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Lesson 4-QR

This lesson provides a review of critical numbers, extreme values, indeterminate forms, inflection points, and theorems related to extrema, derivatives, and the mean value theorem.

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Lesson 4-QR

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  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: Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) 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. Relative Min Relative Max f’ > 0 for x < c f’ < 0 for x < c f’ < 0 for x > c f’ > 0 for x > c x = c x = c 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 Slope - 0 + valley Slope + 0 - hill 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)

  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 Remember: if we have a closed interval, then we have to evaluate the end points! 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

  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 division by zero not allowed negative numbers in an even root

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