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Calculus Midterm 2 Cheat Sheet (DRAFT) by

Formulas for Calculus 1 Midterm 2

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Trigon­ometric Identities

sin2+cos2=1
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)  OR  cos(x)­/sin(x)
tan(x) = sin(x)­/cos(x)
csc(x) = 1/sin(x)
sec2 = tan2+1

Graphing Steps

1. Domain
2. Intercepts
3. Asymptotes
4. Intervals of Increase and Decrease
5. Local Minimums and Maximums
6. Concavity and Inflection Points

Graphing Tips

VA: lim (x->+_­inf­inity) f(x) =_+inf­inity (left and right)
HA: lim (x->+_­inf­inity) f(x) = c at y=c
VA: Find by setting the denomi­nator = 0 and solving for x
HA: y=0 if n<d, ax/bx if n=d, none if n>d
First Deriva­tive: Intervals of increase or decrease + min/max
Second Deriva­tive: Concavity + Inflection Points

Derivative Rules

Product: f'(x)g(x) + g'(x)f(x)
Chain: f'(g(x)) * g'(x)
Quotient: f'(x)g(x) - g'(x)f­(x)­/g(x)^2

Accele­ration and Velocity

Accele­ration is the antide­riv­ative of velocity
(come to stop) 1. Find antide­riv­ative of function
2. Find v(0) or C and set = 0
(for distance) 1. Deriva­tive, solve for t, deriva­tive, plug in
To find t take deriva­tive, to find distance take integral

Evaluating Integrals

a+b/c = a/c + b/c
Indefi­nite: F(x) + C
F(b)-F(a) (find antide­riv­ative and plug in)
 

Unit Circle

Derivative Tests

1st: Positive to Negative: local max
2nd: f'(c) = 0 & f''(c)­>0: local min & concave up
1st: Negative to Positive: local min
2nd: f'(c) = 0 & f''(c)­<0: local max & concave down
Critical points when f'(x)=0
Inflection points when f''(x)=0

Interm­ediate Value Theorem

a<c­<b
Used to find when f(x) has roots
When proving roots, show that one part is positive and the other is negative
To find c, set y=0 and solve for x
To show at most, show that there is 1 critical value and f(x) can only cross x amount of times
Explain that you are using IVT

Areas & Distances

Deriva­tive: rate of change
Antide­riv­ative: total change
n or change t = b-a/n
RHS: E (n i=1) f(ti) change t
LHS: E (n-1 i=0) f(ti) change t
ti = a +i change t

U Subsit­ution

Step 1: Make a "­u-s­ubs­iti­tut­ion­" (let u=)
Step 2: Find du/dx
Step 3: Solve for dx
Step 4: Substitute dx and cancel out terms
Step 5: Integrate with respect to u
*If a definite integral, change the bounds from x bounds to u bounds
*Add C if a indefinite integral
 

Mean Value Theorem

Is continuous and differ­ent­iable
f(a)=f(b)
f'(c)=­f(b­)-f­(a)/b-a
f'(c)=0
How large can this be?
By MVT f'(c) =... for some c in [0,x]. Then do the math. Hence for every x in interval f(x) is whatever the math proves.

Antide­riv­atives

Function
Antide­riv­ative
x^n
x^n+1/n+1
cos(x)
sin(x)
sin(x)
-cos(x)
sec^2(x)
tan(x)
sec(x)­tan(x)
sec(x)

Deriva­tives

Fucntion
Derivative
sin(x)
cos(x)
cos(x)
-sin(x)
tan(x)
sec^2(x)
csc(x)
-csc(x)
sec(x)
sec(x)­tan(x)
cot(x)
-csc^2(x)

Optimi­zation Problems

Usually using two different formulas (like volume and perimeter)
If maximizing volume, solve for one variable and plug that it
Next, solve for derivative and set = 0
After solving for that variable, plug into original (volume) equation
For distance: √(x-a)2 + (y-b)2 & solve for critical point
May need to prove that something is a global min/max

Properties of the Definite Integral

Constant:
Addition:
Pulling a Constant:
Subtra­ction
Splitting