Final Flashcards
Derivative of ln(x)
The derivative is 1/x
The derivative of log_a (x)
1/(x ln(a)) is the derivative
The derivative of a^x
The derivative is a^x ln(a)
The derivative of sinx
The derivative is Cosx
The derivative of cosx
-Sinx
The derivative of tanx
Sec²x
The derivative of secx
Secx tanx
The derivative of csc x
-cscx cotx
The derivative of cotx
-Csc^2 x
The derivative of inverse sinx
1/sqrt(1-x^2)
The derivative of inverse cosx
-1/sqrt(1-x^2)
The derivative of inverse tanx
1/1+x²
The derivative of inverse cscx
-1/(x sqrt(x²-1))
The derivative of inverse secx
1/(x sqrt(x²-1))
The derivative of inverse cotx
-1/(1+x²)
The integral of 1/x
Ln(x) +C
The integral of sinx
-cosx +C
The integral of cosx
Sinx +C
The integral of sec²x
Tanx +C
The integral of secx tanx
Secx + C
The integral of tanx
Ln(|secx|)+ C
The integral of secx
Ln(|secx+tanx|) +C
Integral of 1/(1+x²)
Arctanx +C or inverse tanx +C
The integration by parts formula
The integral of (udv) = uv - the integral of (vdu)
Trigonometric substitution, 3 possibilities…
- Sqrt ( a²-x²) use x=asin(theta)
- Sqrt (a²+x²) use x=atan(theta)
- Sqrt (x²-a²) use x=asec(theta)
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)
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)
Procedure of expressing a Riemann Sum into an integral
- Limit of the sum is replaced by the definited integral
- Xi is replaced by X
- Δx is replaced by dx
How to find the Δx when doing a Rienmann Sum (rectangles)?
Δx= (b-a)/n
How to find Xi when doing a Rienmann Sum?
Xi=a+iΔx
Midpoint Rule
Using midpoint for each intervals while doing the rectangle approximation gives a more accurate answer than the right or left endpoints
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)
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
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”.
The total displacement using integrals
Total displacement = the integral from t1 to t2 of |v(t)| dt
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
Finding the volume of a solid obtained by rotating the region under the curve/between 2 curves
- graph the fcts
- rotate and create the solid
- 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=…
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
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
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
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
sinAcosB =?
1/2[sin(A-B) + sin(A+B)]
sinAsinB=?
1/2[cos(A-B) - cos(A+B)]
cosAcosB=?
1/2[cos(A-B) + cos(A+B)]
cscx=?
1/sinx
secx=?
1/cosx
The Riemann Summation of C =?
nC
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)
Integration of rational fcts by partial fractions
- Is the num of = or larger degree ? Yes= long division
No.. -» 2. Partial fractions
Improper integrals types
- -∞ to b
- a to ∞
- a to b, but one or both are discontinuous (check the domain of the fct)
- a to b, but discontinuity at c (between a and b)
Improper integrals are called conv or div if…
their limit exist = conv
their limit does not exist = div
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 =#
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 )
