1. Lesson 4-QR
Quiz 1 Review

2. Objectives
Prepare for the quiz on sections 4-1 thru 4-4

3. Vocabulary
Critical number – a value of x such that f’(x) = 0 or f’(x) does not exist
Existence Theorem – a theorem that guarantees that there exists a number with a certain property, but it doesn’t tell us how to find it.
Extreme values – maximum or minimum functional values (y-values)
Indeterminate Form – (0/0 or ∞/∞) a form that a value cannot be assigned to without more work
Inflection point – a point (x,y) on the curve where the concavity changes

4. Theorems
Extreme Value Theorem: If f is continuous on the closed interval [a,b], then f attains an absolute maximum value f(c) and a absolute minimum value f(d) at some numbers c and d in [a,b]. Fermat’s Theorem: If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0. Or rephrased: If f has a local maximum or minimum at c, then c is a critical number of f. [Note: this theorem is not biconditional (its converse is not necessarily true), just because f’(c) = 0, doesn’t mean that there is a local max or min at c!! Example y = x³]

5. Closed Interval Method:
To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]:
Find the values of f at the critical numbers of f in (a,b) (the open interval)
Find the values of f at the endpoints of the interval, f(a) and f(b)
The largest value from steps 1 and 2 is the absolute maximum value; the smallest of theses values is the absolute minimum value.

6. Theorems
Mean Value Theorem: Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) then there is a number c in (a,b) such that instantaneous rate of change = average rate of change f(b) – f(a) f’(c) = mtangent = msecant b – a Rolle’s Theorem: c) f(a) = f(b) then there is a number c in (a,b) such that f’(c) = 0

7. Review of 1st and 2nd Derivatives
Function
Extrema are y-values of the function!
First Derivative – f’(x)
Slope of the function
f’(x) = 0 at “critical” values of x
Possible locations of relative extrema
Relative extrema can also occur at endpoints on closed intervals [a,b]
Second Derivative – f’’(x)
Concavity of the function
f’’(x) = 0 at possible points of inflection
IPs are places where there is a change in concavity

8. 1st and 2nd Derivative Tests
First derivative test uses the change in signs of the slopes [f’(x)] just before and after a critical value to determine if it is a relative min or max
Second derivative test uses a functions concavity at the critical value to determine if it is a relative min or max
x = c
f’ > 0 for x < c
f’ < 0 for x > c
Relative Max
Slope
hill
x = c
f’ > 0 for x > c
f’ < 0 for x < c
Relative Min
Slope
valley
Relative Max
Relative Min
x = c
x = c
f’’(x) < 0
Concave down
f’’(x) > 0
Concave up

9. Intervals - Table Notation
f(x) = x4 – 4x3 = x3(x – 4)
f’(x) = 4x3 – 12x2 = 4x2(x – 3)
f’’(x) = 12x2 – 24x = 12x (x – 2)
Intervals
(-∞,0)
0,2) 2
(2,3) 3
(3,4) 4
(4,∞)
f(x) + -
-16
-27
f’(x) - slope 64
f’’(x) - concavity 36 96
Notes: IP
y=0
min

10. L’Hosptial’s Rule
L’Hospital’s Rule applies to indeterminate quotients in the form of 0/0 or ∞/∞
f(x) f’(x)
Lim = Lim (can be applied several times)
g(x) g’(x)
Other indeterminate forms exist and can be solved for, but are beyond the scope of this course

11. Extrema Problem
f(x) = x³ + 4x² -12 f’(x) = 3x² + 8x = x (3x + 8) f’(x) = 0 at x = 0 and x = -8/3 f’’(x) = 6x + 8 f’’(x) = 0 at x = -4/3
Remember: if we have a closed interval, then we have to evaluate the end points!
Intervals
(-∞,-8/3)
-8/3
(-8/3,-4/3)
-4/3
(-4/3,0)
(0,∞)
f(x)
-68/27
-196/27
-12
f’(x) - slope + -
f’’(x) - concavity
Notes:
Rel max IP
Rel min

12. Mean Value Theorem
f(x) = 6x² - 3x + 8 on [0,3] Continuous on [0,3] polynomial Differentiable on (0,3) polynomial 53 – 8 45 f(3) = 53 f(0) = 8 f’(c) = = = 15 3 – 0 3 f’(x) = 12x – 3 = 15 12x = 18 x = 3/2
MVT applies
M secant
M tangent

13. Mean Value and Rolle’s Theorems
Continuous on [a,b] problems where f(x) is undefined division by zero not allowed negative numbers in an even root Differentiable on (a,b) problems where f’(x) is undefined