PTF #AB 21 Mean Value Theorem & Rolle s Theorem

Transcription

1 PTF #AB 1 Mean Value Theorem & Rolle s Theorem Mean Value Theorem: What you need: a function that is continuous and differentiable on a closed interval f() b f() a What you get: f '( c) where c is an x -value in the given interval b a Verbally it says: The instantaneous rate of change = average rate of change Graphically it says: The tangent line is parallel to the secant line Rolle s Theorem (special case of Mean Value Theorem): What you need: a function that is continuous and differentiable on a closed interval AND the y -values at the endpoints to be equal What you get: f '( c) 0 where c is an x -value in the given interval Verbally it says: The derivative equals zero somewhere in the interval Graphically it says: There is a horizontal tangent line (max or min) 1. Let f be the function given by 3 f( x) x 7x 6. Find the number c that satisfies the conclusion of the Mean Value Theorem for f on 1, 3.. Determine if Rolle s Theorem applies. If so, find c. If not, tell why. 4 f( x) x x, for 3. Let f be a function that is differentiable on the interval 1,10. If f () 5, f (5) 5, and f (9) 5, which of the following must be true? Choose all that apply. I. f has at least zeros. II. The graph of f has at least one horizontal tangent line. III. For some c, c 5, then f() c 3

2 PTF #AB Extrema on an Interval Extrema: the extreme values, i.e. the absolute maximums and minimums Extreme Value Theorem: As long as f is continuous on a closed interval, then f will have both an absolute maximum and an absolute minimum. Finding Extrema on a closed interval: 1. Find the critical numbers of the function in the specified interval.. Evaluate the function to find the y -values at all critical numbers and at each endpoint. 3. The smallest y -value is the absolute minimum and the largest y -value is the absolute maximum. 1. Find the absolute extrema of each function for the given interval: c. f x x e x ( ) 1 on 0,3 a. f x ( ) x 1 on 1, b. f() x x cos() x on 0,

3 PTF #AB 3 Finding Increasing/Decreasing Intervals 1. Find the critical numbers.. Set up test intervals on a number line. 3. Find the sign of f '( x ) (the derivative) for each interval. 4. If f '( x ) is positive then f( x ) (the original function) is increasing. If f '( x ) is negative then f( x ) (the original function) is decreasing. 1. Find the intervals on which the function 3 3 f( x) x x is increasing and decreasing. Justify. 3. The derivative, g ', of a function is continuous and has two zeros. Selected values of g ' are given in the table below. If the domain of g is the set of all real numbers, then g is decreasing on which interval(s)? Increasing? x g'( x ) Let f be a function given by 4 f( x) x x. On which intervals is f increasing? Justify.

4 PTF #AB 4 Relative Maximums and Minimums First Derivative Test: 1. If f '( x ) changes from + to -, then x is a relative max.. If f '( x ) changes from - to +, then x is a relative min. Second Derivative Test: 1. If f ''( x ) is neg (the function is ccd), then x is a relative max.. If f ''( x ) is pos (the function is ccu), then x is a relative min. * x must be a critical number* To find the y-value or the max/min and to see if it is an absolute max/min: 1. Take the x -values and plug them back in to the original equation.. Compare. 1. The function defined by f( x) x 3x for all real numbers has a relative maximum at x =? Justify What is the minimum value of f( x) xlnx? Justify.. Find the relative maximum value for f( x) x 3 e x. Justify. 4. If f has a critical number at x and f ''( x) 3, then what can you conclude about f at x?

5 PTF #AB 5 Points of Inflection Points of Inflection: Points on the original function where the concavity changes. 1. Find where y '' is zero or undefined these are your possible points of inflection (PPOIs). Must test intervals to find the actual POIs they are only where the second derivative changes sign! 1. Write the equation of the line tangent 3 to the curve yx 3x at its point of inflection.. Given f ''( x) x 3x 1, find the points of inflection of the graph of y f( x).

6 PTF #AB 6 Finding Concave Up/Concave Down Intervals 1. Find the PPOIs.. Set up test intervals on a number line. 3. Find the sign of f ''( x ) (the second derivative) for each interval. 4. If f ''( x ) is positive then f( x ) (the original function) is concave up (ccu). If f ''( x ) is negative then f( x ) (the original function) is concave down (ccd). 1. Find the intervals on which the function x 1 f( x) 6 3 is concave up or concave down. Justify.. Let f be a function given by 4 3 f() x 3x 16x 4x 48. On which intervals is f concave down? Justify.