Studied by 157 peopleY=f(x) must be continuous at each:
-Critical Point or undefined and endpoints
Local Minimum
Goes (-,0,+) or (-, und, +)
Local Maximum
Goes (+,0,-) or (+, und, -)
Point of Inflection
Concavity Changes
(+,0,-) or (-,0,+)
(+,und,-) or (-,und,+)
D/dx(sinx)
Cosx
D/dx(cosx)
-sinx
D/dx(tanx)
Sec²x
D/dx(cotx)
-csc²x
D/dx(secx)
Secxtanx
D/dx(cscx)
-cscxcotx
D/dx(lnx)
1/x
D/dx(ln(n))
1/n
D/dx(eⁿ)
Eⁿ
∫Cosx
Sinx
∫-sinx
Cosx
∫Sec²x
Tanx
∫-csc²x
Cotx
∫Secxtanx
Secx
∫-cscxcotx
Cscx
∫1/n
Ln(n)
∫Eⁿ
Eⁿ
When doing integrals never forget
Constant (+c)
∫Axⁿ
A/n+1(xⁿ⁺¹)+C
∫Tanx
Ln|secx|+c
Ln|cosx|+c
∫Secx
Ln|secx+tanx|+c
D/dx(sin⁻¹u)
1/√1-u²
D/dx(cos⁻¹x)
-1/√1-x²
D/dx(tan⁻¹x)
1/1+x²
D/dx(cot⁻¹x)
-1/1+x²
With derivative inverses
You plug in the number of the trigonometric function into x
D/dx(sec⁻¹x)
1/|x|√x²-1
D/dx(csc⁻¹x)
-1/|x|√x²-1
D/dx(aⁿ)
Aⁿln(a)
D/dx(Logₙx)
1/xln(a)
Chain Rule
Take derivative of outside of parenthesis
Take derivative of inside parenthesis and keep the original of what was in the parenthesis
For example, sin(x²+1)→ 2xcos(x²+1)
Product Rule
d/dx first times second + first times d/dx second
Quotient Rule
LoDHi-HiDLo/LoLo
Fundamental Theorem of Calculus
∫(a to b) f(x) dx = F(b) - F(a)
Basically saying that F’(x)=f(x)
f relative max→f ‘ goes from
Positive to negative
f relative min→f ‘ goes from
Negative to positive
Intermediate Value Theorem
If I pick an X value that is included on a continuous function, I will get a Y value, within a certain range, to go with it.
Mean Value Theorem
If function is continuous on [a,b] and first derivative exist on interval (a,b), then there is at least one number (c) such that f’(c)=f(b)-f(a)/b-a
Rolle’s Theorem with Mean Value Theorem
If function is continuous on [a,b] and first derivative exist on interval (a,b), and f(a)=f(b) then there is at least one number x=c such that f’(c)=0
Trapezoidal Rule
The average of left hand point method and right hand point method
Theorem of mean value and average value
1/b-a ∫f(x)dx which is the average value
Disk Method
V=π∫[R(x)]²dx
Washer Method
π∫(R(x))²-(r(x))²dx
General volume equation
V=∫area(x)dx
-If no shape is given, then you just take the integral of the function
Distance, Velocity, and Acceleration
Velocity is d/dx of position
Acceleration is d/dx of velocity
Displacement
∫(Velocity)dt
Distance
∫|Velocity|dt
Speed
The absolute value of velocity
Average Velocity
Final Position-Initial Position/Total time
cos(0)
1
sin(0)
0
Pie circle starts with
π/6
tan(π/3)
√3
tan(π/6)
√3/3
tan(90⁰)
Undefined
Double Argument
sin2x=2sinxcosx
cos2x=cos²x-sin²x=1-2sin²x
cos²x=1/2(1+cos2x)
sin²x=1/2(1-cos2x)
Pythagorean Identity
sin²x+cos²x=1
L’Hopital Rule
If limit equals 0/0 or ∞/∞, then you can take the derivative
Note
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