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Calculus II > Final > Flashcards

Final Flashcards

1
Q

Derivative of ln(x)

A

The derivative is 1/x

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2
Q

The derivative of log_a (x)

A

1/(x ln(a)) is the derivative

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3
Q

The derivative of a^x

A

The derivative is a^x ln(a)

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4
Q

The derivative of sinx

A

The derivative is Cosx

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5
Q

The derivative of cosx

A

-Sinx

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6
Q

The derivative of tanx

A

Sec²x

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7
Q

The derivative of secx

A

Secx tanx

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8
Q

The derivative of csc x

A

-cscx cotx

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9
Q

The derivative of cotx

A

-Csc^2 x

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10
Q

The derivative of inverse sinx

A

1/sqrt(1-x^2)

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11
Q

The derivative of inverse cosx

A

-1/sqrt(1-x^2)

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12
Q

The derivative of inverse tanx

A

1/1+x²

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13
Q

The derivative of inverse cscx

A

-1/(x sqrt(x²-1))

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14
Q

The derivative of inverse secx

A

1/(x sqrt(x²-1))

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15
Q

The derivative of inverse cotx

A

-1/(1+x²)

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16
Q

The integral of 1/x

A

Ln(x) +C

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17
Q

The integral of sinx

A

-cosx +C

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18
Q

The integral of cosx

A

Sinx +C

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19
Q

The integral of sec²x

A

Tanx +C

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20
Q

The integral of secx tanx

A

Secx + C

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21
Q

The integral of tanx

A

Ln(|secx|)+ C

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22
Q

The integral of secx

A

Ln(|secx+tanx|) +C

23
Q

Integral of 1/(1+x²)

A

Arctanx +C or inverse tanx +C

24
Q

The integration by parts formula

A

The integral of (udv) = uv - the integral of (vdu)

25
Q

Trigonometric substitution, 3 possibilities…

A
  1. Sqrt ( a²-x²) use x=asin(theta)
  2. Sqrt (a²+x²) use x=atan(theta)
  3. Sqrt (x²-a²) use x=asec(theta)
26
Q

Integration by parts, priority of the “u”

A

LIATE
1. Ln or log
2. Inverse trig (arctan, ect)
3. Algebraic (x², ect)
4. Trig (sin, cos, ect)
5. Exponential fct (anything power x, x+3, ect)

27
Q

How to get the 2 other trigonometric identities from cos^2(x) + sin^2(x)=1 ?

A

divide the first identity by sin(x) or cos(x)

28
Q

Procedure of expressing a Riemann Sum into an integral

A
  1. Limit of the sum is replaced by the definited integral
  2. Xi is replaced by X
  3. Δx is replaced by dx
29
Q

How to find the Δx when doing a Rienmann Sum (rectangles)?

A

Δx= (b-a)/n

30
Q

How to find Xi when doing a Rienmann Sum?

A

Xi=a+iΔx

31
Q

Midpoint Rule

A

Using midpoint for each intervals while doing the rectangle approximation gives a more accurate answer than the right or left endpoints

32
Q

The fundamental theorem of Calculus Part 1

A

g(x)=the integral from a to x of f(t) dt
-where x is between a and b
-f(t) is the curve and g(x) the area under the curve
-g’(x)=f(t)

33
Q

The fundamental theorem of Calculus part 2

A

The integral of f(x) dx from a to b = F(b)-F(a)
- where F(a) and F(b) are the antiderivative of f

34
Q

The Net change theorem

A

The integral from a to b of F’(x) dx = F(b)-F(a)
- F’(x) =f(x) and it represents the rate of change of f(x) (which is the y axis). So F(b)-F(a) is the change is “y” when x goes from a to b, it is the NET CHANGE in “y”.

35
Q

The total displacement using integrals

A

Total displacement = the integral from t1 to t2 of |v(t)| dt

36
Q

The area between 2 curves

A

A=lim as x approaches infinity of the Summation (n and i=1) of [f(xi)-g(xi)] Δx

or

A= the integral from a to b of [f(x)-g(x)] dx

-where f(x) is bigger or equal to g(x) (it has to be on top in the graph)
-If one fct is not always on top, we need to slit the region into 2 or more section, find the value of each area and add them at the end. ( |f(x)-g(x)|= f(x)-g(x) when f is bigger and g(x)-f(x) when g is bigger

37
Q

Finding the volume of a solid obtained by rotating the region under the curve/between 2 curves

A
  1. graph the fcts
  2. rotate and create the solid
  3. V= integral from a to b of A(x) dx
    A(x) = π (r)^2
    r= radius, which is the fct of the solid

if it’s a washer, A(x)= π (r)^2 (outer fct) -π (r)^2 (inner fct)

Sometimes we need to rotate about the y axis instead, transform your fct from y=… to x=…

38
Q

The average value of a fct

A

favg=f(c)=(1/b-a) the integral from a to b of f(x) dx

f(c) (b-a) = the integral from a to b of f(x) dx (A=bh), this does a rectangle

39
Q

Trigonometric integrals
first thing to do

A

Theres is 3 pairs that do not simplify:
1. sinx and cosx
2. secx and tanx
3. cscx and cotx

if you have a mix of these pairs, start by simplify, put everyting back into sinx and cosx

40
Q

Trig integrals of sinx and cosx

A

If one of them is odd, save a power and convert the sin^2(x) or the cos^2 (x) using the trig identity. Then, u substitution

if both are even, use the half-angle identities provided (ex: cos^2 (x) = 1/2(1+cos2x))

It may help to also use this : sinx cosx = 1/2 sin2x or 2sinxcosx= sin2x

41
Q

Trig integrals of secx and tanx

A

If secant is even, factor out sec^2 (x) and expressed the remaining factors with trig identity. Substitute u=tanx

If tangent is odd, save a factor secxtanx and use trig identity with the rest. Substitute u=secx

42
Q

sinAcosB =?

A

1/2[sin(A-B) + sin(A+B)]

43
Q

sinAsinB=?

A

1/2[cos(A-B) - cos(A+B)]

44
Q

cosAcosB=?

A

1/2[cos(A-B) + cos(A+B)]

45
Q

cscx=?

A

1/sinx

46
Q

secx=?

A

1/cosx

47
Q

The Riemann Summation of C =?

A

nC

48
Q

Trigonometric substitution verification fact

A

When you do your triangle, use the fct you are working with to find the first 2 sides (sinθ, tanθ or secθ) and the with Pythagor find the 3rd (it should give the sqrt you started with)

49
Q

Integration of rational fcts by partial fractions

A
  1. Is the num of = or larger degree ? Yes= long division
    No.. -­» 2. Partial fractions
50
Q

Improper integrals types

A
  1. -∞ to b
  2. a to ∞
  3. a to b, but one or both are discontinuous (check the domain of the fct)
  4. a to b, but discontinuity at c (between a and b)
51
Q

Improper integrals are called conv or div if…

A

their limit exist = conv
their limit does not exist = div

52
Q

The Maclaurin and Taylor Series definition

A

f(x)=summation (infinity and n=0) of Cn (x-a)^n

Where Cn= (f^(nth derivative)of (a))÷ n!

The Maclaurin series is centered at a=0

The Taylor series is centered at a =#

53
Q

Trig circle

A

30° = π/6 = (√3/2, 1/2 )
45° = π/4 = (√2/2,√2/2 )
60° = π/3 = ( 1/2, √ 3/2 )
90° = π/2 = ( 0,1 )

Calculus II (1 decks)

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