate?
When f(x) is concave down.

When f(x) is concave up.
What's the midpoint formula?
((b-a)/n)*(the midpoints)
What's the leftsum formula?
((b-a)/n)*[(f(x1)+f(x2)+f(x3)+...
+f(x(n-1)]
What's the rightsum formula?
((b-a)/n)*[(f(x1)+f(x2)+f(x3)+...
+f(x(n-1)+f(xn)]
tan^-1(0)=
0
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.
sin^2(x) oscillates between what 2 numbers?
0 and 1
What should we take out of ∫1/sqrt(x^3+x)
Take out x^3. In the denominator, the larger exponents matter less.
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.
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.
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.
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.
Why is ∫1/(4x-2)dx from 0 to 1 improper?
Because 0.5 is located between 0 and 1, and f(0.5) is an assymptote.
What has to be true of L'Hopital's Rule?
g'(x) (the denominator) can't = 0.