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௫→
Test 1 Notes:
lim
௫→
[
)]
= lim
௫→
± lim
௫→
Limit Laws:
lim
௫→
[݂ܿ (ݔ)] = ܿ lim
௫→
lim
௫→
[
)]
= lim
௫→
(ݔ) lim
௫→
lim
௫→
= lim
௫→
/(lim
௫→
lim
௫→
= [lim
௫→
(ݔ)]
lim
௫→
lim
௫→
lim
௫→
If ݂ (ݔ) ≤ ݃ (ݔ) ≤ ℎ(ݔ) when x is near a and lim
௫→
(ݔ) = lim
௫→
ℎ(ݔ) = ܮ ,then
lim
௫→
Jump discontinuity: lim
⬚
ݐℎ ≠ lim
⬚
○ Infinite discontinuity: vertical asymptotes
○ Removable discontinuity
A function is continuous at a number c if lim
௫→
- [ ] includes point ; ( ) does not include point ; u is union
- Any polynomial is continuous everywhere
Sandwich Theorem:
Suppose that f is continuous on [a,b] and let N be any number between ݂ (ܽ ) and ݂ (ܾ ) where
݂ (ܽ ) ≠ ݂ (ܾ ) then there exists a number c in (a,b) such that ݂ (ܿ ) =ܰ
Intermediate Value Theorem:
If r > 0 is a national number then lim
௫→ ஶ
ଵ
௫
ೝ
If r > 0 is a rational number such that ݔ
is defined for all x then lim
௫→ஶ
ଵ
௫
ೝ
To compute lim
௫→ ±ஶ
(௫)
(௫)
) where f(x) and g(x) are non-transcendental functions, divide each
number in the numerator and denominator by the highest powered variable that occurs in the
denominator, then apply limit rules
Limit Rules for Infinite Limits:
Final Study Guide:
Friday, December 09, 2011
9:52 PM
denominator, then apply limit rules
*Cheats: if the highest power is in the denominator then = 0 ; if the highest power is the same ; if
the higher number is in the numerator = ±∞
݉ = lim
∆௫→
- Derivative = slope of curve = rate of change = slope of tangent line
ᇱ
ᇱ
ି ଵ
௫
௫
௫
௫
Product Rule:
ௗ
ௗ௫
ᇱ
ᇱ
Quotient Rule:
ௗ
ௗ௫
(௫)
(௫)
(௫)
ᇲ
(௫)ି (௫)
ᇲ
(௫)
(௫)
మ
- Normal lines have slopes that are negative reciprocals of each other
- f(t)= position = s
- s'(t) = velocity = v
- v'(t) = acceleration = a
- a'(t) = jerk = j
- If the sign of a(t) and v(t) are the same its speeding up
- If the sign of a(t) and v(t) are different its slowing down
Test 2 Notes:
ଶ
ଶ
Derivatives of Trig Functions:
Recall:
- State what rate of change is to be used
- Draw a diagram
- State the equation
- Take the derivative of the equation
- Solve for rate of change or variable
- Substitute using given values
Test 3 Notes:
- L(x) = F(a) + f'(a) (x-a)
The differential of x: ݀ ݔ = ∆ݔ = ݔ
ଶ
ଵ
The differentiable of y: ݀ ݕ = ݂
ᇱ
Linearization:
ାଵ
Newton's Method:
If f is continuous on a closed interval [a,b], then f attains an absolute maximum value and an
absolute minimum value in [a,b]
- A critical number of a function f is a number c in the domain f such that either f'(c) = 0 or f'(c)= dne
- Find the function values of all critical numbers in (a,b)
- Find the function values of the end points of [a,b]
The largest function value is the absolute maximum, and the smallest function value is the
absolute minimum
- To find the absolute extreme values of a continuous f function on a closed interval [a,b]
Extreme Value Theorem:
- F is continuous on [a,b]
- F is differentiable on(a,b)
- f(a) = f(b)
Then there is a number c in (a,b) such that f'(c) = 0
Let f be a function that satisfies the following 3 conditions:
Rolle's Theorem:
- F is continuous on [a,b]
- F is differentiable on (a,b)
Let f be a function that satisfies the following 2 conditions:
Then there is a number c in (a,b) such that ݂
ᇱ
()ି ()
ି
To find increasing and decreasing intervals of a function find critical numbers using the first
derivative
- To find concavity of a function find the possible points of inflection using the second derivative
Mean Value Theorem:
- Domain
- Intercepts
- Symmetry (to find symmetry let x= - x if f(x) = f(-x) the function is symmetric)
- Asymptotes
- Intervals of increasing and decreasing
- local extreme values
- Intervals of concavity and points of inflection
Sketching a Curve:
- Read and understand the problem. What is given? Note conditions? What do you need to find?
- Draw a diagram
- Introduce notation/ assign variables
- Set up equation in terms of one variable
- Take the derivative and solve for critical numbers
- Test the critical number to see if it is an extreme value
- Be sure to answer the question.
Applied Optimization Method:
