Lesson 4-QR Quiz 1 Review

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

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

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]

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.

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

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

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 + 0 - hill x = c f’ > 0 for x > c f’ < 0 for x < c Relative Min Slope - 0 + valley Relative Max Relative Min x = c x = c f’’(x) < 0 Concave down f’’(x) > 0 Concave up

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

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

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

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

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