Final Flashcards
1
Q
Derivative of ln(x)
A
The derivative is 1/x
2
Q
The derivative of log_a (x)
A
1/(x ln(a)) is the derivative
3
Q
The derivative of a^x
A
The derivative is a^x ln(a)
4
Q
The derivative of sinx
A
The derivative is Cosx
5
Q
The derivative of cosx
A
-Sinx
6
Q
The derivative of tanx
A
Sec²x
7
Q
The derivative of secx
A
Secx tanx
8
Q
The derivative of csc x
A
-cscx cotx
9
Q
The derivative of cotx
A
-Csc^2 x
10
Q
The derivative of inverse sinx
A
1/sqrt(1-x^2)
11
Q
The derivative of inverse cosx
A
-1/sqrt(1-x^2)
12
Q
The derivative of inverse tanx
A
1/1+x²
13
Q
The derivative of inverse cscx
A
-1/(x sqrt(x²-1))
14
Q
The derivative of inverse secx
A
1/(x sqrt(x²-1))
15
Q
The derivative of inverse cotx
A
-1/(1+x²)
16
Q
The integral of 1/x
A
Ln(x) +C
17
Q
The integral of sinx
A
-cosx +C
18
Q
The integral of cosx
A
Sinx +C
19
Q
The integral of sec²x
A
Tanx +C
20
Q
The integral of secx tanx
A
Secx + C
21
Q
The integral of tanx
A
Ln(|secx|)+ C
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
Trigonometric substitution, 3 possibilities...
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
Integration by parts, priority of the "u"
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
How to get the 2 other trigonometric identities from cos^2(x) + sin^2(x)=1 ?
divide the first identity by sin(x) or cos(x)
28
Procedure of expressing a Riemann Sum into an integral
1. Limit of the sum is replaced by the definited integral
2. Xi is replaced by X
3. Δx is replaced by dx
29
How to find the Δx when doing a Rienmann Sum (rectangles)?
Δx= (b-a)/n
30
How to find Xi when doing a Rienmann Sum?
Xi=a+iΔx
31
Midpoint Rule
Using midpoint for each intervals while doing the rectangle approximation gives a more accurate answer than the right or left endpoints
32
The fundamental theorem of Calculus Part 1
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
The fundamental theorem of Calculus part 2
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
The Net change theorem
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
The total displacement using integrals
Total displacement = the integral from t1 to t2 of |v(t)| dt
36
The area between 2 curves
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
Finding the volume of a solid obtained by rotating the region under the curve/between 2 curves
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
The average value of a fct
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
Trigonometric integrals
first thing to do
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
Trig integrals of sinx and cosx
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
Trig integrals of secx and tanx
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
sinAcosB =?
1/2[sin(A-B) + sin(A+B)]
43
sinAsinB=?
1/2[cos(A-B) - cos(A+B)]
44
cosAcosB=?
1/2[cos(A-B) + cos(A+B)]
45
cscx=?
1/sinx
46
secx=?
1/cosx
47
The Riemann Summation of C =?
nC
48
Trigonometric substitution verification fact
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
Integration of rational fcts by partial fractions
1. Is the num of = or larger degree ? Yes= long division
No.. -» 2. Partial fractions
50
Improper integrals types
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
Improper integrals are called conv or div if...
their limit exist = conv
their limit does not exist = div
52
The Maclaurin and Taylor Series definition
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
Trig circle
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 )
