# Math Chapter 3

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1. Requirements for relative max/min?
Must be continuous and if its on a closed interval, it cannot be an endpoint.

• f " > 0 relative MIN
• f " < 0 relative MAX
2. Critical point
When f '(c) = 0 or undefined.
3. Location of Extrema
When derivative is undefined, or 0. So the same as a critical point
4. Finding Extrema on Closed Interval
• Find Critical Numbers (f '(c))
• Evaluate f(c) for each critical number
• Evaluate f(c) for endpoints
• High is MAX, Low is MIN

• NOTE: If it says a polynomial to the 3rd degree, there can only be 2 relative numbers.
• ex: 4th degree = 3 relative numbers etc
5. What types of functions are always continuous and differentiable?
• Polynomials
• Rational Functions
• Trig Functions
6. Rolle's Theorem
• Is f(x) continuous on [a,b]? why?
• Is f(x) differentiable on (a,b)? Why?
• Does f(a) = f(b)?

If all of these apply: find the critical points at f '(x).
7. Does Rolle's Theorem work for Absolute Values?
Only if the closed interval does NOT include the vertex. Because it is not differentiable at a sharp point
8. Mean Value Theorem (MVT)
If f is continuous on [a,b] and differentiable on (a,b) then there exists c on (a,b) such that f '(c)= [f(b) - f(a)] / (b-a)
9. Rates of Change and Slopes
• Average Rate of Change = Instantaneous Rate of Change
• Slope of Secant = Slope of Tangent
10. Increasing/Decreasing
• f '(c) > 0 f is increasing
• f '(c) < 0 f is decreasing
• f '(c) = 0 f is constant

• When f ' changes from - to +, relative MINimum
• When f ' changes from + to -, relative MAXimum
11. Velocity and Acceleration General Info

Speeding Up/Speeding Down?
• s(t) = position > 0 right of origin / < 0 left of origin
• s'(t) = velocity > 0 moving right / < 0 moving left
• s"(t) = acceleration

• Find when s '(t) = 0 and s "(t) = 0.
• Put on number line.
• Plug numbers in for BOTH velocity and acceleration.
• When signs are same it is speeding UP
• When signs are different it is slowing DOWN
12. Find polynomial of least degree such that
(given relative min/max)
f(x) = ax^4 + bx^3 + cx^2 + dx + e

• Plug in (x,y) in each f equation.
• Find derivative for equation and plug in (x,y)
• Solve for a,b,c,d,e
13. Concavity
• Concave UP = f ' is INCREASING
• Concave DOWN = f ' is DECREASING

• Concave UP = f " > 0
• Concave DOWN = f " < 0
14. Point of Inflection
When concavity changes from + to - or - to + at a point

• f " = 0 or undefined
• TANGENT MUST EXIST
• (vertical/horizontal line is usually the case)
15. 2nd Derivative Test
• When f '(c) = 0
• f "(c) < 0 Relative MAX
• f "(c) > 0 Relative MIN

• Steps:
• find f '(x)
• plug in results to f "(x)
16. Limits of Infinity (oo)
Divide by highest power of the DENOMINATOR

• Denominator > Numerator f(x) = 0
• Denominator = Numerator f(x) = coefficients
• Denominator < Numerator f(x) = +/- infinity (oo)

lim (x => oo) (sinx)/x = 0
17. L'Hopital's Rule
ONLY USE WHEN IT IS 0/0 OR oo/oo !!!

• Always substitute first to find out ^
• Find derivative, and substitute again.
• Keep going until it works

NOTE: do NOT use quotient rule!
18. Indeterminate Forms
(!= means does not equal)

• oo - oo != 0
• 0 x oo != 0
• 1^oo != 1
• oo^0 != 1
• 0^0 != 1
19. Determinate Forms
• nonzero/0 = +/- oo
• 0^-oo = oo
• 0^oo = 0
• -oo - oo = - oo
• oo + oo = oo

• 1/0 = oo
• 1/oo = 0
20. Graphing
• If f is increasing, f ' > 0 (above axis)
• If f is decreasing, f ' < 0 (below axis)

If f '(x) = 0, there is a HORIZONTAL TANGENT

If f '(x) does not exist, there is a SHARP TURN or VERTICAL TANGENT

• If f " > 0, concave UP
• If f " < 0, concave DOWN

If f ' does not exist, then it is a VERTICAL TANGENT

When f " = 0, there is an inflection point (CONCAVITY CHANGES)
 Author: craziieegurl13 ID: 45578 Card Set: Math Chapter 3 Updated: 2010-10-28 04:51:05 Tags: applications differentiation rolles theorem extrema math Folders: Description: Chapter 3: Applications of Differentiation Show Answers: